Jaakko Hintikka

REFORMING LOGIC (AND SET THEORY)

1. Frege’s mistake

Frege is justifiably considered the most important thinker in the development of our

contemporary “modern” logic. One corollary to this historical role of Frege’s is that his

mistakes are found in a magnified form in the subsequent development of logic. This

paper examines one such mistake and its later history. Diagnosing this history also

reveals ways of overcoming some of the limitations that Frege’s mistake has unwittingly

imposed on current forms of modern logic.

Frege’s mistake concerns the semantics (meaning) of quantifiers. The mistake is

to assume that this semantics is exhausted by the quantifiers’ (quantified variables’)

ranging over a class of values. These values are the members of the domain (universe of

discourse) of the language to which the quantifiers belong. The entire job description of

the quantifiers is to indicate whither or not at least one member of the domain has a

certain (possible complex) predicate (existential quantifier) and to indicate whether all of

them have one (universal quantifier). In other words, quantifiers are higher order

predicates indicating whether or not a given lower-order predicate is nonempty or

exceptionless. This is in fact precisely how Frege proposes to treat quantifiers in his

logical theory. (See Frege 1984, pp. 153-154, pp. 26-27 of the original.)

This is obviously part of the semantical task of quantifiers. However, it is not the

only one. Quantifiers have another function in language. There is a task that any

language must be capable of fulfilling if it is to serve as a language of science and for that

matter as a language suitable for innumerable purposes in everyday life. This task is to

C:\Hintikka.Reforming logic.and set theory.0408.doc.7/30/2008

indicate what depends on what, more explicitly, to express relations of dependencies and

independencies between variables. It is easily seen that the only way of expressing such

dependencies in an ordinary logical language on the first-order level is through formal

dependencies and independencies between quantifiers. That the variable y depends on x

(in the sense of ordinary-life dependence) is expressed by the fact that the quantifier

(Q1y) to which formally depends on the quantifier (Q2x) to which x is bound. Thus in an

(interpreted) sentence of the form

(1.1) (∀x)(∃y)F[x,y]

the variable y depends on the variable x, as is seen e.g. from the fact that the truth-making

value (“witness individual”) of y depends on the value of x. (Re witness individuals, see

also sec. 9 below.)

Such dependence can be expressed on the second-order level by quantifiers

asserting the existence of a function that embodies this dependence. For instance, (1.1) is

equivalent with

(1.2) (∃f)(∀x)F[x,f(x)]

Here f picks out as its value b=f(a) a truth-making value b of y that corresponds to the

value a=x of each x. It will turn out that this way of expressing the dependence of

variables can also be expressed on the first-order level by means of the dependence

relations of first-order quantifiers. This can be done in IF logic; see section 2 below.

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This independence of the two aspects of the semantics of quantifiers of one another is

vividly seen in many-sorted quantification theory. The two quantifiers can range over

different and even exclusive domains, and yet be either dependent or independent of each

other, as the case may be.

It is not anachronistic to call Frege’s neglect of the role of quantifiers as

expressing such dependencies a mistake. Frege’s own co-discoverer of the logic of

quantifiers, C.S. Peirce, was fully cognizant of this dimension of their semantics. In

practice, its most basic manifestation is the importance of quantifier ordering. In Peirce,

this ordering comes up in the form of the distinction between the two players of the

semantical games and quantifiers of whose importance Peirce was aware. Peirce’s pen-

pal Ernest Schröder struggled with the problems of coping with the same aspect of the

meaning of quantifiers in less vivid terms. (See here Hintikka 1996 (b) and the references

given there.)

2. IF logic and scope

One consequence of Frege’s mistake has been pointed out earlier and corrected, at least

in part. (See e.g. Hintikka 1996.) Since part of the task of quantifiers is to express

dependencies between variables, our logic should be able to do this job completely. In

other words, we should be in a position to express any possible pattern of dependencies

and independencies between variables. These interpreted dependencies between

variables are expressed by the formal dependencies between the quantifiers to which they

are bound. Now how are these formal dependencies codified in the usual logical

notation? The obvious answer is: By the nesting of quantifier scopes. But this nesting

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relation is of a rather special kind. It is among other features transitive and

antisymmetric. Furthermore, it is linear in the sense that the scopes of two quantifiers

cannot overlap only partially. Hence only such dependence patterns can be formulated in

the received logic of quantifiers when the dependence relation has these special

properties. As a consequence, only some of all possible patterns of dependence and

independence can be expressed in the received first-order logic. Hence this logic does

not fulfill its whole job description. Frege’s mistake thus gave rise to a flaw in the

received first-order logic.

This flaw is corrected in what has come to be called IF logic (For it, see e.g.

Hintikka 1996 (a), Hintikka and Sandu 1996.) This can for most purposes be

accomplished by introducing an independence-indicating / (“slash”) that makes a

quantifier (Q2y/Q1x) (replacing (Q2y) independent of another quantifier (Q1x) even when

it occurs in the syntactical scope of (Q1x).

It is thus seen that IF logic is not a special logic alternative to the received logic of

quantifiers. On the contrary, it is our usual Frege-Russell first-order logic that is

unnecessarily restricted in its expressive power and hence should be considered a special

logic among alternatives. In contrast, IF logic is the unrestricted logic of quantifiers.

In this essay, IF logic is not discussed further and is not relied on, either, except as

an object lesson. It is nevertheless in order to point out some consequences of its very

existence.

