logic and the classical theory of mind

PETER NOVAK

LOGIC AND THE CLASSICAL THEORY OF MIND1

ABSTRACT. I extract several common assumptions in the Classical Theory of Mind(CTM) – mainly of Locke and Descartes – and work out a partial formalisation ofthe logic implicit in CTM. I then define the modal (logical) properties and relationsof propositions, including the modality of conditional propositions and the validity ofargument, according to the principles of CTM: that is, in terms of clear and distinctideas, and without any reference to either possible worlds, or deducibility in an axiomaticsystem, or linguistic convention.

1. FIVE WAYS OF DEFINING LOGICAL MODALITY

The Classical Theory of Mind – of, primarily, Descartes and Locke –stands and falls with a certain account of propositional logical modality:namely, the account in terms of clear and distinct ideas, which I willelaborate in detail in this paper. In modern Analytic Philosophy, therehave been roughly four kinds of account of logical modality on offer.Firstly, the best known of these is the account in terms of possible worlds,according to which a proposition is, say, logically true just in case it istrue in all possible worlds. Secondly, logicians have often favoured theaccount in terms of deducibility in a formal system, according to whicha proposition is logically true just in case it is deducible as a theorem insuch-and-such an axiomatic or deductive system. Thirdly, linguisticallyorientated philosophers have tended to think of logical or necessary truthas a matter of mere linguistic-cum-behavioural convention and resistanceto revision. Fourthly, very much out of favour has been the notion thata proposition is analytically or logically true just in case it is true byvirtue of its meaning. The first three accounts have been characteristic ofAnalytic Philosophy, and have been often inter-mingled: not uncommonlya philosopher and logician would hold that necessary truth is both amatter of truth in all possible worlds and of deducibility, and that it isall but a behavioural convention. The fourth account has been typicallyseen as the Conservative mentalistic position, and as irrevocably defeatedby Quine’s (1951) attack on the analytic-synthetic distinction. The CTMapproach we are about to embark on, in terms of clear and distinctideas, could be regarded as spelling out the allegedly discredited notionof analytic truth as truth by virtue of meaning. But I am not happy to

Journal of Philosophical Logic 27: 389–434, 1998.c© 1998 Kluwer Academic Publishers. Printed in the Netherlands.

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regard it so; not because I would have to combat Quine (for that wouldnot be too difficult; see MS, Chapters 4 and 5), but because the notionof analyticity as truth by virtue of meaning is so truncated as to be aforegone conclusion. In general, Quine’s notions of classical mentalism– that the meaning of a word is an idea in the mind, that an analytictruth is truth by virtue of meaning, a synthetic truth is truth by virtueof fact, that “the a priori” is knowledge independent of experience, “thea posteriori” knowledge dependent on experience, and so forth – are leftin so poor a state it is impossible to make any coherent sense of them.Rather than being embroiled in such notions, I will turn to the ClassicalTheory of Mind itself; and, in order to make its exposition as plain aspossible, I will design in this paper a simple formal model of CTM, inwhich the mentalistic notions of meaning and logical modality can beclearly investigated. The following quotation will serve to set the theme:

Another Faculty, we may take notice of in our Minds, is that of Discerning and dis-tinguishing between the several Ideas it has. . . On this faculty of Distinguishing onething from another, depends the evidence and certainty of several, even very generalPropositions, which have passed for innate Truths; because Men over-looking the truecause, why those Propositions find universal assent, impute it wholly to native uniformImpressions; whereas it in truth depends upon this clear discerning Faculty of the Mind,whereby it perceives two Ideas to be the same, or different. (Locke 1975: IIxi1)

The propositions Locke speaks of are classical logically necessary truths:for example, that everything is identical to itself; that it is impossible forthe same thing, at the same time, to be so-and-so and not to be so-and-so;and so forth. Although Locke’s main purpose is to show that there is noneed to suppose such propositions innate and universal for the humanmind, his arguments for this claim reach beyond the concerns of psy-chology. Implicit throughout the Essay is a logical theory which Lockeuses, on the one hand, to defend his psychology against the doctrine ofinnate propositional knowledge; and, on the other hand, to account forsuch properties of propositions as necessary – or, to put it in Locke’snomenclature – demonstrative truth.

The main goal of this paper is to clarify what that implicit logicaltheory is, and how it can enlighten contemporary issues regarding thenature of logical modality, including the modality of conditional proposi-tions and the validity of argument. However, my exposition of the theorywill not strictly follow Locke’s Essay. Section 2 will outline the basictenets of the Classical Theory of Mind (CTM); and CTM will be taken toinvolve some common assumptions in the psychology of Descartes andKant, as well as Locke. Section 3 will set out an account of representa-tion, or the meaning of mental symbols, drawn primarily from Locke butcommon to CTM. Section 4 will review the classical theory of epistemic

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evaluation, or determination of truth-value, of propositions; specifical-ly, those modes of evaluation whereby the mind acquires what Lockecalls “intuitive” and “demonstrative knowledge”. Section 5 will put forthan account, proceeding from the notion of intuitive and demonstrativeknowledge, of such modal properties of propositions as logical necessity,contingency, possibility, etc. Section 6 will extend this account to cov-er the modalities of conditional propositions, and Section 7 will makeuse of the foregoing notions to form a CTM-based account of deductivevalidity. Section 8 will summarise and conclude the paper with somereflections on the significance of these issues for the project of AnalyticPhilosophy.

2. THE CLASSICAL THEORY OF MIND

CTM rests on a certain construal of common-sense, introspective psy-chology: it regards the mind as a system of (instances of) operations,to be identified with believing, desiring, expecting, supposing, and soforth, on (tokens of) the sentences of a representational mental code,or propositions; and it regards propositions as composed of (tokens of)mental terms, or ideas. The following clauses, 2.1–2.5, will spell out –in as much detail as is necessary for our purposes – the key assumptionsunderlying CTM.

2.1. There is a Finite Basis of Semantically Simple Ideas (Taken asTypes of Symbol), or Simple Terms of the Mind’s RepresentationalCode

Comments. Locke recognises several categories of simple ideas: thereare ideas of sensation and ideas of reflection; in each class, there areideas representing primary qualities, and those representing secondaryqualities (there are no simple ideas of tertiary qualities); ideas of sensationare further classified as visual, auditory, olfactory, ideas of bodily statessuch as pain and hunger, and so on. I need not take a stance on the exactcomposition and size of the basis of simple ideas, but I will assume, inaccord with not only Locke but also Descartes and Kant, that:

(i) each basic idea – i.e., one belonging to the basis – represents amode; that is, a non-relational, monadic property (so that there areno basic ideas standing for either relations or particular objects);

(ii) the basis is empirical, in that it comprises only ideas representingeither sensory or reflective modes.

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Formal model. In order to begin to construct a rudimentary formal systemillustrating the logic of CTM, I will assume that the empirical basisconsists of twenty simple ideas, signified by the first twenty letters ofthe alphabet:

A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T .Further, I will assume that a basic idea may be tokened either as a gram-matical predicate, or as a subject, or object (i.e., grammatical object);and I will allow of an unlimited number of tokens, as opposed to types,of any idea (basic or other). When a basic idea is tokened as a subjector object (for example, in the proposition blue is not identical to yel-low, the basic ideas blue and yellow are tokened, respectively, as subjectand object), I will represent it by one of the twenty lower-case letters:a, b, c, . . . When it is tokened as a predicate, I will represent it by anupper-case letter followed by a variable: Ax,Ky,Qz, . . . The last threeletters of the alphabet, with numerical subscripts if necessary, will bereserved as variables: x, y, z, x1, y1, . . . (The remaining letters – u, v,w,with subscripts if needed – will be used as individual constants; seeSection 5.1.)

Finally, I will suppose that the mind has an input analyser, the functionof which is to distribute variables over predicate tokens in an appropriateway: e.g., when an organism perceives, at the same time, an apple whichis red and green, and a pear which is brown and yellow, its mind willtoken the ideas red and green (say) x-wise, and the ideas brown andyellow (say) y-wise. In this way, the input analyser will itemise or, ifyou please, articulate the environment which the mind thinks about.

2.2. There is a Generative Mechanism Comprising Operations for theProduction of Infinitely Many Complex Ideas from the EmpiricalBasis

Comments. Locke distinguishes three generative operations for complexideas: namely, composition, abstraction, and comparison. Compositionworks to combine several ideas into one compound idea; abstractionworks to separate an idea from one or more complex ideas that containit in common; and comparison works to generate ideas of relations –for example, the idea of identity – by comparing two or more otherideas in regard of their semantic content, or meaning. In Locke’s view,these operations are not idea-laden: that is, though they are operations onempirical ideas, they do not involve or require any further, non-empiricalideas in order to function.

Formal model. For the sake of my model of CTM, I will depart fromLocke in the following two respects. Firstly, contrary to his empiricism

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– according to which there are no simple ideas except those comprisingthe basis – I will assume that the generative mechanism is idea-laden,consisting of two simple, non-empirical ideas representing the standardtruth-functions: viz., ∧ and ¬; and that other ideas of truth-functions – ∨,⊃, ≡, etc. – are definable from the two simples. The mechanism will thengenerate complex ideas by variously compounding the basic empiricalpredicate-tokens, much as do the formation rules of predicate logic:

(Ax ∧Bx), (Fy ⊃ ¬Gz),((Kx ∨ Jy) ≡ (¬Jy ⊃ ¬¬Kx)), and so on.

Secondly, I will suppose that the Lockian operation of comparison worksto generate the idea of identity, =, by comparing any two tokens of thesame type of idea; and I will regard = as a simple idea, though not asbelonging to the empirical basis of simple ideas. In other words, I willsuppose – contrary to Locke’s empiricism, which takes all ideas derivedby comparison, including the idea of identity, as complex – that the mindhas the capacity for a hierarchy of simple ideas, beginning with the basicsimple ideas, and forming further, non-empirical, relational simple ideas,among them the idea of identity, by the operation of comparison. (It isplausible to think of comparison as forming complex as well as simplerelational ideas, but we need not include such ideas in the model.)

2.3. There is a Generative Mechanism Comprising Operations for theProduction of Infinitely Many Propositions from the Stock ofSimple and Complex Ideas

Comments. These operations organise ideas into propositions, and arenecessary for the mind’s capacity to represent the external or internalenvironment as instantiating states of affairs, to judge the environmentto be so-and-so, etc.

Formal model. I will take it that the generative mechanism for propo-sitions is, like that for complex ideas, idea-laden, comprising the non-empirical, simple idea (Cyrillic “S”), expressible by the word “some”;and I will allow three kinds of proposition to be formed:

(i) propositions of identity between any two basic ideas: e.g.,a = a, a = b, j = k, . . .

(ii) quantified propositions: (δ)Γδ, with δ a variable, and Γδ anypredicate-token, simple or complex, in which all occurrences ofδ are free, and all occurrences of other variables, if there are any,bound.

(iii) compound propositions: ¬α, (α∗β), where α and β are any propo-sitions, and ∗ is any dyadic truth-functional idea.

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(Aδ)Γδ – where A (Cyrillic “A”) is the idea of universal quantification,expressible by “all” – will be defined as usual by ¬(δ)¬Γδ. Clause(ii) should be read as allowing that Γδ may be a complex predicate-token involving other quantifiers: e.g., (Fx ∧ (Ay)(Fy ⊃ y = x)), etc.The provisions for Γδ are to ensure that there are no propositions withvacuous quantifiers, or quantifiers binding no variable in their scope, andno propositions with free variables.

The idea could be represented by the standard Russellian sign “∃”,and A by “∀”; but there are good reasons for changing the notation. “∃”is typically regarded as an existential quantifier, as well as an indefinitequantifier expressible by “some”; so, if I retained Russell’s notation inmy model of the mental code, I would have to take “∃” for an idea ofexistence, expressible by “is” or “exists”; however, the notion SOMEis very different from the notion IS or EXISTS, and this should bereflected in the ideas one introduces into the formal model: one shouldhave a distinct idea for the notion SOME, and a distinct idea for thenotion IS. In the present introductory model, I will use for SOMEand A for ALL, and I will allow the notion IS to be merely implicitin the model, without being explicitly expressed by a distinct idea (seeSection 3.3 for the semantics of and A); but in a fuller model, theidea of existence should be explicitly distinguished from . (A furtherreason for changing the Russellian notation is that part of this project isto restore logic upon its psychological foundations; in other words, torescue it from the Russellian anti-mentalistic, upside-down and backwardstance; and what better measure to begin with, than to turn the symbolsface-to-front and on their feet again.)

Both the generative operations for complex ideas and those for propo-sitions comprise ideas, simple and complex, which are non-empirical,and which become psychologically active – in the formation of com-plex empirical ideas and propositions – only when some basic ideas areoccasioned by, and formed in the mind in response to, experiential stim-ulations. In this sense, I will regard these operations as constituting, inpart, the a priori faculty of the mind; and the non-empirical ideas ladenin the operations – together with the semantically simple, non-empiricalidea of identity – I will regard as a priori ideas. Thus I will have twokinds of idea involved in my model of CTM:

(a) empirical (or a posteriori) ideas, which are either of sensation or ofreflection, and either simple or constructed from the simples;

(b) a priori ideas, which are non-empirical, either simple or complex,and the function of which is, in part, to organise simple empiricalideas into complex ideas and propositions.

