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A Suppositional Theory of

Conditionals

1 Abstract

Supposition and conditionals appear closely related. For example, (1) and (2)seem to play similar roles in discourse:

(1) Suppose that the butler did it. Then the gardener is innocent.

(2) If the butler did it, then the gardener is innocent.

Suppositional Theories take this observation as a starting point, appealingto supposition to provide an account of the natural language conditional (e.g.,(Mackie (1972)); (Edgington (1995)); (Barker (1995); a.o.). For example, here isJ.L. Mackie:

“The basic concept required for the interpretation of if-sentences isthat of supposing [. . . ] To assert ‘If p, q ’ is to assert q within thescope of the supposition that p”(1972, 92-93).

This paper develops a suppositional theory of conditionals. However, it differsfrom extant theories in (i) arguing for a precise semantic connection betweeninstructions to suppose and conditional antecedents, and (ii) providing novellinguistic data in favor of that theory.

2 Supposition and ‘If’-Clauses

2.1 Conditional Inferences

Consider the following three inference patterns:

(Pres) φ ψ ⇒ (φ ∧ ψ) Preservation

(DA) (φ ∨ ψ) ¬φ⇒ ψ Direct Argument

(CT) φ⇒ (ψ ⇒ χ), ψ φ⇒ χ Conditional Telescoping

(Pres), (DA) and (CT) are often taken to be (intuitively) valid for indicativeconditionals. Consider, e.g., (3)-(5):1

1(NB: throughout, ; is used for subjunctives, 99K for indicatives, ⇒ for non-specific (i.e.,subjunctive or indicative) conditionals and ⊃ for the material conditional).

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A Suppositional Theory of Conditionals

(3) Ada is drinking red wine. (So) if she’s eating fish, she’s eating fish anddrinking red wine.

(4) Claude is either in London or Paris. (So) if he’s not in London, he’s inParis.

(5) If Lori is married to Kyle, then if she’s married to Lyle, she’s a bigamist.She’s married to Lyle. (So) if she’s married to Kyle, she’s a bigamist.

In contrast, the same inference patterns are standardly taken to be (intuitively)invalid for subjunctives:

(6) Ada is drinking red wine. (So) if she were eating fish, she’d be eatingfish and drinking red wine.

(7) Claude is either in London or Paris. (So) if he weren’t in London, he’dbe in Paris.

(8) If Lori were married to Kyle, then if she were married to Lyle, she’d be abigamist. She’s married to Lyle. (So) if she were married to Kyle, she’dbe a bigamist.

Counter-instances to (6)-(8) are easily identified. For example, circumstances inwhich Ada is drinking red wine and eating beef, but would be drinking whitewine were she eating fish will constitute counter-instances to (6); circumstancesin which Claude is in London, but might be in Rome were he not will constitutecounter-instances to (7); and circumstances in which Lori is married to Lyle, butwould not be, were she to be married to Kyle will constitute counter-instancesto (8).

Notably, however, embedding the rightmost premise under ‘Suppose’ leads eachsubjunctive inference pattern to improve considerably:

(9) Suppose Ada were drinking red wine. (Then) if she were eating fish,she’d be eating fish and drinking red wine.

(10) Suppose Claude were in London or Paris. (Then) if he weren’t in London,he’d be in Paris.

(11) If Lori were married to Kyle, then if she were married to Lyle, she’d be abigamist. Suppose she were married to Lyle. (Then) if she were marriedto Kyle, she’d be a bigamist.

That is, where the non-conditional premise is supposed — rather than asserted— subjunctive instances of (Pres), (DA) and (CT) appear valid. Two briefobservations are in order: first, note that in (9)-(11) the discourse particle‘then’, rather than ‘so’, as in (3)-(8), must be used to indicate entailment.Imperative clauses headed by ‘suppose’ behave like ‘if’-clauses in this respect.Like the former, the latter also license the occurrence of ‘then’, (and, correlatively,preclude the occurrence of ‘so’). Second, in addition to being embedded under

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suppose, the non-conditional premise occurs with an additional layer of pasttense morphology in each of (9)-(11). The way in which morphological markinginteracts with the entailment patterns is discussed in further detail in §5.4.

Related to (PRES), (DA) and (CT) are their deduction theorem equivalents,below:

(Pres⇒) φ⇒ (ψ ⇒ (φ ∧ ψ))

(DA⇒) (φ ∨ ψ)⇒ (¬φ⇒ ψ)

(CT⇒) φ⇒ (ψ ⇒ χ) ψ ⇒ (φ⇒ χ)

Indicative instances of (Pres⇒), (DA⇒) and (CT⇒) are standardly taken tobe valid. Indeed, this follows from the acceptance, for indicatives, of (Pres),(DA), (CT) and the Deduction Theorem.

(DT) φ ψ iff φ⇒ ψ Deduction Theorem

If the deduction theorem were accepted for subjunctives, we would expect each of(Pres⇒), (DA⇒), and (CT⇒) to have false subjunctive instances, correspondingto the invalid instances of the respective inference patterns (6)-(8). However,the relevant instances, (12)-(14), appear valid.

(12) If Ada were drinking red wine, then if she were eating fish, she’d beeating fish and drinking red wine.

(13) If Claude were in London or Paris, then if he weren’t in London he’d bein Paris.