Once we realize that the nesting of syntactical scopes is not an ideal method of

expressing dependence and independence, we realize also that we have to be careful of

the traditional notion of scope as an explanatory notion in semantics. (Cf. here Hintikka

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1997.) The traditional notion combines two things that per se have nothing to do with

each other. Syntactical scope is used to indicate the dependence and independence of

quantifiers and other logical operators of each other. (This might be called dependence

scope or priority scope.) But it also makes the syntactical segment of a sentence (or

discourse) where a variable is bound to a given quantifier. (Binding scope.)

Once the difference between these two is understood, certain problems in the

semantics of natural language are solved. A case in point is the semantics of the so-

called donkey sentences.

(2.1) If Peter owns a donkey, he beats it.

(2.2) If you give each child a gift for Christmas, some child will open it today.

The meaning of (2.1)-(2.2) cannot be expressed in the notation of the received first-order

logic. But if a binding scope is expressed by parentheses ( ) and dependence scope by

brackets [ ], the logical form of these two will be

(2.3) [(∃x)(O(p,x)] ⊃ B(p,x))

(2.4) [(∀x)((∃y)G(x,y)] ⊃ (∃z)O(z,y))

The apparent difficulty with such “donkey” sentences as (2.1) – (2.2) is largely

due to the very same mistake we saw Frege committing. What distinguishes expressions

like (2.3)-(2.4) from familiar ones is conspicuously the use of the dependence-indicating

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brackets [ ]. A failure to use them is accordingly not to give the dependence-identifying

role of quantifiers their full due.

Much of what has been said of dependence relations between quantifiers can be

said of dependence relations of other logically active notions, including propositional

connectives, epistemic and modal operators etc.. For instance, epistemic logic was held

back for years before it was realized that wh-knowledge can only be adequately

expressed by means of quantifiers that are independent of clause initial epistemic

operators, as e.g. in “It is know who is F” whose logical form turns out to be

(2.5) K(∃x/K)F[x]

where the stroke / expresses independence. (See here Hintikka 2003.)

In general, by freeing the conventions governing the scope we can achieve the

same result as by introducing an independence indicator. In this way, we will be able to

express patterns of dependence and independence between quantifiers (and propositional

connectives) and constants that cannot be expressed in the received first-order logic.

(Constants may also have to be included in the arguments of Skolem functions.) The fact

that we can thus carry out the liberation of quantifiers by changing only the punctuation

of logical sentences is vivid evidence for the naturalness and indeed indispensability of IF

logic.

It is even possible in this way to turn Tarski’s T-schema into a truth definition.

Let us assume that x is a variable for the Gödel numbers x = g (S) of sentences S.. Then

Tarski’s T-schema summarizes all sentences of the form

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(2.6) T(a) ↔ S[a]

where T(x) is a truth predicate. Tarski is right in that we cannot have

(2.7) (∀x)(T(x) ↔ S[x])

As I have pointed out on other occasions, this failure is due to the fact that quantifiers and

other logical operators in S[x] should not depend on the variable x, which has a purely

syntactical role in S[x]. Such dependencies can be ruled out by writing instead of (2.7)

(2.8) (∀x)([T(x)] ↔ S[x])

Of course, this is no longer equivalent to any ordinary first-order sentence. The same

thing can be expressed in IF logic by making all the quantifiers and propositional

connectives in (2.8) (other than (∀x)) independent of the initial universal quantifier (∀x).

Either way, our liberated notation enables us to do what Tarski proved impossible to do

by means of the received Frege- Russell first-order logic: convert the T-scheme into a

genuine truth definition.

3. From existential instantiation to functional instantiation

Another consequence of Frege’s mistake that is (perhaps unwittingly) repeated by later

logicians looks so insignificant that it has not attracted much attention. It concerns the

formulation of the rules of inference for our basic first-order logic. There it looks very

much as if the meaning of quantifiers is done full justice to (in a context of deduction) by

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the usual rules of instantiation. The rule of existential instantiation applies to a formula

(∃x)F[x] with an initial existential quantifier. It allows the replacement of this formula

by F[β] where β can be thought of as standing for a possibly unknown individual of the

kind the given formula says is instantiated. This obviously captures the force of the

existential quantifier as expressing non-emptiness.

Intuitively, the term β operates just like the “John Does” and “Jane Roes” of

lawyers’ jargon. (Wallis thought that historically such legal usage was the historical

model for algebraic symbols; see Klein 1968. p. 321.) Formally, the term β can be a

“dummy name” or in our deductive practice simply a new individual constant.

Likewise, the usual rule of universal instantiation might seem to capture

adequately the semantical force of a universal quantifier as expressing universality

(exceptionlessness).

But even though these instantiation rules express truth and nothing but the truth

about the meaning of quantifiers, they do not tell us the whole truth. One at first sight

inconspicuous feature of theirs is that they apply only to sentence-initial quantifiers.

They do not apply to quantifiers inside a formula, not even if this formula is assumed to

be in the negation normal form. (This assumption is routinely made in this paper.) Every

logic instructor who has taught to her students the usual rule of existential instantiation is

likely to find herself later correcting students who are proposing to apply it to quantifiers

inside a formula, perhaps within the scope of universal quantifiers. At this point, a clever

student could try to embarrass the instructor by asking: “Since the rule of existential

instantiation is obviously based directly on the meaning of the existential quantifier,

surely it ought to be applicable independently of the context. What happens in such an

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application is that we merely choose one individual of a certain kind among existing ones

for our attention.”

If the instructor is up to her task, she will point out that the choice of the

“arbitrary individual” β is not absolute, not a once-and-for-all matter, but depends on

other individuals. More specifically, it depends on the values of the universal quantifiers

within the scope of which the existential quantifiers occurs (in a sentence that is in the

negation normal form).