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In the following sections, I will gradually specify the a priori faculty fur-ther. In Section 5.1, I will treat of the mind’s a priori analytic knowledgein the formal model; a priori synthetic knowledge will not be discussedor modelled in this paper (but see MS: Chapter 9).

2.4. There are Finitely Many Basic Psychological Operations onPropositions

Comments. The psychological propositional operations are broadly oftwo kinds: viz., cognitive and emotive. Emotive operations I will notdeal with, beyond noting that for the cognitive operations to work – andfor psychological processes to ensue in action – there must be someemotions providing the “motive force”. Concerning the basic cognitiveoperations, Locke’s position is that they are epistemic and evaluative;that is, they are operations by which the mind confirms or disconfirms(or, in general, determines the truth-values of) propositions; and all suchepistemic-evaluative operations are founded, as the opening quotationin Section 1 indicates, on the mind’s faculty of discerning the semanticidentity or non-identity between ideas.

Although discerning underlies all evaluative operations (includingsuch as the mind may deploy to confirm propositions without demonstra-tive certainty), there is one sort of evaluation especially relevant to ourconcerns: namely, evaluation by which the mind acquires the Lockianintuitive and demonstrative knowledge. To take the most trivial illus-tration, the mind can determine, with certainty, that any two tokens ofthe same type of idea are semantically identical, and any two tokens ofdifferent types of idea are semantically non-identical; hence it can deter-mine that, for example, a = a is true, a = b is false, etc. (We shouldrecall that a and b stand for – or, in our formal model of the mentalcode, are – semantically simple empirical ideas such as blue or yellow;see Section 2.1. So in saying that the mind can determine that a = ais true, a = b is false, etc., I say that it can determine, by semanticanalysis alone, the truth-values of such propositions as blue is identicalto blue, blue is identical to yellow, and so forth; and it can make suchdeterminations because it discerns the semantic identity, or meaning, ofeach simple idea, thus knowing with certainty that any two tokens of theidea blue have the same meaning, and any two tokens of the ideas blueand yellow, respectively, have different meanings. More on ideationalmeaning in Section 3, and on epistemic evaluation solely from meaningin Section 5.)

To spell out the Lockian epistemic-evaluative operations for intuitiveand demonstrative knowledge – and their significance for our notions of

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modal properties of propositions, including conditional propositions andarguments – is one of the goals of this paper. However, as in the caseof the generative mechanisms for complex ideas and propositions, I willdepart from Locke’s empiricism – which holds that these operations donot require any non-empirical or other ideas in order to function – inassuming that the operations are idea-laden, comprising several classesof a priori ideas; in particular, ideas of modal properties and relationsof propositions, ideas of epistemic values (i.e., truth-values), and ideaswhich may be regarded as individual constants (with some reservations,to be mentioned later).

I will explain what these ideas are, and how they work in the epistemic-evaluative operations, in Section 5. For the moment, we should note onlythat the a priori faculty consists of at least three components: the twogenerative mechanisms for complex ideas and for propositions; and theepistemic-evaluative operations on propositions, by which the mind ac-quires intuitive and demonstrative knowledge. The last component com-pletes the a priori faculty as a faculty of judgement, enabling the mindto assess the logical modality of any proposition (pace the moderns whobelieve in undecidability), and to determine the epistemic values of suchpropositions as may be known with intuitive or demonstrative certainty.

2.5. There is a Generative Mechanism for the Production of ComplexPsychological Operations on Propositions

Comments. Arguably, the complex cognitive operations include expect-ing, doubting, recalling, concluding, and so forth; on the emotive side,there are, arguably, fearing, hoping, aspiring, etc. We need not worryabout the exact identity of any of these operations, except in as much aswe want a general notion of a propositional mental state, and a propo-sitional mental process:

(i) a particular mental state, such as believing that bats are birds, is aninstance of the belief-operation on a proposition (i.e., a token of themental sentence) that means that bats are birds;

(ii) a particular mental process (e.g., thought, deliberation) is a causalsequence of instances of such operations.

So much for the five key assumptions underlying CTM. To summariseour model of CTM thus far, we have a representational mental codewhich is akin to the usual language of monadic QT=. One of the differ-ences is that our basis of simple symbols is finite, and a basic symbolmay be tokened either as a grammatical subject, or as an object, or as apredicate. But the most notable differences will concern the semantics forthe code, as we shall see in the next section. Further, our model differs

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from Locke’s version of CTM, in that our basis of empirical (or a pos-teriori) ideas does not exhaust the class of semantically simple ideas:there are other, relational simple ideas (viz., the idea of identity); and thegenerative operations for complex ideas and for propositions, as well asthe epistemic-evaluative operations on propositions, also involve a pri-ori, non-empirical simple and complex ideas. The a priori ideas laden inthe epistemic-evaluative operations will be introduced in Section 5; andother such ideas will be defined in Sections 6 and 7.

3. THE CLASSICAL THEORY OF REPRESENTATION

The following clauses, SP1–SP3, express the semantical principles implic-it in CTM:

SP1 Semantics is term-based: i.e., the primary bearers of meaningare mental terms, or ideas; and the meaning of a complexmental symbol, idea or proposition, is built from the mean-ings of the simple constituents of the symbol.

SP2 The meaning of a symbol, simple or complex, consists in thatthe symbol denotes, or represents, a certain universal: that is,a property, relation, or – in the case of propositions – a stateof affairs; in other words, meanings are not (sets of) par-ticular objects, but relations of denotation, or representation,between a symbol and a universal.

SP3 The universals denoted, or denotata, are nominal: that is, itis the mind, rather than the environment, which determinesthe identity of the property or relation or state of affairsrepresented by a symbol; which is to say, one cannot get tounderstand the real or noumenal nature of the denotatum ofa symbol merely by knowing the symbol’s meaning.

There are several exceptions to principle SP3. For Descartes, simple ideasof divine perfections (more generally, simple ideas the mind needs to con-template divine perfections) represent real, mind-independent properties.For Locke, simple ideas of primary qualities – for instance, ideas ofextension, duration, solidity, existence, unity, etc., in the case of physicalobjects, and of consciousness, representation, duration, existence, unity,etc., in the case of mental objects – represent, and are resemblances of,real properties. For Kant, all ideas, empirical or non-empirical, representnominal rather than real or noumenal properties. Locke’s view is, moreprecisely, that all simple ideas represent essences which are both real andnominal; but simple ideas of primary qualities are, in addition, veridical

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and resemblances, whereas simple ideas of secondary qualities are notresemblances, though still veridical, in that they signify some real exis-tence from which the original tokenings of the ideas were derived, andwhich is causally sufficient to occasion the ideas. (See, e.g., (IIviii14–15;IIIiv2–3).)

In summary, the common formula of the semantics of CTM is that themeaning of a symbol – whether simple or complex, empirical or a priori –consists in the relation of denotation (representation) between the symboland a certain nominal universal. I will express this by writing: M(#) =df

D(#, [#]), where # is any symbol, [#] is the universal represented by thesymbol, and M and D are the Cyrillic counterparts of the letters “M”and “D”, standing for “meaning” and “denoting”, respectively. We shallnow apply this precept to our model of the mental code, as it has beenset out so far in Sections 2.1–2.3.

3.1. Basic Ideas

Each basic empirical idea represents a mode; that is, a monadic, non-relational, empirical property. In other words:

Z M(α) =df D(α, [α]), where α is any basic idea (which may betokened as a grammatical subject, object or predicate), and [α]

is the empirical mode represented.

3.2. Complex Ideas, and the a priori Ideas Generating Them

There are four groups of symbols falling under this head: the simplea priori ideas ¬ and ∧; the complex a priori ideas ∨, ⊃, ≡, etc.; thesimple a priori predicate =; and the complex empirical ideas generatedby the operations of the a priori ideas (or a priori operations).

3.2.1. The Simple a priori Ideas ¬ and ∧<¬ M(¬) =df D(¬, [¬]), where [¬] is an epistemic property of a

proposition of the form ¬α, such that ¬α is true iff α is nottrue, iff it is not the case that α (with α any proposition).

<∧ M(∧) =df D(∧, [∧]), where [∧] is an epistemic property of aproposition of the form (α ∧ β), such that (α ∧ β) is true iffboth α is true and β is true, iff it is the case that α and it is thecase that β (with α, β any propositions).

Comments. The (non-trivial) difference between CTM-based semanticsand the standard truth-conditional semantics for these terms is that theusual definition of a truth-function does not, in the former, by itself

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specify the meaning of any truth-function idea, only the identity of thetruth-function denoted; the meaning consists in that the idea denotes thefunction.

Also, the way the functions are set out does not matter; any conve-nient method, such as truth-tabular, would do equally well. I have chosento describe [¬] and [∧] as epistemic properties of propositions of suchand such a form, consisting in that the propositions are true iff. . ., etc., toemphasise the epistemic character of the functions (in that they are func-tions of epistemic values), and to express the clauses in a way suitablefor the epistemic-evaluative operations on propositions, to be describedin Section 5. In general, we shall see in this section and in Section 5 that,in this model of CTM, whilst basic ideas denote empirical properties, alla priori ideas, with the exception of =, denote epistemic properties.

3.2.2. The Complex a priori Ideas ∨, ⊃, etc.

<∨ M(∨) =df D(∨, [¬(¬α ∧ ¬β)]);

<⊃ M(⊃) =df D(⊃, [¬(α ∧ ¬β)]), with α, β any propositions.

Comments. Note that although the functions [¬(¬α∧¬β)] and [¬(α∧¬β)]are specified contextually – as they must be, since they are relationalproperties – the semantics for the ideas representing the functions areterm-based, in accord with Principle SP1.

3.2.3. The Simple a priori Predicate =

<= M(=) =df D(=, [=]), where [=] is a relation between any twoitems ζ and η, whether particular or universal, such that ζ isidentical to η.

Comments. Since [=] cannot be defined in terms of properties representedby other ideas available in the model code, we take = as semanticallysimple. This implies that the mind has only to think the idea, without anyanalysis, to understand its semantic content. However, to be thought, theidea must be tokened together with some subject and object ideas, in amental state or process, and therefore cannot be understood in isolation.Thus although the meaning of = does not depend on the meanings ofother ideas, understanding it does depend on understanding other ideas.

3.2.4. Complex Empirical IdeasComplex empirical ideas are constructs from basic ideas, generated byoperations of a priori ideas; their meanings are therefore defined by themeanings of the basic and the a priori ideas. I will set out clauses forcomplex empirical ideas which are negations and conjunctions of basic

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ideas; the meanings of more complex empirical ideas will be definableanalogously.

M(¬Υφ) =df D(¬Υφ, [¬Υφ]), where [¬Υφ] is an empirical propertycompounded of [Υφ] and [¬], such that it is not the case that φ is Υ(with Υ a basic idea and φ a variable).

M(Υφ ∧ Ψω) =df D(Υφ ∧ Ψω, [Υφ ∧ Ψω]), where [Υφ ∧ Ψω] is anempirical property compounded of [Υφ], [Ψω], and [∧], such that it isthe case φ is Υ and it is the case that ω is Ψ (with Υ and Ψ any basicideas, and φ and ω any variables).

Comments. Informally, we may read the clauses as saying that the mean-ing of ¬Υφ consists in its denoting the property of not being Υ, and themeaning of (Υφ∧Ψω) consists in its denoting the property of φ’s beingΥ and ω’s being Ψ.

3.3. Propositions, and the a priori Ideas Generating Them

We have three kinds of symbol to deal with in this section: the simplea priori idea of indefinite quantification, ; the complex a priori ideaof universality, A; and propositions generated by the operations of the apriori ideas.

3.3.1. The Simple a priori Idea

< M() =df D(, []), where [] is an epistemic property of aproposition of the form (δ)Γδ, such that (δ)Γδ is true iffsome item, in the mind’s domain of epistemic evaluation ofthe proposition, partakes of [Γ] (with δ a variable, and Γδ anypredicate-token, whether simple or complex, in which all occur-rences of δ are free, and all occurrences of other variables, ifthere are any, bound).

Comments. The restriction to the mind’s domain of epistemic evaluationis included in the clause since the determination of epistemic value is,in the case of contingent propositions, not independent from the envi-ronment in which the propositions are tokened. For example, when amind considers the proposition some bats are in the orchard, whetherthe proposition is true or false depends on the environment wherein themind is embedded in an epistemically relevant way.