(14) If Lori were married to Kyle, then if she were married to Lyle, she’d bea bigamist. (So) If Lori were married to Lyle, then if she were marriedto Kyle, she’d be a bigamist.

Summarizing, whereas the three inference patterns are invalid for subjunctives intheir basic form, they improve substantially if the leftmost premise is embeddedeither (i) under supposition or (ii) in the antecedent of a subjunctive in whichthe conclusion is nested. In this respect, supposition and subjunctive antecedentsappear to have similar effects.

2.2 Epistemic Contradictions

Sequences of the form ¬φ;3φ (and, equally, 3φ;¬φ (though cf. Groenendijk andStokhof (1991))) are widely agreed to give rise to infelicity. Call such sequencesEpistemic Contradictions. When their elements are conjoined, epistemiccontradictions pattern with unembedded Moore sentences — conjunctions of theform ¬φ ∧ ¬(KnowsSpeaker(¬φ)).

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(15) ??It isn’t raining but it might be.

(16) ??It isn’t raining but I don’t know that.

(15) and (16) are equally infelicitous. However, as noted by Yalcin (2007),(15) and (16) pattern differently when embedded under supposition and in theantecedents of subjunctives:

(17) ??Suppose it isn’t raining but it might be.

(18) Suppose it isn’t raining but I don’t know that.

(19) ??If it weren’t raining but it might’ve been, I’d have brought an umbrellaneedlessly.

(20) If it weren’t raining but I didn’t know that, I’d have brought an umbrellaneedlessly.

This contrast constitutes a second, independent connection between the twoenvironments. Whereas both have an ameliorative effect on the felicity ofMoore sentences within their scope, they each inherit the infelicity of embeddedepistemic contradictions.

2.3 Counterfactual Usage

The connection between supposition and conditional antecedents is furtherreinforced by consideration of a third body of data. As has been widely noted,unlike indicatives, subjunctives can be used counterfactually (see, e.g., Stalnaker(1975), von Fintel (1999)). As demonstrated in (21)-(22), the latter, but not theformer, are acceptable in discourse contexts which entail the negation of theirantecedent.

(21) The butler didn’t do it. ??If he did it, he used the candlestick.

(22) The butler didn’t do it. If he’d done it, he would’ve used the candlestick.

However, the second discourse degrades substantially if the first sentence is eitherembedded under supposition (e.g., (23)) or in the antecedent of an embeddingconditional (e.g., (24)).

(23) Suppose that the butler hadn’t done it. ??If he’d done it, he would’veused the candlestick.

(24) ??If the butler hadn’t done it, then if he’d done it, then he would’ve usedthe candlestick.

That is, when the antecedent of a subjunctive is inconsistent with (i) an earliersupposition, or (ii) the antecedent of an embedding subjunctive, subjunctivespattern with indicatives in being incompatible with counterfactual uses.

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2.4 Summary

The pattern observed in §§2.1-3 is suggestive. Intuitively, the inference patterns(Pres), (DA) and (CT) improve due to the fact that supposition and subjunctiveantecedents both require information conveyed by their subordinate clauses tobe preserved when evaluating ‘downstream’ subjunctives.For example, consider(9). Having supposed (rather than merely asserted) that Ada is drinking redwine, we hold this fact fixed when evaluating the conditional in the conclusion.Accordingly, counter-instances (such as the one considered for (6)) cannot arise.Similar considerations will also explain (10)-(11). If we assume that subjunctiveantecedents effect subjunctives in their antecedent in the same way as supposition,we can account for (12)-(14) in the same way.

This rough picture generalizes to other kinds of example. For example, considerthe unavailability of counterfactual uses in (23)-(24). After a bare assertionthat the butler is innocent, the subjunctive in (23) allows us to evaluate itsconsequent at some (minimally different) possibilities in which he was guilty.However, if supposition and subjunctive antecedents require us to hold fixed hisinnocence, downstream subjunctives whose antecedents entail his guilt will beexpected to impose conflicting constraints.

The remainder of the paper develops a new suppositional theory of conditionals(both indicative and subjunctive) which implements this rough picture to accountfor the data. Informally, the idea is as follows. Supposition has a dual effect ondiscourse context: (i) it induces a minimal revision to the possibilities underconsideration, to incorporate the supposed information; and (ii) it modifies whatwill count as a minimal revision in the future, ensuring that the informationsupposed will be preserved. The conditional form (i.e., ‘If. . . (then). . . ’) is thentreated as expressing a strict conditional, but one in which the informationconveyed by the antecedent is supposed, rather than added to the contextdirectly.

§3 discusses extant theories which draw a close connection between ‘if’-clausesand supposing; §4 introduces a dynamic framework in which contexts representpossibilities under consideration (via an information state) and the structure ofmodal space (via a selection function), and articulates a uniform theory of con-ditionals within it. §5 demonstrates how the indicative/subjunctive distinctioncan be explained within this framework, and how it accounts for the data. §§6-7conclude.