This answer points to an important truth. Existential instantiation can take place

inside larger formulas, if we use as an instantiating term a function term that takes into

account the dependence of the existential quantifiers to which it is applied on other

quantifiers in the same sentence. If we heed those dependencies, we can generalize the

rule of existential instantiation. The generalized formulation might run as follows:

Assume that S is a sentence in the negation normal form and that the formula

(3.1) (∃x)F[x]

occurs somewhere in S=S[(∃x)F[x]]. Then S may be replaced by

(3.2) S[F[f(y1,y2,…)]

where (∀y1), (∀y2),… are all the universal quantifiers which the scope of which (∃x)

occurs in S, and f is a new function constant. If there are no such universal quantifiers,

the function term f(y1, y2, …) is replaced by a new individual constant. The old rule of

existential instantiation is thus a special case of the new one, viz. the case of sentence-

initial existential quantifiers.

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More generally, we can stipulate that (Q1y1), (Q2y2), … are all the quantifiers in S

on which the quantifier (∃x) depends on there. This formulation can be used also in IF

logic.

Notice that this is a first-order rule in the crucial sense that no quantification over

higher-order entities is involved. The reason why we have considered instantiation by

functions rather than individuals should be obvious. It reflects the fact that witness

individuals may depend on other witness individuals.

By the same toke the rule of existential generalization has to be liberated. It will

allow the replacement of any function term of the form f(x,y1,y2,…) to be replaced by a

variable z bound to an existential quantifier (∃z). This quantifier must occur within the

scope of all the quantifiers (∀y1), (∀y2),… . Otherwise its location is free, assuming only

that we are dealing with a formula in the negation normal form

4. Uses of the rule of functional instantiation

The relative neglect of the generalized rule existential instantiation can be taken to be an

instance of the same mistake as has been here attribute to Frege. But is it a mistake in the

present context? Defenders of status quo can try to claim that the rule of functional

instantiation is dispensable, and that its neglect is therefore justified, perhaps in the

interest of theoretical economy.

Admittedly, the rule of functional instantiation is redundant in the received

treatment of first-order logic. In this logic, we can let an existential formula wait in our

logical argumentation until by means of applications of other rules it has been brought to

the surface of our formulas, in other words until it has been brought to a sentence-initial

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position. But in principle we have to ask whether this dredging process affects the

semantics of an existential quantifier, including its dependence relation to other

quantifiers. Logicians have been victims of bad luck in that the process of bringing an

existential quantifier to the surface of a sentence does not affect its deductive function in

the received first-order logic. This is bad luck in that it has directed their attention away

from those aspects of the logic of quantifiers that are due to dependence and

independence relations between them, thus making this instance of Frege’s mistake a

mistake.

An example can illustrate the way in which functional instantiation helps to make

logical proof s shorter and more natural. Consider the conditional

(4.1) (∀x)(∃y)(∀z)(F(x,y) & G(y,z)) ⊃ (∀x)(∀z)(∃y)(F(x,y) & G(y,z))

Its proof e.g. by the tableau method would involve six instantiations, three layers of

formulas and two branches on the right side. In contrast, consider an application of the

rule of functional instantiation to the antecedent of (4.1). It yields

(4.2) (∀x)(∀z)(F(x,f(x)) & g(f(x),z))

An application of the rule of existential generalization yields the consequent.

This proof is not only simpler than e.g. a tableau proof. It is obviously far closer to the

ways in which mathematicians actually think. If you do not see this at once, think of the

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ways in which you would express the functional instantiation proof in the jargon of

mathematicians. The antecedent would be read somewhat as follows:

Given (only) x, there is an object y such that for any z, F(x,y) and G(y,z) But if

so, since this object depends only on x, it will trivially satisfy for any x and z the same

conjunction.

This inference would be considered completely trivial. Yet in reality it involves

an appeal to a principle of reasoning too strong in its general form to be accommodated in

the current first-order axiom systems of set theory, as we will see.

We can use functional instantiation systematically and obtain a huge

simplification of many first-order logical proofs. What one can do is to turn a proposition

into a negation normal form and eliminate all existential quantifiers by means of the rule

of functional instantiation. The remaining quantifiers are all universal. They can all be

moved to the beginning of the sentence and largely neglected. The reason is that all the

variables bound to them admit arbitrary substitutions. Without any great loss of

generality, we can assume that all predicates have been replaced by functions, perhaps by

their characteristic functions. (The characteristic function of a one-place predicate A(x)

is a function f(x) such that A(x) iff f(x) = O. This is easily generalized.)

When all this is done, all usual formal first-order logical proofs become literally

symbolic calculations in which all of the logic of quantifiers is reduced to substitutions of

terms (usually function terms) for free variables in equations combined with each other

truth-functionally. It would be interesting to see what a proof theory for such a logic of

equations might look like.

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A logic developed along these lines does not have theoretical interest only. It

yields a proof method which often is in practice incomparably handier that the usual first-

order proof methods. In order to see this, consider an example. Suppose that we have to

prove the proposition about Abelian groups that would usually be expressed as follows

(4.3) x o (z o y) = (x o y) o z

where o expresses the group operator. (The symbol o expresses a two-argument

function.) Even to express (4.3) by means of quantifiers would require seven of them:

(4.4) (∀x)(∀y)(∀z)(∀u)(∀v)(∀w)(∀t)

(((z o y=u & (x ou)=v & (x y)=w & (wo z=t) ⊃ v=t) o

The associative and commutative laws would likewise require several quantifiers. To

deduce from them (4.3) by means of ordinary first-order logic would be a messy

enterprise. In contrast, the functional deduction is trivial:

(4.5) x (z o y) = xo(y o z) = (xoy) o z o

The first identity is justified by the commutativity of , the second by its associativity.