I will take it that every proposition has a non-empty domain of epis-temic evaluation. The very fact that a proposition is tokened in a mentalsystem ensures that something does exist in the mind’s domain of evalua-tion for the proposition: viz., the mind itself, with its symbolic states andprocesses. (This is the Cartesian point that when I think a proposition, I

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can be certain of the existence of at least one thing, namely, myself; tomake it very clear, I can be certain of the existence of myself, as a mind,since the proposition, as a token of a mental sentence, must be tokenedin a symbolic mental system, that is, myself; in general, there could notbe a proposition without a mind which thinks it, and hence every pro-position must have a non-empty domain of epistemic evaluation. Thismakes the proposition nothing exists necessarily false; clearly, though,it does not make every negative existential proposition necessarily false;nor does it make any meaningless.)

3.3.2. The Complex a priori Idea A

<A M(A) =df D(A, [¬(δ)¬Γδ]), with δ and Γδ as in <.

3.3.3. PropositionsLike complex empirical ideas, propositions are generated from the empir-ical basis by operations of the a priori ideas; accordingly, their mean-ings are defined by the meanings of their empirical and a priori con-stituents. We have three classes to consider: propositions of identitybetween any two basic ideas; quantified propositions; and compoundsby truth-functional ideas.

3.3.3.1. Identity Propositions

<IP M(α = β) =df D(α = β, [α = β]), where [α = β] is a state ofaffairs, composed of [α], [β], and [=], such that α is identicalto β (with α and β any basic ideas).

Comments. In Section 5.1, we shall introduce and define the meaningof individual constants, as a priori ideas belonging to the epistemic-evaluative operations on propositions; and we shall allow also of identitypropositions involving the constants.

3.3.3.2. Quantified Propositions

<QP M((δ)Γδ) =df D((δ)Γδ), [(δ)Γδ]), where [(δ)Γδ] is astate of affairs, composed of [] and [Γ], such that some item,in the mind’s domain of epistemic evaluation of (δ)Γδ, par-takes of [Γ] (with δ a variable, and Γδ any predicate-token,simple or complex, in which all occurrences of δ are free andall occurrences of other variables, if there are any, bound).

Comments. It is worthwhile to emphasise again that states of affairs,qua universals represented by propositional mental symbols, are nominal

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rather then real or noumenal universals; in other words, the identity ofa state of affairs is determined by the proposition which represents it– more generally, by the mind and its symbolic system – rather thanby the environment or noumenal world. This is because each simpleidea denotes a nominal universal; and complex symbols, including pro-positions, denote universals which are constructs from universals denotedby the simple constituents of the symbols. As regards the state of affairs[(δ)Γδ], the mere fact that the proposition(δ)Γδ is a complex symbolguarantees that [(δ)Γδ] is nominal, since its identity is fixed by themind’s own construction from the simple constituents of (δ)Γδ. This inturn guarantees that (δ)Γδ retains its semantic identity (i.e., meaning)independently of how the world is, or whether or not [(δ)Γδ] is, or everhas been, instantiated in the environment.

3.3.3.3. Compound Propositions

<NP M(¬α) =df D(¬α, [¬α]), where [¬α] is a state of affairs, com-posed of [¬] and [α], such that it is not the case that α (with α

any proposition).

M(α ∧ β) =df D(α ∧ β, [α ∧ β]), where [α ∧ β] is a state of affairs,composed of [α], [β], and [∧], such that it is the case that α and it is thecase that β (with α, β any propositions).

M(α ∨ β) =df M(¬(¬α ∧ ¬β)).M(α ⊃ β) =df M(¬(α ∧ ¬β)); and so on for other compounds.

This exhausts the class of ideas and propositions belonging to the modelcode, except for the a priori ideas to be defined in Sections 5–7. In thenext section, we shall look at Locke’s position concerning the modesof epistemic evaluation of propositions, which the mind uses to acquireknowledge on the grounds of its symbolic code; in particular, those modesof evaluation whereby it acquires knowledge solely by ideational analysisand independently of any empirical matters of fact.

4. ASPECTS OF THE CLASSICAL THEORY OF KNOWLEDGE

I will start with a review of Locke’s epistemology, sketching briefly hisaccount of intuitive and demonstrative knowledge, of “sensitive knowl-edge”, and what he calls “right judgement”. Then I will focus in detailon the account of intuitive and demonstrative knowledge, and show thatLocke envisioned, but never brought to fruition, a new kind of “Logickand Critick” proceeding from the account. Lastly, I will extract several

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metalogical principles underlying the account of intuitive and demon-strative knowledge, and formulate definitions of basic logical propertiesand relations of propositions, according to the principles.

4.1. Locke on Epistemic Evaluation

Locke distinguishes roughly four modes of epistemic evaluation, or deter-mination of truth-value, of propositions.

Firstly, he takes it that: “Tis the first Act of the Mind, when it has any. . . Ideas at all, to perceive its Ideas, and . . . to know each what it is,and thereby also to perceive their difference, and that one is not another”(IVi4). This allows the mind to evaluate certain propositions solely byits immediate perception of the semantic “agreement or disagreement”among the constituent ideas comprising the propositions. For example,blue is not yellow is evaluable by an immediate perception of the mean-ing of the constituent ideas; or again, three is more than two, and equalto one and two is evaluable “at the first sight of the Ideas together, bybare Intuition. . .” (IVii1). Locke calls the knowledge acquired by thismode of epistemic evaluation “intuitive knowledge”: for “. . .in this, theMind is at no pains of proving or examining, but perceives the Truth, asthe Eye doth light, only by being directed toward it” (ibid).

I will spell out Locke’s position on intuitive knowledge in detail inSection 4.2, and again in Section 5; at present, I wish to stress only thatLocke does not claim that intuitively known propositions are understoodand evaluated as “semantical units”, without any analysis whatever. Hesays rather that the mind perceives the semantic identity of the con-stituent ideas of the propositions, and hence is able to determine theepistemic values of the propositions “at first sight of the ideas together”.The opposite claim, that such propositions are known as units, withoutany analysis, entails that they are psychologically innate, precisely theposition Locke stands against.

Secondly, some propositions – though still evaluable from the mind’sperception of the “agreement or disagreement” among ideas – dependnevertheless for their evaluation on intervening proofs, or semantic de-monstrations. The proposition the three angles of a triangle are equalto two right ones is evaluable, according to Locke, not from the mind’simmediate perception of the semantic agreement between the ideas com-prising the proposition, but rather from the its intuitive knowledge, orimmediate perception, of a number of semantic connections between theproposition’s constituent terms and various intermediary ideas. Lockecalls the knowledge acquired by this mode of epistemic evaluation “de-monstrative knowledge”, or sometimes “rational knowledge” (IVii2–13).

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Thirdly, Locke allows that some propositions are evaluable solelyfrom the mind’s sensory perception of the “agreement or disagreement”between, on the one hand, the ideas whereof the propositions consist, and,on the other hand, “the particular existence of finite Beings without us”(ibid, 14). For example, the proposition this marigold is yellow, thoughtof a particular yellow marigold on a particular occasion, is evaluablesolely from the mind’s sensory perception of the agreement between theconstituent ideas comprising the proposition, and “the particular existenceof the finite beings” referred to by the ideas. Locke calls the knowledgeso acquired “sensitive knowledge”.

Lastly, most propositions Locke regards as evaluable neither from themind’s intuitive or demonstrative perception of the “agreement or dis-agreement” among ideas, nor from its sensory perception of the “agree-ment or disagreement” between ideas and the “finite beings without us”,but “only as they more or less agree to Truths that are established inour Minds, and as they hold proportion to other parts of our Knowledgeand Observation” (IVxvi12). Notably, scientific hypotheses purporting torepresent the imperceptible causes of observable events are confirmableonly in larger theoretical contexts; not one-by-one, in isolation from thetheories they occur in. Locke calls this kind of epistemic evaluation “rightjudgement” (or “belief”, “opinion”) (IVxiv4).

4.2. Locke’s Prevision of a New “Logick and Critick”

I will now turn to elaborate on the notions of intuitive and demonstrativeknowledge. One of the best sources to this end is Chapter VII of Book IV,where Locke discusses the nature of our knowledge of logical maxims oraxioms, and their role in the demonstrative sciences. The chapter opensthus:

There are a sort of Propositions, which under the name of Maxims and Axioms, havepassed for Principles of Science: and because they are self-evident, have been supposedinnate, without that any Body (that I know) ever went about to shew the reason andfoundation of their clearness or cogency. It may however be worth while, to enquireinto the reason of their evidence, and see whether it be peculiar to them alone, and alsoexamine how far they influence and govern our other Knowledge. (IVvii1)

Locke raises three questions here: the first concerns the “reason of evi-dence” of logical maxims or axioms; the second, whether the certaintyof such propositions is “peculiar to them alone”; the third, how far they“influence and govern our other Knowledge”. The rest of the chapteris organised to answer these questions: in particular, Section 2 answersthe first, Sections 3–7 answer the second, and Sections 8–11 the thirdquestion. I will set out the answers in the same order.

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Locke says that axioms or maxims the mind knows to be true withcertainty since it can perceive – intuitively, without involved reasoning– the agreement or disagreement among the semantically simple ideaswhereof such propositions consist. This intuitive perception of the agree-ment or disagreement among ideas is possible only because each ideahas a clear and distinct – that is, well-defined and unique – semanticidentity (cf. (IIxxix, passim)). The semantic clarity and distinctness ofthe simple ideas “is so absolutely necessary, that without it there couldbe no Knowledge, no Reasoning, no Imagination, no distinct Thoughts atall. By this the Mind clearly and infallibly perceives each Idea to agreewith it self, and to be what it is; and all distinct Ideas to disagree, i.e.the one not to be the other” (IVi4).

In short, the evidence for the logical truth of an axiom or maximis drawn solely from its semantically simple constituent ideas; and thisis possible because each simple idea has a clear and distinct semanticidentity, so that the mind can discern infallibly that any two tokens ofthe same type of idea are semantically identical, and any two tokens ofdifferent types of idea are non-identical.

As regards the second question, whether this kind of evidence is“peculiar to axioms or maxims alone”, Locke’s answer is negative; hesays that it is common to all intuitively and hence, by extension, demon-stratively true propositions (since the latter are provable via intermediateideas, where each connection is known with intuitive certainty). Thatblue is not yellow, that two bodies cannot be in the same place at thesame time, or even that a hill is higher than its valley, are equally certain,and the nature of their evidence is the same as that of, say, the principleof non-contradiction.

Concerning the third question, “how far logical axioms or maximsinfluence and govern our other knowledge”, Locke’s view is that eachlogically true proposition, including such as are sometimes taken asaxioms, is entirely independent of any other proposition, whether log-ically true, false, or contingent; in other words, it does not provide evi-dence to, or receive evidence from, any other proposition whatever. Thisis potentially the most controversial feature of Locke’s view. We shalldo well to look at it closely.

Here is how Locke recapitulates the issue:

. . . let us consider, what influence these received Maxims have, upon the other parts of ourKnowledge. The Rules established in the Schools, that all Reasonings are ex præcognitis,et præconcessis, seem to lay the foundation of all other Knowledge, in these Maxims,and to suppose them to be præcognita; whereby, I think, is meant these two things: First,That these Axioms, are those Truths that are first known to the Mind; and, secondly,That upon them, the other parts of our Knowledge depend. (IVvii8)

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Locke sets out to deny these two scholastic claims, discussing – amongother aspects of the claims – axiomatic reasoning in arithmetic, wherebydistinct propositions are derived one from another, with some of the mostgeneral ones serving as axioms. Locke’s objection to such reasoning isthat a particular arithmetical proposition, such as “1 + 2 = 3”, the mindknows with certainty before any general axioms, and therefore musthave some other way of proving it, this way “being nothing else butthe perception it has of the agreement, or disagreement of its Ideas. . .”(IVvii9). Locke then proceeds to strengthen his objection, referring to allpropositions knowable with certainty:

. . . this is true of them, that they are all known by their native Evidence, are whollyindependent, receive no Light, nor are capable of any proof one from another; muchless the more particular, from the more general; or the more simple, from the morecompounded: the more simple, and less abstract, being the most familiar, and the easierand earlier apprehended. . . [T]he Evidence and Certainty of all such Propositions is inthis, That a Man sees the same Idea to be the same Idea, and infallibly perceives twodifferent Ideas to be different Ideas. . . For a Man cannot confound the Ideas in his Mind,which he has distinct: That would be to have them confused and distinct at the sametime, which is a contradiction. . . (IVvii10)

For the Schoolmen of Locke’s day, this position was difficult to accept(see Section 11, ibid.). One of its startling consequences is that thedemonstrative sciences – logic, mathematics, and perhaps other (Lockeregarded geometry and also the foundations of ethics and theology asdemonstrative) – could be done equally well or better by pure ideationalanalysis, and without any axiomatic reasoning, or any proposition-basedreasoning. A transition to such demonstrative sciences could be contem-plated only if the project of CTM, of spelling out in detail the structureand function of the mind – as a system of symbols and symbolic oper-ations – were brought to fruition. Locke does not claim to have accom-plished this, or even to have formulated the project fully in its logicalfacets. But he expresses his confidence that such a project is feasible, andthat it would “afford us another sort of Logick and Critick, than what wehave been hitherto acquainted with” (IVxxi4). He refers to this “Logickand Critick” at times as “the original way of Knowledge” (IVxvii4), or– borrowing a phrase from Richard Hooker – as the “right helps of Art”;and he leaves the task of working out the details to others, “to cast aboutfor new Discoveries, and to seek in their own Thoughts, for those rightHelps of Art. . .” (IVxvii7).