3 Suppositional Theories: A Brief Overview

A theory of conditionals which makes essential appeal to supposition has beendefended (in various, closely related, forms) by a number of authors, includingMackie (1972), Edgington (1995), Barker (1995), DeRose and Grandy (1999),

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and Barnett (2006). What is common to variants of the theory is a commitmentto the claim that, in uttering a sentence of the form ‘If φ, ψ’ (where φ, ψ areclauses with declarative mood), an agent performs a speech act equivalent (insome respect) to sequentially: (i) supposing that φ, and (ii) asserting that ψ.Callthis the Speech-Act Suppositional Theory. Below is the formulation of the theoryby three of its proponents:

“[I]t [the Speech Act Suppositional Theory] explains conditionals interms of what would probably be classified as a complex illocutionaryspeech act, the framing of a supposition and putting somethingforward within its scope.” (Mackie (1972, 100))

“To assert or believe ‘if φ, ψ’ is to assert (believe) ψ within the scopeof the supposition, or assumption, that φ.” (Edgington (1986, 5))

“The pragmatic theory of ‘if’ states that utterance of ‘if φ, ψ’ is suchan assertion of ψ grounded on supposition of φ where [the speaker]implicates via the presence of ‘if φ’ that their assertion of ψ is sogrounded.” (Barker (1995, 188))

Neither Mackie or Edgington provides a non-metaphorical gloss of their talk ofone speech act occurring within the scope of another. However, the intendedposition appears relatively clear. Performing a speech act of supposing that φresults in a new discourse context (which differs from the discourse context whichwould result from asserting that φ). An assertion of ψ in this new context maydiffer (in its felicity, its illocutionary effects, etc.) from an assertion of ψ in theprior context. The Speech-Act Supposition Theory says, then, that the primaryconventionally determined contribution of an ‘if’-clause is to indicate that thespeaker is performing a speech act equivalent to asserting the consequent in thediscourse context created by supposing that the antecedent.

The Speech-Act Suppositional theory has a number of appealing features. Mostnotably, it accounts for the apparent substitutability in context of (1)-(2).However, the theory also faces substantial, well-known challenges:

Embeddability: Conditionals can occur felicitously in sub-sentential envi-ronments. Amongst other examples, they can be embedded under negation(e.g., (25)), in the complements of attitude verbs (e.g., (26)), and in theconditionals consequents (e.g., (27)):

(25) It isn’t the case that if Lea rolls a six, she’ll win.

(26) Jacob (believes/knows/doubts) that if Lea rolls a six, she’ll win.

(27) If Caroline rolls a five, then Lea will win if she rolls a six.

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This behavior is appears to generate a problem for the Speech-Act Supposi-tional Theory. It is standardly assumed that speech acts cannot be performedusing a clause in an embedded environment (though cf. Krifka (2001, 2004)).The proponent of the theory must, accordingly, provide an account of (i)what the conditional contributes to the content of a sentence when it occursin an embedded environment, and (ii) what the role of the ‘if’-clause is insuch environments, if not to mark a speech act.

Validity: A minimally adequate theory of conditionals ought to be able tobe supplemented with a notion of validity in order to generate predictionsabout how conditionals interact with other ‘logical’ vocabulary. The statusof inference patterns relating conditionals and expressions such as negation,disjunction and conjunction are amongst the core subject matter of the studyof conditionals. Any satisfactory theory should, at least in principle, becapable of adjudicating questions of these kinds.

The problem is that, under the Speech-Act Suppositional Theory, the expres-sions belong to fundamentally different semantic categories. On the theory’sstandard version, negation, disjunction, conjunction, etc. are assigned theirclassical, truth-functional meaning. Accordingly, their logical properties areto be understood in terms of their effect on a sentence’s truth-conditions. Incontrast, any logical properties ascribed to the conditional will arise fromits effect on the speech acts which can be performed by an utterance of asentence, rather than on the truth-conditions of that sentence. Indeed, onthe standard version of the theory, sentences with a conditional at widestscope cannot be attributed truth conditions at all.

Accordingly, any notion of validity capable of making predictions about theinteraction of conditionals with other logical vocabulary needs to be formu-lated as a relation between (classes of) speech acts. It is not at all clear whatsuch a notion of entailment will look like. In particular, since utterance ofconditionals lack truth conditions on the Speech-Act Suppositional Theory, itcannot amount to mere relation of truth preservation between the assertionof the premises and the assertion of the conclusion.

Speech Act theories have attempted to address both issues, though they divergein how they do so. In response to the first problem, Mackie (1972, 103) optsfor a disjunctive approach, on which conditionals may (sometimes) express aconditional proposition in embedded contexts (where it is a matter of contextwhat proposition that is). Edgington (1995, §7.3) denies that sentences like(25)-(27) are, despite superficial appearances, examples of acceptable embeddedconditionals. Barker accepts the data at face value, and appeals to a combinationof metalinguistic negation, sui generis rules and iterated supposition (withoutaddressing embedding under attitude verbs). In response to the second problem,many variants of the speech act suppositional theory have proposed a probabilistic

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treatment of validity (following Adams (1975)), on which an argument is validiff the high probability of each of the premises entails the high probability ofthe conclusion. Where the probability of the conditional is associated with theconditional probability of its consequent on its antecedent, this yields positivepredictions about interaction with connectives. However, it does so only at thecost of giving up classically valid principles governing other logical vocabulary,such as n-ary conjunction introduction.

Rather than providing a detailed evaluation of the prospects of these responses,I wish to show how an alternative form of suppositional theory can avoid bothproblems altogether. The need for a new form of theory is, in part, independentlymotivated by the observations in §§2.1-3, since no extant version of the speechact theory accounts for the complex pattern of data surveyed there.