But not only does functional instantiation facilitate formal logical proofs, tacit

instantiation plays a pervasive role in ordinary human reasoning. Take, for instance the

old chestnut of a puzzle that I have used earlier to illustrate reasoning in ordinary life:

o

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(4.6) A gentleman and his sister are sitting on a bench in a park. Pointing to a child

playing nearby, he says: “That’s my niece. His sister says, “But not mine.” How is it

possible for both of them to be right?

How do we solve in real life such problems? Let us try to do so, and watch

ourselves in process. The child is the brother’s niece if and only if she is female and

(4.7) (∃x)(S(b,x) & P(x,c))

Here c = the child, s = the sister, b = the brother, S(x,y) = x and y are siblings, and P(x, y)

= x is a parent of y. Obviously, (4.7) is tacitly obtained from the definition of a niece.

Our singular terms b and c are tacitly instantiating certain variables (say y and z) in such

a definition. In order to argue further, we obviously have to instantiate the x in (4.7). This

should introduce a function term p(y,z) for the so far unidentified parent of c. But of

course our reasoning practice suggests its dependence of y and z, and argue in terms of it

as if it were a simple term p, and argue simply as follows: Both s and p are siblings of b,

while p and s are not siblings. This is possible only if p=s.

Here it is seen how we spontaneously argue in terms of function terms as if they

were constants. In contrast, a conventional first-order proof would be so complicated as

to tax severely one’s patience, and would not be halfway as übersichtlich.

In more general terms, the rule of functional instantiation thus allows an

automatic concrete interpretation of what is going on in a purely formal or “symbolic”

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proof. It can be viewed as a codification of the idea behind mathematicians’ time-

honored locution for an existential quantifier: “One can find.” This verbal formula leaves

unexpressed the crucial question: What has to be known before one can find it?

This interpretability is relevant to the philosophical problem of understanding

formal logical proofs. Wittgenstein was especially keenly attuned to this problem, but

never found a solution that would have satisfied him. Here we can see what kinds of

interpretations of logical arguments might have satisfied him. (I can imagine Frank

Ramsey surviving and forcing Wittgenstein to see the point.)

5. Functional instantiation is a first-order rule

It is worth emphasizing that a first-order logic amplified with a rule of functional

instantiation is still a first-order logic. Considered alone, such logic is precisely as

strong as the received first-order logic, not any stronger. Moreover, it is first-order in the

crucial sense that it involves no quantification over any higher-order entities is involved.

We all know Quine’s quip “to be is to be a value of a bound variable”. In the present

context, it is much more that a clever slogan. I am convinced Hilbert was right in

thinking that our difficulties in the foundations of mathematics are due to problems

concerning the existence of higher-order entities. (Those problems are e.g. instantiated

by a problem of choosing the axioms of set theory.) A first-order logic that includes

functional instantiation is free from all such problems. The fact that in the rule of

functional instantiation we introduce function constants over and above individual ones

merely reflects the trivial fact that the witness individuals that show (in the sense of

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displaying) the truth of a quantificational sentence can depend on other such witness

individuals.

By the same token, we need not worry about the consistency of the rule of

functional instantiation.

Another indication of the first-order status of the rule of functional instantiation is

that this rule is a valid logical principle of independence-friendly first-order logic. Even

the dispensability of the rule of functional instantiation the received first-order logic is

interesting in the present context. It can be considered a proof of the fact that the rule of

functional instantiation expresses a purely logical principle, and a first-order one to boot.

6. Functional instantiation and the axiom of choice

It might nevertheless seem that the main role of the rule of functional instantiation is to

provide us with a way of improving first-order logic, but not anything relevant to

foundational issues. This can perhaps be said if first-order logic is considered only by

itself. When it is used in wider context, it turns out to have remarkable powers.

For one thing the rule of functional instantiation is no longer dispensable in IF

logic. The dependence and independence relations admitted there are more sensitive than

those to which received logic confines us, so sensitive that they can be disturbed in the

process of bringing an existential quantifier to a sentence-initial position.

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7. Axiom of choice — the axiom of choice

The rule of functional instantiation might still seem to be only a handy tool in improving

the theory and practice of our basic first-order logic. In reality, its most striking

repercussions lie in the foundations of mathematics, especially in set theory.

In dealing with these foundations, we have to go beyond first-order logic. The

received first-order logic is too weak for the purpose, and therefore has to be considered

as a part of a larger enterprise, be it set-theory or higher-order logic. Now what happens

if our modified first-order logic that now includes the rule of functional instantiation

operates as a part of second-order logic? Obviously, we have to assume that this second-

order logic includes the usual unproblematic second-order quantifier rules, including

universal instantiation. Consider then, a sentence of the following form

(7.1) (∀x)(∃y)F[x,y] ⊃ (∃f)(∀x)F[x,f(x)]

Given the rule of functional instantiation, (7.1) is logically true. In order to see that it is,

consider its negation

(7.2) (∀x)(∃y)F[x,y] & (∀f)~(∀x)F[x,f(x)]

An application of the rule of functional instantiation to the first conjunct of (7.2) yields a

formula of the form

(7.3) (∀x)F[x,g(x)]

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Universal instantiation as applied to the second conjunction yields

(7.4) ~(∀x)F[x,g(x)]

which contradicts (7.2). Hence (7.1) is logically true.

In a similar way we can obviously prove any conditional of the form

(7.5) (S ⊃ S(sk))

where S is a first-order sentence and S(sk) the second-order sentence that asserts the

existence of a full array of the Skolem functions for S.