4.3. Metalogical Principles and Definitions

The account of intuitive and demonstrative knowledge can be put con-cisely by the following three clauses, MLP1–MLP3, which I will take as

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metalogical principles for my model of CTM. Three points of nomen-clature before setting out the clauses. Firstly, a proposition will be saidto have such and such a modal or logical property – e.g., to be log-ically true – just in case it is epistemically evaluable, by the mind’scognitive-evaluative operations, as true solely from the meanings of itsconstituent simple a priori ideas (and regardless of any other evidence).The term “evaluable” will be read as saying that there is, not that theremay be, a way of evaluating the proposition solely from the meaningsof its constituent simple ideas. Secondly, the expression “solely fromthe meanings of its constituent simple ideas” I will abbreviate by thescholastic phrase “ex terminis”, which will indicate that the evaluationis to proceed from the ends of semantic analysis; that is, from the mean-ings of the semantically simple, ultimate constituent ideas, not just fromthe lexical constituents of the proposition. (The term “ex terminis” isused by Leibniz (1981: IVvii1), in commenting on Locke’s position con-cerning our knowledge of logical maxims or axioms. Leibniz points outthat “scholastic philosophers have said that such propositions are evi-dent “ex terminis” – from the terms – as soon as they are understood”.However, Leibniz does not regard such evidence as capable of constitut-ing a proof. He thinks that a proof, even in the mind’s representationalcode, must be proposition-based rather than term-based: i.e., such thatsome propositions – the axioms – are indemonstrable, and the mindknows them with certainty innately and immediately, without any analy-sis; whereas all other propositions knowable with certainty are derivablefrom the axioms. This contrasts with Locke’s view that the mind hasno innate propositional knowledge, and that no propositions knowablewith certainty are epistemically prior to any other propositions.) Third-ly, the expression “solely from the meanings of its constituent simple apriori ideas” I will abbreviate by the phrase “ex terminis and a priori”,or sometimes, where the context disambiguates it, merely by the phrase“a priori”. (We should note that ex terminis evaluation may be a poste-riori and therefore contingent, relying on a posteriori or empirical ideas,in addition to a priori ideas; see MS, Chapter 9; again, we should notethat a priori evaluation may be deductive and proposition-based, ratherthan term-based, and therefore not strictly ex terminis; however, anydeductive, proposition-based evaluation will be reducible to ex terminisevaluation; see Section 7.)

Some further comments regarding my usage of the term “ex terminisand a priori”: I just said that “ex terminis and a priori” is short for“[evaluable] solely from the meanings of [a proposition’s] constituentsimple a priori ideas”, and this is how the term is to be understood in

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this paper; for here I will deal only with uncomplicated examples of pro-positions, the ex terminis and a priori evaluation of which need not relyon any other simple a priori ideas except those which are constituents ofthe propositions. However, in a general treatment of the ex terminis anda priori method of proof, I will want to say that such an evaluation of aproposition may draw upon, in addition to the proposition’s constituentsimple a priori ideas, the mind’s total resources of semantically simplea priori ideas; in other words, that the term “ex terminis and a priori”is short for “[evaluable] solely from the meanings of the mind’s simplea priori ideas” (dropping the restriction to the constituent simple a prioriideas of the proposition under evaluation). The key aspect of the exterminis and a priori evaluation of a proposition is therefore not thatit is done from the proposition’s constituent simple a priori ideas, butrather that it is done from simple a priori ideas as such; and these maybe any simple a priori ideas belonging to the symbolic system of themind. Still, to repeat, all examples of ex terminis and a priori evaluationto be discussed in this paper are of the restricted sort, relying solely onthe constituent simple a priori ideas of the proposition being evaluated.

With these provisions, the metalogical principles of CTM may beexpressed thus:

MLP1 Each simple idea, whether empirical (a posteriori) or a pri-ori, has a clear and distinct semantic identity (i.e., meaning).

MLP2 The logical truth of a proposition consists in that the pro-position is evaluable as true ex terminis and a priori, by thecognitive-evaluative operations of the mind.

MLP3 Each logically true proposition is evidentially independent ofany other logically true proposition.

I will finish this section with a list of informal definitions of elementarymodal properties and relations of propositions, given according to theprinciples of CTM (and without reference to possible worlds or any suchentities commonly invoked in contemporary accounts).

A proposition is logically (analytically, necessarily):

(1) true iff evaluable as true ex terminis and a priori;(2) false iff evaluable as false ex terminis and a priori;(3) contingent iff evaluable neither as true nor as false ex terminis and

a priori;(4) possibly true iff not evaluable as false ex terminis and a priori;(5) possibly false iff not evaluable as true ex terminis and a priori.

Two or more propositions are logically:

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(6) compatible iff their conjunction is possibly true;(7) incompatible iff their conjunction is logically false;

and similarly, mutatis mutandis, for other such relations. Of special inter-est is the relation of logical indifference; in the light of MLP3, two pro-positions are logically:

(8) indifferent iff at least one is either logically true or logically false, orthe conjunction of each one of them with the negation of the otheris contingent.

We shall now return to our model of CTM.

5. MODAL PROPERTIES IN THE MODEL CODE

This section will resume the topic of Section 2.4, which is to define thecognitive-evaluative operations by which the mind confirms or discon-firms such propositions as may be known by ex terminis, a priori analysisalone, with intuitive or demonstrative certainty. Section 5.1 will set outthe analytic evaluative operations in general, for the model of CTM.Section 5.2 will make clear, by working out a couple of examples ofex terminis and a priori analysis, just how the operations function. Sec-tion 5.3 will discuss some consequences of the a priori analytic methodfor problems concerning psychological innateness and a priori knowl-edge. (As mentioned earlier, a priori synthetic evaluation, and ex terminisa posteriori synthesis, are dealt with in MS: Chapter 9.)

5.1. Analytic Evaluation in the Model of CTM

I pointed out in Sections 2.4 and 4.2 that, according to Locke, all epis-temic evaluation – logical evaluation in particular – is founded on themind’s capacity of discern the semantic identity of each simple idea, andhence ascertain the identity of any two tokens of the same type of idea,and the non-identity of any two tokens of different types of idea. I willaccept Locke’s insight as a starting point.

However, as in the case of his generative operations for complex ideasand for propositions, Locke takes it that the cognitive-evaluative opera-tions, which rest on the mind’s faculty of discerning, are not idea-laden:that is, although they are operations on propositions constructed fromempirical ideas, they require no non-empirical ideas in order to function.Accordingly, Locke does not distinguish between empirical and non-empirical ideas, either as concerns the generative operations for complexideas and for propositions, or as concerns the evaluative operations onpropositions.

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Contrary to Locke, I will assume that the mind’s evaluative opera-tions are idea-laden, and regard the ideas involved in the operations, likethose of the generative mechanisms, as a priori; in other words, as ideaswhich are non-empirical, and which become psychologically active, inthe epistemic evaluation of propositions, only when some propositionsare formed in the mind in response to experiential stimuli. A priori ideasare thus active in the formation and processing of complex mental sym-bols in at least three stages: firstly, there are a priori ideas that generatecomplex ideas from the basic empirical ideas; secondly, there are a pri-ori ideas that generate propositions from the complex as well as simpleideas; thirdly, there are a priori ideas active in the evaluative operationson propositions.

I will distinguish five evaluative operations: namely, assuming anepistemic value of the proposition under evaluation, α; inferring the val-ues of the atomic constituent propositions of α; discerning the semanticidentity of the atomic propositions in terms of their semantically sim-ple constituent ideas; judging the epistemic value of α; and judging thelogical modality of α. Clauses AO1–AO5 will explain the operations indetail, including the a priori ideas laden in them. I will ask the reader totreat these explanations patiently. It may not be immediately clear howthe operations are to be integrated in an evaluative procedure; but what-ever obscurity may remain after setting out the clauses will be removedby way of examples in Section 5.2. (The symbol “AO” is short for “ana-lytic operation”; in MS, Chapter 9, I match the analytic operations withcorresponding synthetic operations and a synthetic evaluative procedure.)

AO1 Assume the proposition under evaluation has a certainepistemic value, either true or false.

Comments. Using the symbols τ for the value true, ϕ for the value false,and “AssumeV(. . .) = . . .” for the operation of assuming the value,we can write AO1 as “AssumeV(α) = τ (or ϕ)”, where α is any pro-position. The symbol “AssumeV(. . .) = . . .” stands for a propositionaloperation rather than an idea of any sort; however, the symbols τ and ϕstand for – or are, in our model of CTM – a priori ideas, and as suchneed to be given formal semantical clauses, with ϕ defined in terms of τ :

<τ M(τ) =df D(τ, [τ ]), where [τ ] is the epistemic value true of aproposition α, such that α is true iff it is the case that α.

<ϕ M(ϕ) =df D(ϕ, [ϕ]), where [ϕ] is the epistemic value false ofa proposition α, such that α is false iff ¬α is true, iff it is notthe case that α.

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AO2 Infer the epistemic value of each atomic constituent of theproposition under evaluation, by the semantical clauses<#.

Comments. I will write AO2 as “InferV(α) = τ/ϕ(∗,<#)”, where α isany constituent proposition of the proposition under evaluation, ∗ standsfor the number(s) of the line(s) from which the inference is drawn, and<# indicates the semantical clause(s) by which it is drawn. The purposeof AO2 is to resolve the proposition under evaluation into its atomicconstituents and to assign a truth-value to each, given the assumptionmade by AO1 (and making subordinate assumptions if needed). Atomicconstituents are propositions of the form Θλ, or ζ = η, or the negationsthereof, where Θ is any basic predicate idea, λ any individual constant,and ζ and η are either any basic ideas, or any individual constants. Anindividual constant I will regard as an a priori idea laden in, or belongingto, the operation of inferring, tokened when the inference is drawn by <

from a proposition of the form (δ)Γδ, and denoting an item, whetherparticular or universal, which the mind supposes, on the strength of theassumption made by AO1, to exist in the domain of epistemic evaluationof (δ)Γδ, and to partake of [Γ]. In other words, an individual constant,λ, is an a priori idea, tokened in the operation InferV(Γλ) = τ(∗,<),where ∗ assigns the truth-value [τ ] to a proposition of the form (δ)Γδ;and the meaning of λ consists in its representing an item [λ] (supposed,on the strength of AO1, to exist in the domain of evaluation of (δ)Γδ,and to partake of [Γ]). I will take it that the mind can form an unlimitednumber of individual constants, and use the lower-case letters u, v, w –with numerical subscripts if necessary – to represent them in the modelcode; strictly speaking, I should take it that the mind can form a finitenumber of semantically simple individual constants, and form furthersuch ideas by some generative mechanism; but I need not complicatethe issue at this stage. (What I call “individual constant” is not quite thesame as an individual constant of Russellian logic; in the latter, individualconstants are proper names standing for individual objects, and these aredistinguished from predicates standing for properties defining sets ofobjects; in contrast, my individual constants are not proper names, butrather something like pseudonyms for individual objects, since the ideasu, v, w, etc., are used only within a process of evaluation, and may be re-used within another process of evaluation; thus, instead of being propernames, these ideas are used in a certain symbolic procedure, or – so tospeak – for a certain literary sake, as pseudonyms are characteristicallyused.)

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The semantical clauses for these ideas, and for the atomic propositionsin which they occur, are as follows:

<IC M(λ) =df D(λ, [λ]), where λ is any individual constant, and[λ] is the item, particular or universal, represented.

<AP1 M(Θλ) =df D(Θλ, [Θλ]), where [Θλ] is a state of affairs, com-posed of [Θ] and [λ], such that λ partakes of the property [Θ](with Θ any basic empirical predicate, and λ any individualconstant).

<AP2 M(ζ = η) =df D(ζ = η, [ζ = η]), where [ζ = η] is a state ofaffairs, composed of [ζ], [η], and [=], such that ζ is identicalto η (with ζ , η any constants).

AO3 Discern the meaning of each atomic proposition in termsof its simple constituent ideas, and determine whethereach idea has a clear and distinct semantic identity, giv-en the assignment of epistemic value to the atomic pro-position.

Comments. I will use the phrase “DiscernM(#) =df D(#, [#])” for theoperation of discerning, where # is either a semantically simple con-stituent idea of an atomic proposition, or the atomic proposition itself;and I will form a list of such phrases, to include every simple idea occur-ring in each atomic proposition. The purpose of discerning is to find:

(a) exactly what the atomic propositions say, in terms of their semanti-cally simple constituent ideas;

(b) whether each simple idea of an atomic proposition has a clear anddistinct – that is, well-defined and unique – semantic identity, giventhe assignment of epistemic value to the atomic proposition by AO2.