4 Update Semantics with Selection Functions

The present suppositional theory is not a speech act theory, at least in the senseof the previous section. Rather than treating the introduction of suppositionsas a specific type of conversational act, I propose instead that instructionsto suppose express a specific type of sentence meaning. Since instructions tosuppose are distinguished by their effects of discourse context, implementingthis idea requires a framework in which the conversational effect of uttering asentence is encoded in the meaning of the sentence uttered. The dynamic systemwhich adopted below provides one such framework.

Given an object language enriched with a sentential supposition operator, wecan model the introduction of supposition at the level of compositional semanticcontent. The conditional of natural language is then ascribed the meaning of astrict conditional with a embedded supposition operator (following the informalgloss provided in §2.4). Crucially, this approach allows us to avoid the issues withembeddability and validity which arise for traditional suppositional theories.

4.1 Updating Update Semantics

In static semantics, the meaning of a sentence is a proposition — a functionfrom points of evaluation to truth values. In dynamic semantics, the meaning ofa sentence is a context change potential (CCP) — a function from points ofevaluation to points of evaluation.

In standard dynamic frameworks, points of evaluation model discourse contexts(states of a conversation). By identifying meanings with CCPs, dynamic seman-tics models both how an utterance’s evaluation is dependent upon context andhow an utterance changes the context at which later utterances are evaluated.

In update semantics (Veltman (1996)), points of evaluation are identified with

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information states — sets of worlds. Informally, an information state, c,represents the possibilities under consideration at a given point in a conversation.Let L0= |A|¬φ|φ∧ψ|φ∨ψ|2φ|. A world w,w ′. . . is a total function from atomicformulae of L0 into {0,1}.

Definition 1 (| · |).

i. c|A|=c ∩ {w|w(A) = 1}.ii. c|¬φ|=c/c|φ|.

iii. c|φ ∧ ψ|=c|φ||ψ|.iv. c|φ ∨ ψ|=c|φ| ∪ c|ψ|.

v. c|2φ|=

{c, if c|φ| = c;

∅, otherwise.

|A| eliminates any non-A-worlds from c; |¬φ| eliminates any worlds from c whichsurvive update of c with |φ|; |φ∧ψ| returns the result of updating c sequentiallywith |φ| and |ψ|; |φ ∨ ψ| returns the union of c|φ| and c|ψ|; |2φ| is a test whichchecks whether c is a fixed point of |φ|; if so, it returns c; if not, it returns ∅.Stipulatively, φ ⊃ ψ =def ¬(φ ∧ ¬ψ) and 3φ =def ¬2¬φ.

Definition 2 ( us ).

i. c us φ iff c|φ| = c.

ii. ψi, . . . ψj us φ iff, for all c, c|ψi|, . . . |ψj | us φ

We say that c supports φ (c us φ) iff c is a fixed point of |φ|; ψi, . . . ψj (update-

to-test) entail φ (ψi, . . . ψj us φ) iff for any c, c|ψi|, . . . |ψj | is a fixed point of|φ|.

In update semantics, a point of evaluation represents the possibilities underconsideration. This enables the framework to model the way that utterancesin a language equipped with modals are both sensitive to — and can have aneffect on — what is considered possible at a state of a conversation. However,in order to model the phenomena observed in §§2.1-3, we also need to be ableto represent the way in which utterances are sensitive to — and can have aneffect on — what is counted as a minimal revision of the possibilities underconsideration in a conversation. That is, our points of evaluation need to alsorepresent the structure of modal space.

In static variably-strict theories of conditionals, the structure of modal space ata context can be modeled by a classical selection function (Stalnaker (1968),Lewis (1973)); that is, a function from a static point of evaluation (a world)and a static sentence meaning (a proposition) to a static point of evaluation.2

2At least for Stalnaker. Lewis’s selection functions are functions from a point of evaluationand meaning to a set of points of evaluation.

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Given minimal constraints, a classical selection function induces a total orderover the domain of worlds (relative to each world). The dynamic frameworksketched above differs from the static framework assumed by Stalnaker andLewis in its choice of both points of evaluation and meanings. Thus, in thepresent setting, the structure of modal space is better modeled as a functionfrom an update semantic point of evaluation (an information state) and anupdate semantic sentence meaning (an update semantic CCP) to an updatesemantic point of evaluation. An update semantic selection function f is a(potentially partial) function, which maps a pair 〈c, |φ|〉 to an information state c′,where defined. Intuitively, f〈c, |φ|〉 can be thought of as the minimal revision toc which incorporates the information conveyed by φ. Different update semanticselection functions (hereafter, simply ‘selection functions’) will correspond todifferent choices about what information is to be retained. We can consider arange of constraints on such selection functions, analogous to those imposed byStalnaker and Lewis.