What is remarkable about (7.1) is that it is an application of what is usually called

the axiom of choice. Indeed the schema instantiated by (7.1) is sometimes used as a

formulation of the axiom of choice. Since (7.1) are provable by using only first-order

principles including the rule of functional instantiation (over and above trivially valid

ones), it follows that the (so-called) axiom of choice is a valid first-order logical

principle.

This conclusion is reinforced by the fact that the axiom of choice is valid in first-

order IF logic. For instance, it is easily seen that the counterpart of (7.1) in IF logic is a

logical truth there. In IF logic, the consequent of (7.1) becomes

(7.6) (∀x1)(∀x2)(∃y1/∀x2)(∃y2/∀x1)(((x1=x2) ↔ (y1=y2)) & F[x1,y1] & F[x2,y2))

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This is logically equivalent with the second-order sentence

(7.7) (∃f1)(∃f2)(∀x1)(∀x2)((x1=x2) ↔ (f1(x)=f2(x)) & F[x1,f1(x1)] & F(x2,f2(x2)])

Here the first conjunct says that f1 and f2 are the same function. Hence (7.7) is equivalent

with

(7.8) (∃f)(∀x)F[x,f(x)]

which is the consequent of (7.1).

In short, the rule of functional instantiation is tantamount to a strong form of the

axiom of choice. In the rest of this paper much of the discussion is formulated in terms

of the axiom of choice. It should not be forgotten that we shall be in effect talking about

the first-order rule of functional instantiation.

In view of what has been found about the first-order status and the consequent

indispensability of this rule, the nature and status of the axiom of choice have to be

reconsidered. Indeed the first-order status of the axiom of choice is in stark contrast to the

ways it is usually dealt with. Usually, it is considered a set-theoretical principle. Often,

this principle is codified into the axiom system of set theory. This is where the term

“axiom” in “axiom of choice” comes from. Even though this term will be seen to be

inappropriate, it will nevertheless be used in what follows.

What has been seen is that the rule of functional instantiation has the effect of

turning the “axiom” of choice into a first-order logical truth. This is interesting also in

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view of the history of foundational studies. The ideal that the great Hilbert had was to do

mathematics entirely on the first-order level. (He blamed all the ills in the foundations of

mathematics on the use of higher-order conceptualizations.) (Hilbert 1922, pp.162-163.)

The first and foremost example of an indispensable higher-order mode of

reasoning is the axiom of choice. Hilbert’s ε-calculus was an attempt to bring the axiom

of choice down to the first-order level. (Se Hilbert and Bernays 1934-39.) It was not a

complete success in this respect. For one thing, it did not facilitate consistency proofs for

elementary arithmetic.

One can even pinpoint the crucial shortcoming of Hilbert’s epsilon-technique. He

was on the right track in using choice terms, but he failed to indicate explicitly what the

choices in question depend on. It is a variant of the mistake we found in Frege: A failure

to appreciate fully the role of dependence relations in first-order logic.

We have now seen that this mistake is not inevitable. By showing that the “axiom” of

choice is a first-order logical principle, we have realized an important part of Hilbert’s

hopes. This has repercussions for the evaluation of Hilbert’s foundational work in

general. For instance, when elementary number theory is based on IF first-order logic

instead of the received one, it becomes possible to prove its consistency by arguably

elementary means. (See Hintikka and Karakadilar 2006.)

8. Axiom of choice vs. axiomatic set theory

The first-order character of the axiom of choice means that it is inappropriate to construe

it as an axiom of a nonlogical mathematical theory, viz. axiomatic set theory. It ought to

be instead a part of the logic which is used in set theory and in terms of which proofs in

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set theory are being couched. The failure of logicians to do that is attributable to the

disregard of dependence relations between quantifiers that has been called Frege’s

mistake. Indeed, from (7.1) one can see how the axiom of choice is a matter of spelling

out what the dependence of an existential quantifier or a universal one means.

It might at first look as if it were merely a matter of terminology whether the

assumption we are dealing with is called a first-order logical principle or a set-theoretical

axiom. However, calling it an axiom of a mathematical theory has a point only if this

axiom makes a difference in the sense of ruling out otherwise conceivable alternatives.

The claim that is made here is therefore that a set theory without the axiom of choice

involves serious interpretational problems. Later in this essay, it will be discussed how

these difficulties are manifested in the foundations of set theory.

But what is the difficulty here? A version of the axiom of choice is included in

the usual axiom systems of set theory. And there does not seem to be any difficulties in

formulating the first-order logic that is used as a basis of set theory so as to include a rule

of functional instantiation. Here the issue seems to be merely a matter of philosophical

emphasis.

Things are not so simple, however. Here we meet the feature of the problem

situation that has not been completely unknown but whose full significance has not been

appreciated. Logicians are here facing a dilemma. On the one hand, the form of the

axiom of choice that is used in first-order axiomatic set theories does not capture the

same full force of the principle that is among other formulations captured by (7.1). On

the other hand, if we try to incorporate assumptions codifying this force in the usual first-

order axiom systems of set theory, they become uninterpretable and even inconsistent.

21

This happens independently whether the strong axiom of choice is introduced by a

separate set-theoretical axiom or whether it is introduced by strengthening the underlying

logic used in set theory by incorporating the role of functional instantiation in it.

The fact that the full force of the axiom of choice cannot be stated in first-order

axiomatic set theories without making them uninterpretable as set theories can be seen in

different ways. Any set theory AX that can serve as a basis of mathematical theories

should allow the reconstruction of elementary arithmetic. Hence we can use Gödel

numbering or an equivalent technique to discuss the syntax of AX in the very same set

theory based on AX. Among other things, we can then formulate a numerical predicate

K(g(s)) = K(x) that says that the sentence with the Gödel number x=g(S) does not have

all its Skolem functions. By the diagonal theorem there is then a sentence S of the form

K(n)=K(g(S)) with the Gödel number n. Here n is the numeral expressing n. Intuitively

(although slightly inaccurately) S could be taken to say, “My Skolem functions do not all

exist” in the same sense as the famous Gödel sentence which says, “I am unprovable.”