Whenever there is a semantic conflict between the simple constituentideas of one or more atomic propositions – given the assignments of epis-temic values to the propositions – I will indicate it by writing“DiscernM(#1, #2, . . .)NC&D(∗1, ∗2, . . .)”, where #1, #2, . . . are thesemantically simple ideas involved in the conflict, “NC&D” stands for“not clear and distinct”, and ∗1, ∗2, . . . refer to the lines in which the ideasoccur. If no semantic conflict is discerned, I will write “DiscernM(#1,#2, . . .)C&D(∗1, ∗2, . . .)”, where #1, #2, . . . list all the semantically sim-ple constituent ideas of the proposition under evaluation, the symbol“C&D” stands for “clear and distinct”, and ∗1, ∗2, . . . refer to the linesin which the simple ideas occur in atomic propositions.

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AO4 Judge the epistemic value of the proposition under eval-uation.

Comments. I will write AO4 as “JudgeV(α) = τ/ϕ”, where α is theproposition under evaluation. The operation is to work thus:

If, in the scope of the assumption made by AO1 (and each of its sub-ordinate assumptions, where there are any), the meanings of some simpleideas are discerned to be not clear and distinct, then the assignment ofepistemic value made by AO1 is judged to be the opposite: in otherwords, supposing AssumeV(α) = ϕ, AO4 will yield JudgeV(α) = τ ;supposing AssumeV(α) = τ , it will yield JudgeV(α) = ϕ.

If, on the contrary, in the scope of AssumeV(α) = τ (or at least oneof its subordinate assumptions, where there are any), the meanings ofall simple ideas are discerned to be clear and distinct, and if the sameholds also for AssumeV(α) = ϕ, then the judgement of epistemic valueis suspended; which I will write as “JudgeV(α) = ν”, where ν standsnot for a truth-value, but rather indicates that AO4 is unable to determinethe truth-value.

Briefly, AO4 may be set out as follows, where #1, #2, . . . are thesemantically simple constituent ideas of the atomic propositions obtainedby the operation AO2:

(i) if AssumeV(α) = ϕ and DiscernM(#1, #2, . . .)NC&D in all sub-ordinate assumptions, then JudgeV(α) = τ ;

(ii) if AssumeV(α) = τ and DiscernM(#1, #2, . . .)NC&D in all subor-dinate assumptions, then JudgeV(α) = ϕ;

(iii) if AssumeV(α) = τ and DiscernM(#1, #2, . . .)C&D in at least onesubordinate assumption, and if the same holds also for AssumeV(α)= ϕ, then JudgeV(α) = ν.

AO5 Judge the logical modality of the proposition underevaluation.

Comments. I will write AO5 as “JudgeMOD(. . . α) = τ/ϕ”, where α isthe proposition under evaluation, and “. . .” makes room for the a prioriideas of logical modalities laden in the operation: viz., for necessity,♦ for possibility, and O for contingency. The operation may be set outthus:

if JudgeV(α) = τ , then JudgeMOD(α) = τ ;if JudgeV(α) = ϕ, then JudgeMOD(♦α) = ϕ;if JudgeV(α) = ν, then JudgeMOD(Oα) = τ .

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The semantical clauses for the a priori ideas of modal properties of pro-positions will be as follows, with ♦ and O defined in terms of (andwith no reference to possible worlds, etc.):

< M() =df D(, []), where [] is an epistemic property ofa proposition of the form α, such that α is true iff α isevaluable as true ex terminis and a priori, by the evaluativeoperations AO1–AO4 (with α any proposition).

<♦ M(♦) =df D(♦, [♦]), where [♦] is an epistemic property of aproposition of the form ♦α, such that ♦α is true iff ¬¬α istrue, iff α is not evaluable as false ex terminis and a priori, bythe evaluative operations AO1–AO4 (with α any proposition).

<O M(O) =df D(O, [O]), where [O] is an epistemic property of aproposition of the form Oα, such that Oα is true iff (¬¬α ∧¬α) is true, iff α is evaluable neither as true nor as falseex terminis and a priori, by the evaluative operations AO1–AO4 (with α any proposition).

I will assume that once the mind has acquired these ideas, in the evalu-ative operation AO5, it is able to use them in the generative operationsfor complex ideas and for propositions, to generate complex ideas suchas (Fx∨¬Fx), ♦Gy, and so forth, propositions such as (Ax)(Fx∨¬Fx), (y)♦Gy, etc.

Let us now summarise AO1–AO5, before turning to examples. As-sumeV assigns an arbitrary epistemic value to the proposition underevaluation. InferV resolves the proposition into atomic propositions, andassigns to each an epistemic value in accord with AssumeV. DiscernMsorts out just what the atomic propositions say, in terms of the semanti-cally simple ideas comprising them; and whether each idea has a clearand distinct semantic identity, given the assignments of epistemic valueto the propositions. JudgeV determines the epistemic value of the pro-position under evaluation, depending on whether or not the simple ideasretain clear and distinct semantic identity, given the assignments of valueto the atomic propositions. Lastly, JudgeMOD assesses the modal statusof the proposition, so that whenever JudgeV determines the value of theproposition as so-and-so, the proposition is so-and-so with logical neces-sity; and whenever JudgeV cannot determine the value, the propositionis logically contingent.

The gist of the operations is to examine whether the assumed epis-temic value of the proposition under evaluation accords with the semanticidentity of the simple ideas comprising the proposition; if it does not,then the assignment of value must give way to the identity of meaning.

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The simple ideas, to put it in a Lockian way, ‘the mind can neither makenor change’; they are the rock foundation of the mind, and constrain notonly which propositions the mind can form and think, but also which canbe bearers only of truth, which only of falsehood, and which of eithertruth or falsehood. The logical modality of a proposition thus consistsin the constraint the semantic identity of the constituent simple ideasof the proposition imposes on the proposition’s epistemic value. Thisis the notion of modality Locke expresses in his definition of logicalfalsehood: “Contrary to Reason are such Propositions, as are inconsis-tent with, or irreconcilable to our clear and distinct Ideas” (IVxvii23).Again, Descartes relies on this notion of modality when he defines theterm “possible truth” to “mean what everyone commonly means, namely‘whatever does not conflict with our human concepts’. . .” (1984: 107).

The apparent circularity in Locke’s definition I have resolved by myoperations AO1–AO5. It is worth stressing that AO1–AO5 are indeedoperations, not symbols or ideas belonging to the code; as such, they arenot bearers of any semantic properties, and can and should be specified,in the final account, syntactically. Hence although the modal ideas ,♦ and O are defined in terms of InferV, and although the operationsJudgeV and JudgeMOD I have described using clauses of the form“if. . . , then. . .”, the definitions of , ♦, and O are not circular, sincethey involve no unspecified modal ideas.

Lastly, I wish to reiterate that AO1–AO5 are analytic operations,allowing the mind (in this model of CTM) to acquire a priori analyticknowledge: that is, to know with certainty solely by ideational analysis.I show elsewhere (MS, Chapter 9) that this is not the only way the mindcan know with certainty; rational ideational synthesis will also delivercertainty. In general, any necessarily true proposition will be provableeither by rational ex terminis and a priori analysis, or by rational ex ter-minis and a priori synthesis (contrary to Kant’s notion of the analytic-synthetic distinction, but in agreement with Descartes and others).

I will now turn to work out a couple of examples of ex terminis anda priori analysis, taking it as an occasion to answer potential queries,and to remove any remaining obscurity in the account.

5.2. The Trifling Propositions a = a and (Ax)(Fx ⊃ Fx)

Consider first the trifling proposition a = a.

1. AssumeV(a = a) = ϕ.2. InferV(¬a = a) = τ(1,<ϕ).3. DiscernM(a) =df D(a, [a]), where a is a basic idea, and [a] is the

empirical mode represented (2,Z).

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4. DiscernM(=) =df D(=, [=]), where [=] is a relation between anyitems ζ and η, such that ζ is identical to η(2,<=).

5. DiscernM(¬) =df D(¬, [¬]), where [¬] is an epistemic property ofa proposition of the form ¬α, such that ¬α is true iff α is not true,iff it is not the case that α (with α any proposition) (2,<¬).

6. DiscernM(τ) =df D(τ, [τ ]), where [τ ] is the epistemic value true ofa proposition α, such that α is true iff it is the case that α (2,<τ ).

7. DiscernM(¬a = a) =df D(¬a = a, [¬a = a]), where [¬a = a] is astate of affairs, composed of [a], [=] and [¬], such that it is not thecase that a is identical to a (2 − 5,<IP,<NP).

8. DiscernM(a,=,¬, τ)NC&D (2–7).9. JudgeV(a = a) = τ (1–8).10. JudgeMOD(a = a) = τ (1–9).

Comments. I do not attempt to automate the procedure by an algorithm,but I think it is obvious that this could be done; especially Line 8, thediscerning of not clear and distinct meaning of the simple ideas, wouldneed spelling out, to bring it forth that the ideas a, =, ¬ and τ cannotsimultaneously have, under the assumption of epistemic value of Line 1,clear and distinct semantic identity.

Turn now to another trifling proposition, (Ax)(Fx ⊃ Fx).

1. AssumeV((Ax)(Fx ⊃ Fx)) = ϕ.2. InferV((x)¬(Fx ⊃ Fx)) = τ(1,<ϕ,<A,<¬).3. InferV(¬(Fu ⊃ Fu)) = τ(2,<).4. InferV(Fu) = τ(3,<¬,<⊃).5. InferV(¬Fu) = τ(3,<¬,<⊃).6. DiscernM(F ) =df D(F, [F ]), where F is a basic predicate, and [F ]

is the empirical mode represented (4, 5,Z).7. DiscernM(u) =df D(u, [u]), where u is an individual constant, and

[u] is the item represented (4, 5,<IC).8. DiscernM(¬) =df D(¬, [¬]), where [¬] is an epistemic property of

a proposition of the form ¬α, such that ¬α is true iff α is not true,iff it is not the case that α (with α any proposition) (5,<¬).

9. DiscernM(τ) =df D(τ, [τ ]), where [τ ] is the epistemic value true ofa proposition α, such that α is true iff it is the case that α (4, 5,<τ ).

10. DiscernM(Fu) =df D(Fu, [Fu]), where [Fu] is a state of affairs,composed of [F ] and [u], such that it is the case that u partakes of[F ] (4, 6, 7, 9,<AP1).

11. DiscernM(¬Fu) =df D(¬Fu, [¬Fu]), where [¬Fu] is a state ofaffairs, composed of [F ], [u] and [¬], such that it is not the case thatu partakes of [F ] (5–9, <AP1 ,<NP).

12. DiscernM(F, u,¬, τ)NC&D (4–11).

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13. JudgeV((Ax)(Fx ⊃ Fx)) = τ (1–12).14. JudgeMOD((Ax)(Fx ⊃ Fx)) = τ (1–13).

Comments. It is important to bear in mind that each simple idea, whetherempirical or a priori, is clear and distinct when it is considered by itself,in isolation; further, no simple ideas can become not clear and distinctmerely by being combined into a proposition, whatever the propositionmight be. The only way to render some ideas not clear and distinct is,firstly, to combine them into a certain proposition – such as (Ax)(Fx ⊃Fx) or a = a – and, secondly, to assign a certain epistemic value tothe proposition (in our examples, the value ϕ). In short, it is not ideasas such, nor propositions, which can be rendered not clear and distinct,but ideas combined into a proposition under a certain assumption ofepistemic value.

Notice also that DiscernM works only on the constituents of theatomic propositions and on the simple idea τ . Hence the only ideas thatcan enter into a semantic conflict in DiscernM are the finitely many basicempirical ideas, the simple ideas τ , ¬, and =, and individual constants.This indicates that the operation of discerning could be abbreviated, andthe evaluative procedure much simplified.

However, the procedure cannot be so abbreviated as to stop simplyat the level of inferring the epistemic values of the atomic propositions(Lines 4 and 5); that would be to rely solely on the Aristotelian principleof non-contradiction, which is proposition-based rather than term-based,and which rests on the uniqueness of truth-value rather than unique-ness of meaning. The point of the logic of CTM is precisely that itresolves the Aristotelian criterion of the uniqueness of truth-value ofa proposition to the criterion of the uniqueness of meaning, under anassumption of truth-value, of the semantically simple ideas comprisingthe proposition (a criterion which may be regarded as Platonic, in agree-ment with the many Platonic features implicit in CTM, and with thegeneral anti-Aristotelian push common to all early-modern CTM theo-rists, including Locke).