(Succ) f〈c, |φ|〉|φ| = f〈c, |φ|〉 Success

(Trns) f〈c, |φ|〉 = f〈c, |2φ|〉 Transparency

(Simp) f〈f〈c, |φ|〉|, |ψ|〉| = f〈c, |φ|�|ψ|〉| if ∃c′ : c′ us φ ∧ ψ Simplification

(Min) f〈c, |φ|〉 = c|φ| if c|φ| 6= ∅ or c = ∅ Minimality

Success says that f〈c, |φ|〉 is a fixed point of |φ|. That is, the minimal revision toc incorporating the information conveyed by φ should left unchanged by successiveupdate with φ. Transparency says that the minimal revision to c incorporatingthe information conveyed by φ is the same as the minimal revision incorporatingthe information conveyed by 2φ. Simplification says that, where φ are ψ arecoherent, if f〈c, |φ|〉=c′, then f〈c, |φ|�|ψ|〉 is identical to f〈c′, |ψ|〉.3 That is, theminimal revision to c incorporating the information conveyed, sequentially, byφ and ψ, is the same as the minimal revision incorporating the informationconveyed by ψ to the minimal revision incorporating the information conveyed byψ to c. If, where defined, f satisfies success, transparency and simplification, callf adequate. Every adequate US-selection function will induce a partial orderover information states. Minimality says that, if φ is compatible with c or ifc=∅, f〈c, |φ|〉 is simply the result of updating c with |φ|. That is, if updating cwith φ does not result in moving from a non-absurd state to an absurd state, thenthe minimal revision to c incorporating the information conveyed by φ is simplyc|φ|. In the limiting case where c=∅, minimality says that f〈∅, |φ|〉 = ∅ for all φ.That is, the minimal revision to ∅ incorporating the information conveyed byany sentence is ∅. If, where defined, f satisfies minimality and is adequate, call fproper. Where c|φ| 6= c|φ||φ|, no proper selection function will be defined onf〈c, |φ|〉, since success and minimality cannot be simultaneously satisfied. Theidea, implemented below, will be that every conversation starts with a proper

3i�j is the composition of i and j ; i.e., i�j (x)=j (i(x)).

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selection function, but that updates over the course of conversation can result ina selection function which is improper, yet adequate.

To get an intuitive grasp on the operation of minimal revision, it may be helpfulto see how it can be decomposed. Minimal revision with φ can be understood asa two-step process, in which: (i) the input information state is expanded so thatit is possible to reach a φ-state from it; and, (ii) that state is contracted, so thatit incorporates the information conveyed by φ.

In more concrete terms, imagine an ordering on information states in terms ofthe strength of information they incorporate. c incorporates at least as muchinformation as c′ just in case c′ is a superset of c — that is, c4c′ iff c⊆c′.

A retraction operation is an operation which maps each pair 〈c, |φ|〉 to aless informative state, c′, where there is some state at least as informative asc′ which incorporates the information conveyed by φ.4 That is, if f Ret is aretraction operation, then c4 fRet〈c, |φ|〉, and there is some c′′ such that (i)c′′4 fRet〈c, |φ|〉 and (ii) c′′|φ|=c′′. Retraction operations can be thought of asexpanding the input information state so that it is capable of being strengthenedinto a φ-incorporating state.

An incorporation operation is an operation which maps each pair 〈c, |φ|〉to a more informative state which incorporates the information conveyed byφ. That is, if f Inc is an incorporation operation, then fInc〈c, |φ|〉 4 c andfInc〈c, |φ|〉|φ| = fInc〈c, |φ|〉. This might, but need not, be equivalent to the resultof updating c with |φ|.

Any proper selection function can be broken down into a pair of retraction andincorporation functions.5 That is, we can think of a proper selection function,given any input pair 〈c, |φ|〉 , as first finding a state weaker than c which is aweakening of a φ-incorporating state, and then finding some strengthening ofthat state which incorporates the information conveyed by φ.

{Ab}

{Ab, AB}

{AB}

f Ret(·) f Inc(·)

f (·)

4Retraction and incorporation operations are the model theoretic generalisations of con-traction and expansion in AGM, respectively (Alchourron et al. (1985)).

5So that f 〈c, |φ|〉 = fInc〈fRet〈c, |φ|〉, |φ|〉.

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For example, in the diagram above, dotted arrows represent selection functions,while solid lines represent the ordering of information states in terms of strength.The proper selection function f, which maps the pair 〈{Ab}, |B|〉 to {AB},decomposes into a retraction operation mapping the pair to the weaker state,{Ab, AB}, and an incorporation function which maps that state and |B| to thestronger state, {AB}.

Having introduced a way of modelling minimal revision, let an enriched pointof evaluation, σ, be a pair 〈cσ, fσ〉 of an information state, cσ, and selectionfunction, fσ. We say, derivatively, that σ is proper (adequate) iff f σ is proper(adequate). [·] is an interpretation function mapping sentences to enriched CCPs(functions from enriched points of evaluation to enriched points of evaluation).Correspondingly, we also need a revised notion of support and entailment. Inthe remainder of the paper we will understand support and entailment in termsof information preservation, as characterized by the definitions below.

Definition 3 ([·]).

σ[φ] = 〈cσ|φ|, fσ〉 (for φ ∈ L0))

Definition 4 ( ).

i. σ φ iff cσ = cσ[φ].

ii. ψi, . . . ψj φ iff, for all σ, σ[ψi], . . . [ψj ] φ (if σ[ψi], . . . [ψj ][φ] isdefined).