Thus S must be true, for if it were false, its Skolem functions would exist. Such existence

is enough to guarantee the truth of S. Hence S will be true but without its Skolem

functions, which is violates the notion of truth.

Moreover, the existence of S is easily proved formally in the set theory in

question. This does not mean that the set theory in question is inconsistent, But it means

that it does not admit of the intended kind of interpretation, that is, an interpretation

where the objects quantified over are sets. For the allegedly true sentence S would be

false in such a model.

22

Furthermore, an incorporation of the full axiom of choice in conventional axiom

systems of set theory would make them inconsistent. In the self-applied set theory we

could form a predicate P that applies to the Gödel number n=G(S) if and only if the

Skolem functions of S all exist. Such a predicate would be a truth condition for set

theory. Alas, from Tarski’s impossibility theorem it follows that such a truth predicate is

impossible on the pain of inconsistency, as it would allow for a truth definition for a first

order theory in the same theory. This observation turns out to touch some of the most

important presumed uses of set theory; see sec.10 below.

The use of a restricted form of the axiom of choice in axiomatic set theory is

sometimes motivated by reference of the distinction between sets and classes that is made

in some set theories. The axiom of choice is taken to be applicable to sets only, not to

proper classes. This is not very satisfactory theoretically, either. There does not seem to

be anything intrinsic to a collection of objects that would make it a proper class instead of

a set. For instance, what is it about the class of all unit sets that makes it a proper class?

Frege even identified this class with the number one. Surely the number one should be

capable of serving as a value of any set-theoretical variable x even in a context like x ∈ c.

9. Set theory vs. model theory

Is there an explanation of this tremendous strength of the rule of functional instantiation?

Yes. Its strength is not accidental. It is based on the very nature of quantificational

discourse. More specifically, it is based on the fact that the existence of a Skolem

function for a quantificational sentence S is the natural truth condition for S.

23

One way of seeing this is in terms of the idea of “witness individuals”

vouchsafing the truth of S. For a sentence of the form (∃x)F[x], a witness individual b is

one satisfying F[x], that is, making F[b] true. For a sentence of the form (∀x)(∃y)F[x,y],

witness individuals a, b must satisfy F[a,b]. But here the choice of b depends on the

choice of a. Hence the existence of suitable witness individuals means the existence of a

function f(x) such that, for each a, a and f(a) can serve as witness individuals. This is

generalizable as a matter of course to the existence of Skolem functions as guaranteeing

the existence of the appropriate witness individuals.

This truth condition is equivalent to any other adequate truth condition. This

explains the significance of the rule of functional instantiation, for it is what provides for

the existence of Skolem functions for any true sentence. If those Skolem functions do not

always exist for a true sentence, truth is not expressible in the language in question. In

this sense, the ultimate reason why the strong form of the axiom of choice which is

codified in the rule of functional instantiation is not available in first-order

axiomatizations of set theory is Tarski’s impossibility theorem: such a strong form of

axiom of choice would make truth expressible in those axiomatizations.

Some philosophers have earnestly tried to find “truthmakers”, that is entities of

some kind or other that serve to make true sentences true. The search has not revealed

unproblematic truthmakers. Now we can see what the true truthmakers of a

quantificational sentence S are. They are the Skolem functions of S.

The persuasiveness of this answer is enhanced by the game-theoretical

interpretation of first-order logic. There Skolem functions are codifications of those

strategies that enable a verifier in a semantical game always to win. Such a win may be

24

considered as a tentative verification of the sentence S which is the object of the

semantical game G(S) associated with S.

The failure of the rule of functional instantiation in an axiomatized set theory

therefore means that truth is not definable in it. It may look as if truth may be definable

in a suitably formulated axiomatic set theory on the first-order level. Such appearances

are deceptive, however. What happens in such cases is that the pseudo-definition yields

sometimes wrong results. In particular, these will be in any model (in the first-order

sense of a model) of first-order axiomatic set theory where allegedly true sentences

whose Skolem functions do not exist and others therefore are not true on a set-theoretical

interpretation of the model. Now the availability of a truth predicate is a condition sine

qua non for any realistic model theory. The failure of all truth predicates in a first-order

axiomatic set theory therefore means that first-order axiomatic set theory is an inadequate

framework for model theory of itself.

This is a striking result in that it contradicts the widespread idea of axiomatic set

theory as the natural medium of all model theory. This idea is simply wrong. For any

halfway adequate model theory you need the notion of a truth, which just is not available

in a set theory using traditional first-order logic. First-order axiomatic set theories are

poor frameworks even for their own model theory. A fortiori, there are likely to be poor

frameworks for any theory formulated in their terms.

The inadequacies of first-order axiomatic set theory as a framework of model

theory are made especially serious by the role of metatheoretic conceptualizations in

modern mathematical practice. Philosophers often seem to entertain an oversimplified

picture of a mathematician as a chap who sets up axiom systems and then draws logical

25

conclusions from them. Perhaps this oversimplification is not peculiar to philosophers

only. Even some practicing mathematicians think that all mathematics can do is to draw

conclusions from the axioms of ZF set theory. (Cf. Ruelle, 2007, pp. 63, 73-74.) Model-

theoretic questions are on this view a superstructure that may perhaps be the business of

logicians and philosophers rather than mathematicians per se.