5.3. Comments on Innate and a priori Knowledge

Locke calls propositions such as a = a, (Ax)x = x, (Ax)(Fx ⊃ Fx)“trifling propositions”, since they “add no Light to our Understandings,bring no increase to our Knowledge” (IVviii1). The proposition (Ax)x =x Locke often mentions as a case of putative innate knowledge; that is,as a proposition which allegedly could not allow of any demonstrationor logical analysis, and hence – on the psychological side – had to betaken as known innately and immediately, as a semantical and logical

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unit. This is the doctrine Locke sets out to refute; and his chief argumentis that even though such propositions are known intuitively, in that themind is in “no pains of proving or examining” in evaluating them as true,they are nevertheless not known as units, without any analysis whatever;rather, the mind knows the clear and distinct semantic identity of thesimple constituent ideas of the propositions, and therefore is able todetermine that the ideas semantically “agree one with another”. Such“agreement or disagreement” I have sought to capture in my model ofCTM; and it has this significance for the classical philosophy of mind,that it explains how the mind could demonstrate any analytically trueproposition, no matter how trifling, so that the proposition need not besupposed innate but rather known by a proof, however trivial the proofmay seem.

As to innate ideas rather than propositions, Locke’s first concern isto show that no ideas need be supposed psychologically active prior toand independently of some or other experiences. Locke thereby rejectsthe Cartesian view that the soul can think regardless of any experiences(e.g., in a disembodied state, as Descartes might have it). Another of hisconcerns is that each simple idea should answer adequately and veridi-cally to some aspect of the environment (whether external or internal).These are the foremost reasons why Locke denies the distinction betweenempirical and non-empirical ideas, taking all simple ideas to be empiri-cal; for he fears that admitting non-empirical ideas would be tantamountto admitting non-veridical representations in the very constitution of themind.

I will suggest, in closing this section, that Locke could have afford-ed himself of the useful and correct distinction between empirical andnon-empirical or a priori ideas, such as I have drawn in my modelof CTM, even holding that all mentation depends on and follows aftersome experiences (as Kant held), and even holding (contrary to Kant)that the non-empirical ideas have an adequate and veridical applicationto the world as it is; which would have allowed him to avoid thoroughempiricism and make use of the theoretical benefits of rationalism. Thesebenefits are already apparent in this model of CTM, notwithstanding itssimplicity; but as we gradually enrich the model, adding ideas of space,time, of causal relations, of number, of the self – and also, should wefollow the classical theorists, of the good and bad, etc. – the distinctionbetween empirical and a priori ideas will become still more prominentand indispensable.

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6. THE COMPLEX IDEAS OF IMPLICATION

There is a sense in which only in this section we shall come to the properbusiness of this paper, which is the logic implicit in the Classical Theoryof Mind; for our exposition of the logic would not be complete – in fact,would hardly have begun – without a treatment of the complex a prioriand a posteriori ideas of implication, and of valid deduction. We shalltreat of ideas of implication in this section, and of that of valid deductionin the next; and we shall require all the material covered in Sections 2–5to do it.

I will call propositions of the form “if α were true, then β would betrue” “strong conditionals”, and those of the form “if α were true, thenβ might be true” “weak conditionals”; and I will distinguish two kindsof either strong or weak conditional: necessary conditionals, which areepistemically evaluable solely by ex terminis and a priori analysis; andcontingent conditionals, the evaluation of which depends in part on a pos-teriori ideas. The propositional relation represented by “if. . ., then. . .”,whether in a strong or a weak conditional, I will call “implication”; and,in this model of CTM, I will use the symbol → for the idea of strongnecessary implication, for the idea of weak necessary implication, ⇒for the idea of strong contingent implication, and for the idea of weakcontingent implication.

The purpose of this section is to define the semantic identity of theseideas, in accord with the metalogical principles of CTM. But firstly, weshall need to consider the significance of these principles for the cognitiveoperations of assuming the antecedent and implying the consequent in atrue conditional. This will be the topic of Section 6.1; Section 6.2 willformulate the four semantical clauses, and Section 6.3 will conclude withsome comments concerning the normativity of complex ideas, ideas ofimplication specifically, in CTM-based logic.

6.1. Assuming and Implying in a True Conditional

We have seen in Sections 4.2–4.3 that, according to CTM, all necessarypropositions, true or false, are logically independent, and hence neitherimply nor are implied by any other proposition; they are “known bytheir native Evidence, are wholly independent, receive no Light, nor arecapable of any proof one from another; much less the more particular,from the more general; or the more simple, from the more compounded”(op. cit., Section 4.2). This is a very extraordinary aspect of CTM-basedlogic, so it will be worthwhile to look at it in detail, with regard to the

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issue of assuming the antecedent and implying the consequent of a trueconditional.

6.1.1. Assuming a Logical FalsehoodA proposition is logically false just in case the mind has a way of eval-uating it as false ex terminis and a priori. Therefore the mind cannot,without violating the principle of clear and distinct ideas, assume a logi-cal falsehood as true; for in assuming it as true, the mind has the capacityto work out, by its cognitive-evaluative operations, that the constituentideas of the proposition do not have, under the assumption, clear anddistinct semantic identity, and hence that the proposition cannot be true.It follows that – though any proposition may be assumed as true, inso-far as assuming is merely a cognitive-evaluative operation, ultimately tobe defined syntactically – not any proposition may be assumed as theantecedent of a true conditional. In particular, no conditional with a log-ically false antecedent can be true; for in assuming such an antecedent astrue, the mind is in a position to discern that the simple constituent ideasof the antecedent do not have, under the assumption, clear and distinctsemantic identity, and so judge that the conditional as a whole cannot betrue.

Let us now work over the same claim, and the same argument, using asimple example. Take ¬a = a as the antecedent, and β as any consequent,of the conditional ¬a = a → β (equivalently, use , ⇒ or insteadof →). Even before knowing the precise semantics for the idea → (or, ⇒, ), we can tell, by the principles of CTM, that the conditionalcannot be true. For in assuming the antecedent ¬a = a as true, we areable to show – by an ex terminis, a priori analysis analogous to thatwhereby we proved in Section 5.2 that a = a is logically true – thatthe simple constituent ideas of the antecedent (viz., ¬, a, and =) do nothave, under the assumption, clear and distinct semantic identity, so thatthe antecedent ¬a = a, and hence the conditional ¬a = a → β as awhole, must be false. In general, we can show likewise that since themind cannot assume a logical falsehood as true without violating theprinciple of clear and distinct ideas, no conditional with a logically falseantecedent can be true.

6.1.2. Implying a Logical FalsehoodSimilarly, no conditional with a logically false consequent can be true.For such a consequent is evaluable as false ex terminis and a priori,and therefore the mind cannot – without violating the principle of clearand distinct ideas – imply it as true, whatever antecedent is assumed.Thus β → ¬a = a, where β is any antecedent, must be false, since the

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consequent ¬a = a cannot be implied as true whilst preserving the clearand distinct semantic identity of the simple ideas ¬, a, and =.

6.1.3. Assuming a Logical TruthIn the mental code (as opposed to public-language discourse, where gen-erally some aspects of the thought a speaker intends to convey are leftunexpressed or ambiguous), the truth of the antecedent of a true strongconditional, whether necessary or contingent, is sufficient for the truth ofthe consequent; and the truth of the consequent is necessary for the truthof the antecedent. When the consequent is false, so that the necessarycondition for the truth of the antecedent is not satisfied, the antecedentis also false. In other words, the conditional α → β (or α ⇒ β) istruth-functionally equivalent to ¬β → ¬α (or ¬β ⇒ ¬α); contraposi-tion holds, in the mental code, for strong conditionals. Further, giventhat contraposition holds for the strong conditionals, it holds a fortiorifor weak conditionals; that is, α β and α β are truth-functionallyequivalent to, respectively, ¬β ¬α and ¬β ¬α.

It follows that, by the principles of CTM, no conditional propositionwith a logically true antecedent can be true. For such a conditional – forinstance, a = a → β, where β is any consequent – is truth-functionallyequivalent to ¬β → ¬a = a, which we know to be false, since itsconsequent cannot be implied as true without violating the clear anddistinct semantic identity of the simple ideas ¬, a, and = (Section 6.1.2).

6.1.4. Implying a Logical TruthAnalogously, it follows that no conditional with a logically true conse-quent can be true. For such a conditional – e.g., β → a = a, where β isany antecedent – is truth-functionally equivalent to ¬a = a → ¬β, whichwe know to be false, since the antecedent ¬a = a cannot be assumedas true without violating the clear and distinct semantic identity of thesimple ideas ¬, a, and = (cf. 6.1.1).

Putting 6.1.1–6.1.4 together, we may judge that a necessary conditionfor the truth of a conditional proposition is that both its antecedent andits consequent be logically contingent.

This necessary condition is clearly enunciated by Locke, as I haveshown in Section 4.2. Versions of it have appeared in modern Analyt-ic Philosophy with Strawson, Von Wright, Geach, Smiley, and Watling.But none of the modern versions have contained cogent defences ofthe position, and all are baffled by the apparent conflict between, onthe one hand, the view that a necessary proposition neither impliesnor is implied by any other necessary proposition, and, on the oth-er hand, the common practice of asserting conditionals with necessary

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antecedents and consequents, and – as in reductio ad absurdum argu-ments, or in axiomatic proofs – of implying necessary conclusions onthe grounds of necessary premisses. I will now turn to show that thisconflict is indeed only apparent, and does not at all contravene the logicof CTM.

The view that a necessary proposition neither implies nor is impliedby any other proposition holds, as I have argued in Sections 6.1.1–6.1.4,for the mental code; in the mental code, any necessary proposition isin principle provable, independently of any other proposition, by theepistemic-evaluative operations of the mind. I have illustrated how thisis done in Section 5.2, using as examples the trifling propositions a = aand (Ax)(Fx ⊃ Fx).

However, when an ex terminis, a priori proof of a proposition in themental code is, so to speak, translated into and communicated in a publiccode – that is, in a public-language discourse (whether formalised, as inmathematics and formal logic, or “ordinary”) – we are inclined to expressthe sequence of the epistemic-evaluative operations on the proposition asif it were a sequence of if-then conditionals, such that the consequent ofeach conditional is logically dependent on the antecedent. For example,in the ex terminis, a priori proof of (Ax)(Fx ⊃ Fx), the mind may beginwith the operation AssumeV((Ax)(Fx ⊃ Fx)) = ϕ, and continue withthe operation InferV((x)¬(Fx ⊃ Fx)) = τ . But when it communicatesthis step in a public language, it is wont to express it by saying thatif ¬(Ax)(Fx ⊃ Fx), then (x)¬(Fx ⊃ Fx). This, though it seemsinnocuous, is misleading, since it gives the impression that the mindimplies the necessary consequent from the necessary antecedent, whichin fact, in its ex terminis and a priori method of proof of (Ax)(Fx ⊃Fx), it does not. Strictly speaking, then, the mind should endeavourto mirror as closely as it can, in its public expressions of the a prioriproof, the procedure of the proof, which involves only the epistemic-evaluative operations on the proposition and on its constituent ideas, withno implications from necessary antecedents to necessary consequents.This is what I have done in my own public-language formulation of themodel of CTM, and of the ex terminis and a priori method of proof inthe model.

In general, the apparent conflict between the classical mentalistic viewthat a necessary proposition neither implies nor is implied by any oth-er proposition, and the common public practice of asserting condition-als with necessary antecedents and consequents, and of drawing neces-sary conclusions from necessary premisses, disappears when the publicexpressions of the mind’s ex terminis and a priori method of evaluation

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and proof mirror – as they should, insofar as the primary function ofpublic language is to express thought – the structure of that method.

6.2. The Complex Ideas →, , ⇒, and

I will now turn to define the semantic identity of the ideas of implication.These are complex ideas (like ♦ and O) which I class as belonging tothe operation of judging logical modality, AO5. The a priori ideas ofnecessary implication are simpler than the a posteriori ideas of contingentimplication, and will be set out first.

6.2.1. The Complex a priori Ideas of Necessary Implication

<→ M(→) =df D(→, [→]), where [→] is an epistemic property ofa proposition of the form (α → β), such that (α → β) is trueiff (¬(α ∧ ¬β) ∧ Oα ∧ Oβ) is true; in other words, iff:

(a) (α ∧ ¬β) is evaluable as false ex terminis and a priori,(b) α is evaluable neither as false, nor as true, ex terminis and

a priori,(c) β is evaluable neither as false, nor as true, ex terminis and

a priori,

by the evaluative operations AO1–AO4 (with α, β anypropositions).