σ supports φ iff the information states of σ[φ] and σ are identical. ψi, . . . ψjStrawson entail φ iff for any σ, the information states of σ[ψi], . . . [ψj ] andσ[ψi], . . . [ψj ][φ] are identical (where both are defined). We say that σ is absurdiff cσ = ∅. The logics of L0 generated by | · | and [·], given their respectiveentailment relations, are the same.

4.2 Supposition

Our first task is to employ the enriched framework to model the effects ofsupposition. To do so, we will extend our language with a monadic operator,Sup(·). On the picture sketched in §2.4, after φ is supposed, any revision tothe possibilities under consideration must return an information state whichincorporates the information φ conveys. Accordingly, we need to define an updateoperation on selection functions. Let + be a function which maps formulae to afunction from selection functions to selection functions.

Definition 5 (+).

f+φ〈c, |ψ|〉=f〈c, |φ|�|ψ|〉.

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Intuitively, f+φ is the selection function just like f, but which preserves theinformation conveyed by φ. That is, it only ever returns an information statewhich supports φ. f+φ〈c, |ψ|〉 is the f -minimal revision to c which supportssequential update with φ and ψ. If f is proper, then f+φ will be adequate.However, f+φ may be improper despite f being proper. This will occur if c iscompatible with ψ, but not with φ ∧ ψ.

Let L1={Sup(φ)|φ ∈L0}. L0∪ L1 contains every formula of L0 and the result ofembedding those formulae under Sup(·).

Definition 6 (Sup(·)).

σ[Sup(φ)] = 〈f〈cσ, |φ|〉, f+φσ 〉.

Sup(φ) has a dual effect on σ: first, it replaces cσ with the f -minimal revision ofcσ incorporating the information conveyed by φ. Second, it replaces fσ with theselection function just like it, but which preserves the information conveyed by φ.Informally, Sup(φ) has the effect of (minimally) changing the set of possibilitiesunder consideration so that it entails the information φ conveys, and ensuringthat the result of any further suppositional changes also entail this information.

Supposition is veridical. After supposing φ, the resulting information stateincorporates any information entailed by the information conveyed by φ. It is alsoaccumulative. After a sequence of suppositions, the resulting information stateincorporates all of the information conveyed by each. Finally, it is conservative.If φ is consistent at σ, then updating with Sup(φ) and φ have the same effect onthe information state.

(Ver) Sup(φ) ψ, if φ ψ Veridicality

(Acc) Sup(φ), . . . Sup(ψ) (φ ∧ ψ) Accumulativity

(Con) cσ[Sup(φ)] = cσ[φ], if cσ[φ] 6= ∅ Conservativity

Veridicality and accumulativity are desirable properties for any operator mod-elling the English ‘Suppose (that)’ construction (as (28)-(29) suggest). Conserva-tivity yields less simple predictions for natural language, but as will be shown in§5, is crucial for accounting for the behavior of conditionals under supposition.6

(28) Suppose Claude were in London. (Then) he would be in Britain.

6Conservativity is characterized by the dual requirements that: (i) 3(φ ∧ ψ), Sup(φ) 3ψ;

and (ii) 2¬(φ ∧ ψ), Sup(φ) 2¬ψ The apparent acceptability of the inferences in (†.a-b)suggest these are desirable:

(†) a. The Mets might beat the Cubs and the Cardinals. Suppose they were to beat theCubs. Then they might beat the Cardinals.

b. The Mets can’t beat the Cubs and the Cardinals. Suppose they were to beat theCubs. Then they wouldn’t beat the Cardinals.

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(29) Suppose Ada were drinking red wine, Suppose she were eating fish. (Then)she would be drinking red wine and eating fish.

Before proceeding, it is important to highlight a central feature of suppositionwhich the present treatment fails to address. The effects of supposition arepersistent (they endure beyond its syntactic scope), but they are not irreversible.

(30) a. Suppose that it’s raining. Then, the park will be wet.b. . . . But suppose that it isn’t. Then, the park will be dry.b′. . . . Suppose that we go for a picnic. Then we’ll be miserable.

The discourse in (30.a) can be felicitously extended with (30.b). For this to occur,the former supposition’s effect need to be withdrawn: the information stateresulting from the latter supposition must no longer preserve the informationintroduced by the former.

That (30.a-b) are presented as contrasting appears important in triggeringwithdrawal. If (30.a) is followed by (30.b′) instead, no withdrawal is triggered,even if the possibility of going for a picnic while it’s raining is more remote,discourse-initially, than the possibility of going for a picnic while it’s not.

Modeling the reversibility of supposition does not present a substantial formalchallenge. For example, points of evaluation might instead be treated as sequencesof information state/selection function pairs, 〈σi, . . . σj〉. Supposing that φ wouldthen amount to adding a new element to sequence (i.e., 〈σi[φ], σi, . . . σj〉), whilereversing a supposition would amount to eliminating the top element of asequence. However, rather than complicating the framework in this way, I intendto retain its (comparative) simplicity at the cost of restricting its empiricalscope. Since the core features of supposition with which we are concerned areorthogonal to its reversibility, this is a relatively superficial cost (though we willbriefly return to the issue in §5.4).

4.3 Conditionals

Finally, we need to enrich our formal language with a conditional operator. LetL=L0∪L1∪L2 be the extension of L0∪L1 defined so that:

� If φ ∈L0, then φ ∈L2.