This is a radical misrepresentation of current mathematical practice. Not only has

the line between mathematical theories and their model theories become inconspicuous.

Much of what counts as actual mathematical theorizing is in fact model-theoretical.

Consider, as an example, group theory. Only a miniscule part of any work in set theory

consists of deductions from the axioms of the theory. The bulk of the actual work in

group theory is metatheoretical, consisting largely in such things as classifications of

groups of different kinds, representation theorems, and other ways of gaining an

overview of the models of group theory (i.e. groups) of different kinds. Particular

deductive consequences of the axioms do not play a much bigger role in the real theory

than particular numerical equations like Kant’s 5+7=12 play in actual number theory.

This feature of mathematical practice explains a curious episode in the history of

twentieth-century logical theory. (Cf. Hintikka 2004.) Tarski’s preferences in logic were

algebraic rather than geometric or set-theoretical. In the forties, he ganged up with Quine

to criticize Carnap’s attempts to build a model theory in the form of “logical semantics”.

This makes it prima facie surprising that it was Tarski who in the fifties and sixties led

the development of the present-day model theory. The solution lies in the fact that Tarski

was virtually forced to develop a model theory by his pursuits in the theory and

metatheory of different algebraic systems. It was not initially thought of as a separate

26

branch of logical studies, comparable to proof theory or recursion theory. It was created

as part of the metamathematics of algebra.

10. The meaning of quantifiers and the foundations of mathematics

The rule of functional instantiation does not presuppose any particular “standard”

conception of logic, either. In fact, it offers means of implementing such “nonstandard”

variants of logic as constructivistic and intuitionistic ones and also bringing out their

precise differences from the “classical” logic. Indeed, all we have to do is to restrict the

interpretation of the function constants introduced in functional instantiation in some

desired way, for instance to constructive functions or to known functions.

The problems discussed in this paper are thus likely to come up in any reasonable

approach to the foundations. In view of the role the axiom of choice plays in the

arguments marshaled here, it is therefore instructive to see that, for all the lip service to

the contrary, some of the most prominent constructivists among philosophers of logic (for

some reason they call themselves intuitionists) have ended up endorsing the axiom of

choice. They include Michael Dummett (1997, pp. 52-59) and Per Martin-Löf (1984, pp.

50-52). This strikingly illustrates the fact that what is at issue in the axiom of choice is

the meaning of quantifiers, not the interpretation of mathematical truth in general.

This can be generalized. The introduction of the rule of functional instantiation

has striking consequences for the understanding of what is referred to as “mathematical

practice” and what has recently become a revered holy cow in semi-popular philosophy

of mathematics. In spite of the attention ostensibly paid to this practice, some of its

significant features have not been noted. One of them is the fact that mathematicians

27

routinely use functional instantiation in their reasoning. As soon as objects of a certain

kind exist, mathematicians introduce symbols for them, even though those objects depend

on others,. Often that dependence is not explicitly indicated.

For one simple example, one of the axioms of group theory could be expressed as

(10.1) (∀x)(∃y)(x o y = e)

But nobody in actual practice (other than a logic student) starts a proof from (10.1). A

mathematician immediately introduces a symbol, e.g. x-1 for the y. This is but an

application of functional instantiation.

Likewise, in defining the continuity of a function f(x) at the value xo in an interval

x1 ≤ xo ≤ x2 by the usual ε–δ method, textbooks write out only one symbol for ε and δ,

respectively, even though δ in reality is a function δ(ε) of ε. Moreover, δ depends also on

xo so that it should strictly speaking be expressed as δ(εi xo), even though in introductory

texts this is never expressed. (If δ actually can be chosen independently of xo, we have a

definition of uniform continuity, instead of continuity simpliciter.)

In many, probably most case, such functional instantiations can b treated as

expository tricks. But this does not change the fact that mathematicians routinely rely on

a rule of inference that is (in suitable contexts) extremely strong, in fact so strong that it is

incompatible with the usual axiom systems of set theory. This in turn refutes the

commonplace belief that first-order axiomatic set theories can be considered a lingua

franca of all mathematics.

28

The fact that axiomatic set theory does not capture certain obviously acceptable

modes of inference must also be considered a serious limitation to the uses of axiomatic

set theory. It seems to me that we should pay much more attention to these limitations.

Thus we should for instance consider Gödel’s and Paul Cohen’s unprovability results as

warning signs, as symptoms of shortcomings, rather than informative achievements

concerning the continuum hypothesis or the axiom of choice. (Cf. Cohen 1966.)

In sum, what the logic is that practicing mathematicians in effect use is a version

of first-order logic that includes functional instantiation. This logic is easily confused in

its applications with ordinary first-order logic. The reason is that when mathematicians

instantiate their (usually tacit) quantifiers, the dependence of the instantiating “arbitrary

object” on other objects is often, perhaps typically, left unexpressed. This is not merely a

matter of exposition. Since mathematicians are frequently using functional instantiations

in contexts involving sets on other higher-order entities, their logic is in fact much

stronger than the received first-order logic and in fact stronger than the usual first-order

axiomatized set theories. This shows how unrealistic these first-order axiomatic set

theories are as frameworks of mathematical practice.

It is no excuse for this failure that its roots may lie in the nature of literally

hardwired human preferences in logic reasoning. In general, human reasoners like to

operate with free variables or other symbols that behave like constants in that their

dependencies on other objects can be disregarded, rather than bound variables or other

symbols whose dependence on others is spelled out. The exception is a variable bound to

a sentence-initial universal quantifier. Such a variable can be thought of as representing

“an arbitrary individual” or perhaps “an unknown individual” about which we can reason

29

in the same way as ordinary known ones. This instinctive preference may be due to the

hardwired characteristics of the human information-processing faculty. (See von

Neumann 1958, Hintikka 1990.) Now we have seen that this preferred method of

reasoning can be made possible by the rule of functional instantiation. This rule has

therefore an important role in any humanly natural system of reasoning.