Comments. The modality [→] may be said to consist in that (α → β) istrue just in case (α∧¬β) is contradictory, in the ex terminis sense, but itscontradictoriness is not due to either of the conjuncts taken individually;rather, it is due to their conjunction; that is the gist of the account. Thiskind of account has a rather interesting recent history. Using the term“entailment” where I have used “strong necessary implication”, Smiley(1959) distinguishes between accounts of entailment in terms of logicalconsequence, and those in terms of deducibility, or deductive systems.The former accounts are similar to my own. For example, Strawson(1948: 186) suggests that “[t]he expression ‘ “p” entails “q” ’ is to beused to mean ‘ “p ⊃ q” is necessary, and neither “p” nor “q” is eithernecessary or self-contradictory’. . .”. Von Wright (1957: 178) agrees withStrawson’s suggestion, but thinks that it “stands in need of further clar-ification”, and that “[t]o account of its meaning is to account of themeaning of entailment”. Von Wright then offers a clarification as fol-lows: “p entails q, if and only if, by means of logic, it is possible tocome to know the truth of p → q without coming to know the falsehoodof p or the truth of q” (p. 181; with → standing for ⊃). In turn, Geach

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(1958) accepts Von Wright’s clarification, but replaces the expression “itis possible to. . .” with “there is a way of . . .”, and the expression “bymeans of logic” with “there is an a priori way”. His account then readsthus: “p entails q if and only if there is an a priori way of getting to knowCpq which is not a way of getting to know either whether p, or whetherq” (p. 164; Cpq stands for p ⊃ q). A similar view is defended by Watling(1958), who has much to say concerning the requirement of contingency.

Smiley considers his own version of this kind of account, which isdesigned to allow entailments such as A&∼A ` A (where ` stands for“entails”, & for ∧, and ∼ for ¬), whilst guaranteeing that A&∼A ` Bholds for not any B whatever. The account is this: “A1, . . . , An ` B ifand only if the implication A1& · · ·&An ⊃ B is a substitution instanceof a tautology A′

1& · · ·&A′n ⊃ B′, such that neither ` B′ nor ` ∼

(A′1& · · ·&A′

n)” (p. 240). Smiley thinks that, save for his account, anytheory involving the requirement that the antecedent and the consequentof a true entailment be contingent has to fail, since “under it reductioad absurdum argument is strictly impossible, for no premiss can on thistheory entail a contradiction or even the two arms of a contradictionseparately” (p. 244). He then proceeds to develop the latter approach, interms of deducibility or deductive systems, which was pioneered by Ack-ermann (1956). This approach has become more influential in subsequentyears; the former approach was framed in terms of possible worlds, butthe condition that the antecedent and the consequent of a true entailmentbe contingent tended to wither. Anderson and Belnap (1975) deal withthe problem of entailment entirely in terms of deducibility: “Since wewish to interpret ‘A → B’ as ‘A entails B’, or ‘B is deducible fromA’, we clearly want to be able to assert A → B whenever there exists adeduction of B from A . . .” (p. 7). However, they still regard the formerapproach as “plausible and interesting” (§§ 5.1.1, 15.1, 20.1).

My CTM-based account falls under “Logical Consequence” ratherthan “Deducibility”; and, as we shall see in Section 7.1, the idea ofvalid deduction will be accordingly defined in terms of that of strongnecessary implication, not conversely. The logical-consequence approachmay be further categorised as involving accounts in terms of possibleworlds, and the early-modern (Lockian and Cartesian) accounts in termsof clear and distinct ideas; my proposal obviously falls into the lattercategory. It is notable that what Smiley took to be the greatest hurdle forthe consequence approach – namely, reductio ad absurdum arguments –makes no trouble for that approach when it is of the latter, early-modern,non-possible-worlds-theoretic variety; I have shown how such argumentswork in Section 5.

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< M() =df D(, []), where [] is an epistemic property ofa proposition of the form (α β), such that (α β) is trueiff (♦¬(α ∧ ¬β) ∧ Oα ∧ Oβ) is true; in other words, iff:

(a) (α ∧ ¬β) is not evaluable as true ex terminis and a priori,(b) α is evaluable neither as true, nor as false, ex terminis and

a priori,(c) β is evaluable neither as true, nor as false, ex terminis and

a priori,

by the evaluative operations AO1–AO4 (with α, β anypropositions).

Comments. The modality [] is that (α β) is true iff (♦¬(α ∧¬β)∧Oα∧Oβ) is true, which in turn is true just in case ((α → β)∨ (O¬(α∧¬β) ∧ Oα ∧ Oβ)) is. From this we can see that ((α → β) ⊃ (α β))is logically true; that is, ((α → β) ⊃ (α β)) is true. However,((α → β) → (α β)) is not true, since the antecedent is either logicallytrue or logically false, and so is the consequent. Again, ((α → β) ⊃(¬β → ¬α)) is true, but ((α → β) → (¬β → ¬α)) is not.

I mentioned in Section 5.1 that the a priori ideas of logical properties– , ♦, and O – once acquired in the evaluative operations, may enter intothe generative operations for complex ideas and for propositions, to formideas such as (Fx∨¬Fx), propositions such as O(x)Fx, (Ay)♦Hy,and so forth. Similarly, the a priori ideas of logical relations – viz., →and – once acquired, may enter into the generative operations, to formideas such as ((Fx ∧ Gx) → Fx), (Fx → (Fx ∨ ¬Fx)), propositionssuch as (Ax)((Fx ∧ Gx) → Fx), which will be true, (Ax)(Fx →(Fx ∨ ¬Fx)), which will be false, etc.

Let us now very briefly consider a form of proposition known as “theBarcan formula”, ♦(x)Ψx ⊃ (x)♦Ψx. It should be clear upon reflec-tion that, depending on the idea Ψ, the antecedent and the consequentof a proposition of that form will be either both logically true, or bothlogically false. For supposing Ψ is such that (x)Ψx is possible true,♦(x)Ψx is necessarily true; but then also Ψ is such that Ψu, for anyindividual constant u, is possibly true, and hence (x)♦Ψx is necessar-ily true (for recall that every proposition must have a non-empty domainof epistemic evaluation; see Section 3.3.1). Conversely, supposing Ψ issuch that (x)Ψx is not possibly true, ♦(x)Ψx is necessarily false; butthen also Ψ is such that Ψu, for any constant u, is necessarily false, andhence (x)♦Ψx is necessarily false. Therefore every proposition of theform ♦(x)Ψx ⊃ (x)♦Ψx is a logical truth. However, no propositionof the form ♦(x)Ψx → (x)♦Ψx is true, since both its antecedent

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and its consequent are necessary, contrary to the requirement that bothmust be contingent. (For those who wish to be reminded, the Barcanformula raises a problem for classical quantificational modal logic, sinceany proposition of that form turns out logically true, under standard(S5) possible-worlds interpretation; yet, construing “if. . . , then. . .” as(. . . ⊃ . . .), the proposition says that from the mere possibility ofthere being something which is Ψ, it follows that there is somethingwhich is possibly Ψ; a consequence not guaranteed, intuitively, by theantecedent. CTM resolves the issue, since although all propositions ofthe form ♦(x)Ψx ⊃ (x)♦Ψx are logically true, those of the form♦(x)Ψx → (x)♦Ψx – i.e., the claims to implication – are false.)

6.2.2. The Complex a posteriori Ideas of Contingent ImplicationThe account in this section will be concerned solely with logical featuresof contingent implication, aiming to show how the mind can form com-plex, clear and distinct ideas thereof; there will be no attempt to confrontthe problem of contingent implication in its a posteriori aspects. The pur-pose of the section is to point out the manner by which the mind canbegin to extend its knowledge of necessary, purely a priori conditionalpropositions, to contingent a posteriori conditional propositions (somefeatures of which may be knowable by a priori means, either by rationalanalysis or by rational synthesis; cf. MS, Chapter 9).

The epistemic evaluation of a contingent a posteriori conditional willinvolve contingent propositions representing either natural laws, or nom-inal laws such as conventional regularities, or any other laws or states ofaffairs relevant to the implication between the antecedent and the conse-quent. I will refer to such propositions as “accessory propositions”, andleave it open, for the present purposes, exactly what they might be. Theideas of strong and weak contingent implication (⇒ and , respectively)may be then defined as follows:

<⇒ M(⇒) =df D(⇒, [⇒]), where [⇒] is an epistemic property ofa proposition of the form (α ⇒ β), such that (α ⇒ β) is trueiff (¬(α∧¬β ∧Ω)∧O(α∧¬β)∧Oα∧Oβ) is true; i.e., iff:

(a) (α∧¬β ∧Ω) is evaluable as false ex terminis and a priori,(b) (α∧¬β) is evaluable neither as false, nor as true, ex terminis

and a priori,(c) α is evaluable neither as false, nor as true, ex terminis and

a priori,(d) β is evaluable neither as false, nor as true, ex terminis and

a priori,

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by the evaluative operations AO1–AO4 (with α and β anypropositions, and Ω a conjunction of true accessory pro-positions).

< M( ) =df D( , [ ]), where [ ] is an epistemic property of aproposition of the form (α β), such that (α β) is true iff(♦¬(α ∧ ¬β ∧ Ω) ∧ O(α ∧ ¬β) ∧ Oα ∧ Oβ) is true; i.e., iff:

(a) (α∧¬β∧Ω) is not evaluable as true ex terminis and a priori,(b) (α∧¬β) is evaluable neither as true, nor as false, ex terminis

and a priori,(c) α is evaluable neither as true, nor as false, ex terminis and

a priori,(d) β is evaluable neither as true, nor as false, ex terminis and

a priori,

by the evaluative operations AO1–AO4 (with α and β anypropositions, and Ω a conjunction of true accessory pro-positions).

Comments. The purpose of subclause (b) – viz., O(α ∧ ¬β) – is to sep-arate the ideas of contingent implication from the ideas of necessaryimplication, so that the former and the latter do not overlap under anycircumstances; it is to put the onus of contingent implication on Ω, as wellas α and β, excluding the possibility that the onus be on α and β alone.

Some of the true accessory propositions in Ω might be other con-ditionals representing either natural or nominal laws, and these may ormay not be evaluable with respect to still further conditionals belong-ing to further sets of accessory propositions. If one adopts a thoroughlyempiricist stance – denying the distinction between empirical and non-empirical ideas, and denying the possibility of a priori knowledge –one should regard the ultimate law-representing conditionals as evalu-able at best by empirical means, involving some mix of holistic andinductive confirmation; and one should regard the choice of ultimatelaw-representing conditionals as pragmatic. But if one adopts a rational-ist stance – drawing a distinction between empirical and non-empiricalideas, and allowing of a priori knowledge – one may regard some of theultimate law-representing conditionals as a priori, whether analytic orsynthetic, and thus not contingent. The Classical Theory of Mind, evenin Locke’s version, has always adopted the latter stance, and according-ly regarded physical science, and knowledge of ultimate natural laws,as having a metaphysical foundation knowable with a priori certainty.(Locke would not have used these terms, but in fact he has drawn thecomparable distinctions; see MS, Section 9.2.1.)

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6.3. Comments Concerning the Normativity of Complex Ideas

The ideas →, , ⇒, and are complex ideas; whether the mind formsthem or not depends on its needs, social and environmental circum-stances, and above all on its self-knowledge. So the mind may or maynot, as a matter of fact, form these ideas. However, with respect to themetalogical principles (MLP1–MLP3), which I take to be implicit in thesystem of mind, the formation of these complex ideas is not arbitrary. Iwill mention two reasons.

Firstly, the component of the semantical clauses which says that astrong conditional cannot hold true unless the antecedent is incompatiblewith the negation of the consequent (or, in the case of weak conditionals,unless the antecedent is incompatible with the negation of the consequent,or the antecedent and the consequent are indifferent) does seem to beuniversal: the mind has no option but to construct its complex ideas ofimplication so as to include at least this much.

Secondly, insofar as the mind needs to construct these ideas withrespect to the principles MLP1–MLP3, it cannot but include in theirdefinitions the constraint that necessary propositions, whether true orfalse, neither imply nor are implied by any other propositions, and hencethat the antecedent and the consequent of a true conditional must becontingent; I have shown the reasons for this constraint in Sections 4.2–4.3 and 6.1.1–6.1.4.

In general, there is an objective and universal normative constrainton the formation of the complex ideas of modal properties and relations,set by the clear and distinct semantic identity of the mind’s simple ideas,as well as by its evaluative operations; and although the mind is ableto create what complex ideas of modal properties it finds suitable oreffective for its purposes, it is nevertheless bound to form such complexideas as accord with the simple ideas and evaluative operations, if thecomplex ideas themselves are to be clear and distinct, free from semanticconfusion and inconsistency.

7. THE COMPLEX A PRIORI IDEA OF VALID DEDUCTIVE INFERENCE

The interest of deductive reasoning lies in that the mind can proceed,by logical means alone, from premisses which, taken in conjunction,are contingent, and which are either assumed or known to be true, tocontingent conclusions which are neither given nor yet known, thereby –to put it in Locke’s idiom – enlarging its knowledge. This is somethingthe mind cannot do with the ideas of implication alone; for, to know the

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truth of a conditional, the mind has to contemplate, by the same token,both its antecedent and its consequent, rather than deduce an unknownconsequent from theantecedent.