� If φ ∈L0∪L1 and ψ ∈L2, then φ→ ψ ∈L2.

� Nothing else is a member of L2

φ→ ψ expresses a generalisation of the dynamic strict conditional, defended bye.g., (Veltman (1985), Gillies (2004, 2009), Starr (ms)). 7

7For φ,ψ ∈L0, φ→ ψ=def2(φ ⊃ ψ) (for φ, ψ /∈L0, 2(φ ⊃ ψ) /∈ L).

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Definition 7 (→).

σ[φ→ ψ] =

{σ, if σ[φ] ψ

〈∅, fσ〉, otherwise.

φ → ψ checks whether update with ψ has an effect on the information stateof σ[ψ]. If not, it returns σ; if so, it returns an absurd state, 〈∅, fσ〉. Statedinformally, φ→ ψ induces a test, which passes iff, after update with φ, ψ conveysno new information.

Stalnaker (1968, 1975, 2009), Strawson (1986), and, more recently, Starr (2014),defend (Uniformity) as a constraint on any minimally adequate theory ofconditional:

(Uniformity) The semantic contribution of the conditional form is invariantacross subjunctives and indicatives.8

(Uniformity) requires that any difference between indicatives and subjunctivesis not attributable to an ambiguity in the conditional form itself. According to(Uniformity), ‘If . . . , (then). . . ’ is univocal across subjunctives and indicatives;it is possible to attribute a single semantic clause to the unspecific conditionalconstruction represented by φ⇒ ψ. The present proposal satisfies (Uniformity).According to the proposal, the natural language conditional form simply expressesa strict conditional in which the antecedent is embedded under supposition.

Definition 8. [⇒]

i. φ⇒ ψ =def Sup(φ)→ ψ

ii. σ[Sup(φ)→ ψ] =

{σ, if σ[Sup(φ)] ψ


Stated informally, Sup(φ)→ ψ induces a test which decomposes into two stages:first, it finds the result of updating σ with [Sup(ψ)]. This update returns the f σ-minimal φ-revision of cσ, and replaces f σ with its φ-preserving revision. Second,it checks that the resulting information state is left unchanged when σ[Sup(φ)]is updated with [ψ]. If so, it returns σ; if not, it returns an absurd state.

8Here’s Stalnaker’s formulation of (Uniformity):

“The semantic rule that gives the truth conditions of the conditional as a functionof the contextual parameter will be the same for both kinds of conditionals.”(Stalnaker (2009, 237, fn20))

Here’s Strawson’s:

“[T]he least attractive thing that one could say about the difference between thesetwo remarks is that [...] ‘if ... then...’ has a different meaning in one remark fromthe meaning which it has in the other.” (Strawson (1986, 230))

.

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As should be evident, Definition 8 constitutes a semantic implementation ofthe suppositional theory sketched in §2.4. A conditional encodes instructions toan agent to perform a test in which they update by supposing the antecedent,and then check the effect of incorporating the information in the consequent.

This conditional has a number of attractive features. It validates Import/Export(Appendix: Fact.1.). It is not equivalent to the material condtional. Accordingly,it follows from the collapse result due to Gibbard (1981) that it must invalidatemodus ponens. However, it can be shown that it does so minimally (Appendix:Fact.3) — that is, the cases in which modus ponens fails are all and only thosewhich are required to fail in order to avoid collapse into the material conditional.

These logical features make it a reasonable basis on which to construct a theory ofthe conditional form in natural language. §5 implements this idea, by developingan account of the indicative/subjunctive distinction compatible with Definition8.

5 The Indicative/Subjunctive Distinction

Indicative and subjunctive conditionals differ in meaning. Intuitively, what isconveyed by (31) is not the same as what is conveyed by (32).

(31) If thieves broke into my apartment, they didn’t take anything.

(32) If thieves had broken into my apartment, they wouldn’t have takenanything.

As (33) demonstrates, subjunctives (but not indicatives) are unacceptable indiscourse contexts entailing their antecedent. Conversely, as (34) demonstrates,indicatives (but not subjunctives) are unacceptable in discourse contexts entailingthe negation of their antecedent.

(33) Thieves broke into my apartment.a. If they broke in, they didn’t take anything.b. ??If they had broken in, they wouldn’t have taken anything.

(34) Thieves did not break into my apartment.a. ?? If they broke in, they didn’t take anything.b. If they had broken in, they wouldn’t have taken anything.

Indicatives and subjunctives also differ in morphological marking. The an-tecedent of (32) contains a layer of past perfective morphology lacking in (31). Iwill assume, following, e.g., Iatridou (2000), Schulz (2007), Starr (2014), a.o.,(though cf., e.g., Ippolito (2003), Arregui (2007, 2009) a.o.) that, in virtue of itsenvironment, this does not function to encode its normal past-tense meaning.Rather, in the antecedent of conditionals, it has a dedicated meaning whichaccounts for their contrast with indicative.

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§4.3 defended an analysis on which the conditional form is univocal. This hasthe virtue of parsimony and explains the absence of, e.g., a subjunctive readingof (31) or an indicative reading of (32) (Starr (2014, fn2)). However, absentsupplementation, the uniform analysis leaves the differences between (31)-(32)unexplained. In this section, I show how the indicative/subjunctive distinctioncan be generated out of a difference in presupposition. Stated simply, indicativespresuppose the possibility of their antecedent; subjunctives presuppose thepossibility of its negation.