Perhaps we can from the vantage point that has been reached also put the large-

scale history of modern logic into an interesting and perhaps ironic perspective. One

characteristic feature of the entire logicist enterprise of Frege, Russell and Whitehead and

their ilk is that they tried to reduce mathematical reasoning to purely logical reasoning.

For instance, in Frege’s axiomatization of his Begriffsschrift there are no

characteristically higher-order assumptions. Frege thought he could formulate the crucial

assumption in terms of the identity conditions of extensions and value-ranges of

propositional functions (Cf. the Basic Law V of his Grundgesetze.) This enterprise might

seem hopelessly unrealistic, in view of the apparent limitations of first-order reasoning.

However, it is now seen that logicists’ reliance on first-order logic is not entirely

misplaced. Unfortunately, what later logicians and mathematicians did was look for the

sources of the missing greater strength outside logic, mainly in set theory, instead of

making the most of what they already had in first-order logic.

If the idea behind the rule of functional instantiation is as simple as it has been

seen to be and yet so consequential, how come it has not been used and studied before?

It does not seem unfair to blame it on the same neglect of the role of quantifiers as

dependence indicators as we initially diagnosed in Frege.

30

One form of this neglect is a failure to pay attention to the assumptions that are

actually made in the reasoning used in mathematical practice. When the axiom of choice

was first formulated, it turned out that it had unwittingly been used frequently in accepted

mathematical arguments, sometimes by the very critics of the axiom. It seems that this

self examination should be continued. It has turned out that what looks like a simple

first-order inference may in fact be an appeal to a strong version of the axiom of choice.

This is important in a foundational perspective for the purpose of understanding

mathematical practice. This practice may involve assumptions that go well beyond, not

only our usual first-order logic, but our usual axiomatic set theory. This provides an

interesting perspective on projects like the “reverse mathematics” of Harvey Friedman. It

is of great interest to see precisely what assumptions an actual mathematical argument

presupposes.

11. Quo vadis?

Where should foundational studies be headed after Frege’s mistake has been rectified?

This is too large and too sweeping a question to be dealt with in one paper. Some

observations nevertheless seem pertinent. For one thing, set theory should in the future

be based on some logic that allows the formulation of a truth predicate for set theory by

the means of set theory itself. One such logic is IF first-order logic. But whatever logic

can serve this purpose presumably must dispense with the law of excluded middle, as IF

logic does. This would necessitate giving up of Frege’s well-known requirement on sets,

viz. that the membership in one of them is well defined, not allowing indeterminate cases.

This would mean a significant change in our very notion of set.

31

One can also ask: In the light of these results, where should the study of set theory

be heading? Or should we rather ask: What should set theory be taken to be? There is an

age-old debate as to what logic really is, a theory (alias “science”) or a conceptual tool

for all sciences, an organon. This question is still very much alive. For instance, are the

axiomatizations of this or that part of logic on a par with the axiomatizations of scientific

theories? The deep differences between the two are sometimes overlooked.

The same question should be asked about set theory. It is often taken to be like

any other mathematical theory. But if so, how can it be a way of codifying logical

principles of reasoning, such as in the axiom of choice? In any ordinary axiomatic theory,

we need some logic by means of which we reason about its models. Now the axiomatic

assumptions in set theory are assumptions concerning those models. How can they at the

same time codify modes of reasoning about those models?

The interpretation of axiomatic set theory as a normal mathematical theory leads

to other strange results. For what are the objects which it theorizes about? All actually

existing sets? But how do we know what there actually exists? Either we have to

postulate an upper floor of our universe populated by abstract Platonic objects or else we

have to envisage a super-universe of possible structures, some sort of “model of all

models”. Neither conception can be easily disproved, but neither has much appeal to a

thinker who takes set theory to claim to have a special foundational role. For surely we

need some subject-independent logic in order to reason about such entities.

What has been found in this essay suggests an unpopular answer. It has been

found that one of the kingpins of set theory, the axiom of choice, must be considered a

logical principle, even a first-order one. This strongly suggests looking at the entire set

32

theory in the same way, as a part of logic rather than as a separate mathematical theory.

This suggestion is supported strongly, virtually conclusively by developments starting

from IF logic. Many mathematical conceptualizations and modes of reasoning that go

beyond the resources of the received logic and hence were typically considered set-

theoretical rather than logical, for instance equicardinality and Kǔnig’s lemma, are

captured by means of IF logic. Indeed, if we are willing to use very strong forms of

tertium non datur, the entire force of second-order logic an be captured in a suitably

enriched first-order logic. Since second-order IF logic arguably catches all the inferences

needed in normal mathematics, set theory becomes dispensable as a foundational

enterprise, unless it merges with the strengthened first-order logic.

A revision of set theory along the lines sketched here is not a retreat. On the

contrary, it opens new opportunities. It was seen that the full force of the rule of

functional instantiation cannot be realized within the framework of first-order axiomatic

set theory. Small wonder, therefore, that important problems such as the truth of the

continuum hypothesis cannot be solved in a system like ZF set theory. With the help of a

logic incorporating the rule of functional instantiation these problems become more

easily accessible already on the first-order level. Hence even if you do not want to give

up first-order axiomatic set theory tout court, you may be interested in examining what

can be done in an alternative approach.

33

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