In Section 7.1, I will introduce the symbol ` to represent the complexa priori idea of valid deductive inference in my model of CTM, anddefine the semantic identity of the symbol according to the principles ofCTM. In Section 7.2, I will consider some of Quine’s objections againstmodal properties and relations. These objections properly belong underthe heading of deductive inference, since they have to do with apparentlyinvalid arguments which standard informal modal logic turns out as valid(as opposed to apparently false propositions, such as the Barcan formula,which it turns out as true).

7.1. The Complex a priori Idea `The ideas of deductive inference and strong necessary implication areclosely related, but not identical. Informally, an argument is deductivelyvalid just in case there is a way of deducing the conclusion from thepremisses by a finite number of applications of a certain sort of deduc-tive rules; and the meaning of the idea of validity consists in that itdenotes this epistemic property. I will assume, to define the meaning of` formally, that there are finitely many basic deductive rules ρ1, . . . , ρm,such that each rule is a schema of a true, strong, necessary condition-al, the antecedent of which the mind regards as the premiss(es) of therule, and the consequent as the conclusion. I need not specify just whichschemata of conditional should serve as deductive rules, since my presentgoal is only to tell how the mind can compose, according to the prin-ciples of CTM, a clear and distinct, complex idea of deductive valid-ity. But there are obvious examples of such schemata: ((ξ ∧ ς) → ξ),(((ξ ∨ ς) ∧ ¬ξ) → ς), and so forth, where ξ and ς are contingent pro-positions, so ensuring that the antecedents and consequents constructedfrom them are contingent. The identity of ` may be then defined asfollows.

<` M(`) =df D(`, [`]), where [`] is the epistemic propertyof valid deductive inference, of an argument of the formα1, . . . , αn ` β, such that α1, . . . , αn ` β is valid iff:

(a) there is a way of deducing β from α1, . . . , αn by a finitenumber of applications of the basic rules ρ1, . . . , ρm;

(b) ρ1, . . . , ρm are such that there is a way of deducing β fromα1, . . . , αn iff ((α1 ∧ · · · ∧ αn) → β) is true

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(with α1, . . . , αn the finitely many premisses, and β the con-clusion of the argument).

Comments. Clause (b) says that the system of deductive rules ρ1, . . . ,ρm must be sound and complete vis-a-vis ex terminis evaluation, in thatwhatever argument is provable as valid by the deductive rules must be soprovable ex terminis, and whatever argument is provable as valid ex ter-minis must be so provable by the deductive rules. Note also that althoughthe clause requires, implicitly, that the conjunction (α1 ∧ · · · ∧ αn) becontingent, it does not require that each of the αj’s taken individuallybe contingent; thus, for example, supposing (ξ → ς) logically true, withξ and ς contingent, (ξ → ς), ξ ` ς is a valid argument.

In summary, valid deductive inference is like strong necessary impli-cation; the purpose of forming the complex a priori idea `, in additionto →, is to enable the mind to deduce contingent consequents not yetgiven or known, something for which the idea → on its own is not suited.Were it not for this deductive aspect, there would be no point in drawingthe distinction between strong necessary implication and valid inference,and in forming distinct ideas of these relations. Of the two, however,→ is closer to ex terminis evaluation and hence more fundamental; forex terminis evaluation is the final arbiter concerning the validity of anydeductive argument, just as it is concerning the logical modality of anyproposition.

7.2. Some of Quine’s Objections Against Modality

Quine (1953) raises doubts about logical modality because of argumentssuch as these:

(i) Necessarily 9 is greater than 7;the number of planets = 9;therefore necessarily the number of planets is greater than 7.

(ii) Possibly the number of planets is less than 7;the number of planets = 9;therefore possibly 9 is less than 7.

(iii) Necessarily if there is life on the Evening Star then there is life onthe Evening Star;the Evening Star = the Morning Star;therefore necessarily if there is life on the Evening Star then thereis life on the Morning Star.

According to an informal conception of standard modal logic, whichQuine is working with, these intuitively invalid arguments turn out as

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valid, using only the substitutivity of identicals as a deductive rule.According to CTM-based logic, in contrast, no such arguments are valid,for at least two reasons. To begin with, each of the arguments has alogically false conclusion. (For whatever modality a proposition has, ithas it with logical necessity; in the case of the conclusions of (i), (ii),and (iii) – that necessarily the number of planets is greater than 7, thatpossibly 9 is less than 7, that necessarily if there is life on the EveningStar then there is life on the Morning Star – the modalities are clearlywrong, on intuitive grounds; so the conclusions must be regarded as logi-cally false.) I showed earlier that the mind cannot infer such conclusionsas true without violating the principle of clear and distinct ideas (seeSection 6.1.2). This alone is sufficient to render the arguments invalid,since there can be no schemata of true strong necessary conditional thatcould serve as deductive rules to validate such arguments.

Secondly, each of the arguments mixes a logically true with a con-tingently true premiss, and draws its conclusion by substituting into theformer on the basis of the latter. But although mixing necessary withcontingent premisses is in itself not a sin (provided the necessary pre-miss is true), substituting into a necessary premiss on the basis of acontingent identity – that is, a contingent proposition of identity – is afallacy. For the logical modality of a proposition consists in the mode ofepistemic evaluation of the proposition, and therefore constrains wherethe evidence for the proposition may come from. In the case of the nec-essarily true premiss, its evidence must be drawn ex terminis, from themeaning of its constituent ideas. Hence one could not truth-preservinglysubstitute into the premiss, unless the replacing idea and the idea beingreplaced were semantically identical, or synonymous (i.e., same in mean-ing); which clearly they are not, in either of the arguments.

The design of this paper has been expository rather than polemical;so I will not engage Quine any further. But I would that my expositorylabour had this much effect on Quinean professors, that when they chal-lenge the notion of logical modality, which we have inherited from theclassical philosophy of mind, they confront the real thing, not – as hasbeen usual with them – a straw proxy of their own making.

8. REMARKS ON CTM AND ANALYTIC PHILOSOPHY

At various places in the Essay, Locke foreshadows a “new sort of Logickand Critick”, other than either axiomatic or syllogistic logic. He tells usthat this “Logick and Critick” should rest on the mind’s “original way ofKnowledge”, underlying both axiomatic and syllogistic reasoning (or any

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conclusive reasoning whatever) and consisting in the mind’s “perceptionof the connexion and agreement, or disagreement and repugnancy of anyof our Ideas” (IVi1). It is plausible, in fact, to read the entire Essay asan attempt to spell out this “original way of Knowledge”; and althoughLocke does not bring the project anywhere near completion, leaving it toothers to carry it farther, a diligent reading of the Essay will show thathe comes a long way.

I will close this paper by highlighting a few features in my elaborationon the Lockian legacy, which seem to me most characteristic of the logicimplicit in it, and in CTM generally. There are three groups of criteriaI will use to this end; these may be regarded broadly as concerning thesyntax, semantics, and epistemology of CTM.

Among the syntactic features I will mention only the most comprehen-sive: that logical symbols are mental symbols, and that constructing a log-ical system requires mapping out (in practice, modelling) the system ofmind, as a system of cognitive-evaluative operations on the propositionsof the mental code; in turn, it requires that propositions be construed ascomposed of tokens of mental terms, or ideas, with complex ideas builtfrom a finite number of basic empirical ideas by a finite number of gener-ative operations. A major point of contention between classical theoristshas been whether the basic empirical ideas exhaust the class of simpleideas, and which ideas are simple. I chose to draw a distinction betweenempirical and non-empirical ideas, and to regard the empirical ideas asa posteriori, whilst non-empirical ideas as a priori, supposing the latterto be psychologically active, only when some basic ideas are acquired,in the generation of complex empirical ideas and propositions, and inassessing the epistemic values and logical modalities of the propositions.

The key semantic features of CTM are that the fundamental bear-ers of meaning are mental terms rather than propositions or structuresof propositions; that the meaning of a symbol consists in its denotingor representing a certain universal (not in the universal itself, let alonein a particular or set of particulars partaking of the universal); and thatthe universals represented are – with some qualifications – nominal: i.e.,fixed by the mind itself, not the environment, whether external or inter-nal. I took each basic idea to represent an empirical mode, whilst eacha priori idea (with the exception of the idea of identity) to represent anepistemic property, thus reflecting the special role of a priori ideas inepistemic evaluation.

The main epistemic features are, firstly, that the evidence for the log-ical truth or falsehood of a proposition is drawn solely from the pro-position’s semantically simple, clear and distinct constituent ideas, by

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certain evaluative operations, including assuming an epistemic value,inferring the values of atomic propositions, discerning the meanings ofsimple ideas, and judging the epistemic value and logical modality; and,secondly, that the criterion for the resolution of an assumption of value isnot the Aristotelian principle of non-contradiction, but rather the principleof clear and distinct semantic identity of simple ideas. The foundation oflogical modality, and of logic and epistemology in general, thus residesin the semantic identity of the simple ideas whereof the mind compos-es its complex ideas and propositions, and in the operations whereby itevaluates the propositions. For the simple ideas and the operations willnot allow just any proposition to be a bearer of truth, or falsehood; somepropositions they constrain to be true, some to be false, regardless ofnon-semantic and non-logical, empirical matters of fact. This constitutesthe objective ground of logic, and puts a normative restriction on the for-mation of clear and distinct complex ideas, including ideas of implicationand valid deductive inference.

Classical formal logic of Frege and Russell, which lies in the heartof Analytic Philosophy, may be viewed as an attempt to inject rigourinto early-modern logic, and Analytic Philosophy overall as an attemptto infuse greater rigour into early-modern philosophy. Frege and lat-er Analytic Philosophers had this much in common, that they sought anobjective ground for logic outside the mind: Frege in a mind-independentrealm of abstract objects; most of his followers, unhappy with this “unsci-entific Platonism”, and convinced that logic could not derive its objec-tivity from the mind, increasingly sought it in public, behavioural, overthabits and conventions (where, after much searching, they did not findit either, concluding there was no such ground). The reasons why Fregeand others felt compelled to abandon mentalism, and to seek anotherground for logical certainty and objectivity, had to do not so much witha clearly understood failure of mentalism, but rather with a failure tounderstand mentalism, as well as a need for a simplification and expedi-ency, making the task of formalisation in logic easier to manage. In myendeavour to revive early-modern mentalism and give it a partial formalexpression, I forego the beguiling benefits of purifying formal logic ofits mentalistic origins, and allow it to be embroiled, once again, in themany profound problems the classical philosophy of mind encounters(the very problems the early formal logicians desired to shake off). Yetthis makes logic truly a central part of philosophy, and restores to it adepth and scope which have disappeared in the modern and post-moderndevelopments. One such problem concerns the nature of ideas, taken notonly as bearers of semantic properties, and not only as components of

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propositions and so subjects of logical and epistemic evaluation, but alsoas discrete forms of consciousness. Here logic touches on the very heartof philosophy, something we ought to appreciate rather than shun.

NOTE

1This paper is a slightly revised version of Chapter 7 of my Mental Symbols: A Defence of

the Classical Theory of Mind (Dordrecht: Kluwer Academic Publishers, 1997). Through-

out, I will refer to the book as MS.

REFERENCES

Ackermann, W. (1956): Begrundung einer strengen Implikation, The Journal of SymbolicLogic 21, 113–128.

Anderson, A. R., Belnap, N. D., Jr., et al. (1975): Entailment: the Logic of Relevanceand Necessity, Vol. 1. Princeton University Press, Princeton.

Descartes, R. (1984): Meditations on First Philosophy. In Cottingham, J., Stoothoff, R.and Murdoch, D. (eds and trans.). The Philosophical Writings of Descartes. CambridgeUniversity Press, Cambridge.

Geach, P. T. (1958): Entailment, The Aristotelian Society: Supplementary Vol. XXXII,157–172.

Leibniz, G. W. (1981): New Essays on Human Understanding, Remnant, P. and Bennett,J. (eds and trans.). Cambridge University Press, Cambridge.

Locke, J. (1975): An Essay Concerning Human Understanding, edited by Nidditch P. H.,Oxford University Press, New York.

Novak, P. (1997): Mental Symbols: A Defence of the Classical Theory of Mind. KluwerAcademic Publishers, Dordrecht.

Quine, W. V. O. (1951): Two Dogmas of Empiricism. In From a Logical Point ofView. Harvard University Press, Cambridge, Mass. 1953. First published: PhilosophicalReview, 1951 LX, 20–43.

Quine, W. V. O. (1953): Reference and Modality. In From a Logical Point of View.Harvard University Press, Cambridge, Mass.

Smiley, T. J. (1959): Entailment and Deducibility, Proceedings of the Aristotelian SocietyLIX, 233–254.

Strawson, P. F. (1948): Necessary Propositions and Entailment-Statements, Mind LVII,184–200.

Von Wright, G. H. (1957): The Concept of Entailment. In Logical Studies. Routledgeand Kegan Paul, London, 166–191.

Watling, J. (1958): Entailment, The Aristotelian Society: Supplementary Vol. XXXII, 143–156.

689 Ford Road, Priestdale, Qld. 4127, Australia(e-mail: [email protected])

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