5.1 Indicatives

On the basis of (21.a) (and following von Fintel (1997) and Gillies (2007)(a.o.)),suppose that indicative conditionals presuppose the possibility of their antecedent.

Definition 9. [99K]

σ[φ 99K ψ] =

{σ[Sup(φ)→ ψ], if σ 3φ

undefined, otherwise.

σ[φ 99K ψ] is defined only if σ is compatible with the information conveyed byφ. In this case, it applies the test induced by Sup(φ) → ψ. It follows fromconservativity that, for unnested conditionals, φ 99K ψ is Strawson equivalent toφ→ ψ .9

(Eqv) φ 99K ψ φ→ ψ ifφ ∈L0 Equivalence

Indeed, we can say something (slightly) stronger. As long as ψ belongs tothe (consequent) closure of L0 under 99K,10 then if φ 99K ψ is defined on σ,σ[φ 99K ψ] = σ[φ→ ψ]. That is, if nested subjunctives are ignored, indicative andstrict conditionals coincide. This is a comforting result, if one is sympathetic tothe thought that logic generated by the dynamic strict conditional is appropriatefor indicatives. It follows directly that (Pres), (DA), and (CT) are all Strawsonvalid for 99K over the restricted language, as are their nested variants.

5.2 Subjunctives

On the basis of (33.b), suppose that, whereas indicatives presuppose the possi-bility of their antecedent, subjunctives presuppose the possibility of its negation.

Definition 10. [;]

9By conservativity, wherever σ 3φ, cσ[Sup(φ)] = cσ[φ]. That is, if σ supports 3φ, thenf σ(cσ , |φ|) = cσ |φ|. Hence, if φ 99K ψ is defined on σ, and ψ ∈L0, cσ[Sup(φ)][ψ] = cσ[φ][ψ]. So,for all σ on which φ 99K ψ is defined, σ[φ 99K ψ] = σ[φ→ ψ].

10As defined in Appendix A.

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σ[φ; ψ] =

{σ[Sup(φ)→ ψ], if σ 3¬φundefined, otherwise.

σ[φ ; ψ] is defined only if σ is compatible with the information conveyed by¬φ. In this case, it applies the test induced by Sup(φ) → ψ. The suggestedpresupposition is the same as that proposed by Kartunnen and Peters (1979),and accounts for the acceptability of counterfactual uses of subjunctives (e.g.,((34.b)).

Unlike indicatives, subjunctives do not undergo presupposition failure in contextswhich support the negation of their antecedent. In such contexts, suppositionof the antecedent returns the minimal-revision of the input information statewhich supports the antecedent (along with a selection function which preservesthe information conveyed by the antecedent). Clearly, this information state willinclude possibilities which are considered counterfactual at the original discoursecontext.11 The test imposed by the subjunctive passes if this information stateis left unchanged after update with the consequent.

Since their presuppositions are compatible, subjunctives and indicatives arenot predicted to be in complementary distribution (see von Fintel (1997) forarguments that this is a desirable result). This allows us to account for theobservation that subjunctives appear to permit non-counterfactual uses; theyare felicitous in discourse contexts which do not entail the negation of theantecedent.

(35) Maybe thieves broke into my apartment. If they’d broken in, theywould’ve taken something.

(35) is predicted to be acceptable as long as the context is unopinionated with

respect the subjunctive’s antecedent (i.e., σ 6 φ, and 6 ¬φ}, ). Note however,that the non-counterfactual use of subjunctives is limited. The account correctlypredicts that they are infelicitous in discourse contexts such as that in (33),which entail their antecedent.

Interestingly, there is reason to think that in discourse contexts unopinionatedabout their antecedents, the indicative/subjunctive distinction collapses. First,note that (36), the licensing conditions of which are satisfied only in unopinionatedcontexts, is marked:

(36) ??If the butler did it, he used a candlestick, but if the butler had done it,he might not have used a candlestick.

That is, the sequence φ 99K ψ;φ; 3¬ψ is inconsistent.12 Second, any minimally

11Indeed, where the antecedent is non-modal, it will be disjoint from the input informationstate

12φi, . . . , φj are inconsistent iff φi, . . . , φj ⊥.

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adequate theory of conditionals should make disjunctions such as that in (37)valid:

(37) Either if the butler had done it, he might have used a candlestick or ifthe butler had done it, he wouldn’t have used a candlestick.

That is, every context should support one (and only one) out of φ ; 3ψand φ ; ¬ψ. Finally, as (38) suggests, subjunctives imply indicatives inunopinionated contexts.

(38) If the butler had done it, he would’ve used a candlestick. (So) if thebutler did it, he used a candlestick.

That is, if a context supports φ ; ψ and 3ψ, then it supports φ 99K ψ. Yet,combined, these three principles imply that, in unopinionated contexts, anindicative is supported iff the corresponding subjunctive is. In unopinionatedcontexts, if φ 99K ψ is supported, then, by the first observation, φ; 3¬ψ is not.Accordingly, by the second observation if φ ; 3¬ψ is not supported, φ ; ψmust be. Hence, in unopinionated contexts, φ ; ψ is supported if φ 99K ψ is.Furthermore, given the third observation, φ 99K ψ is supported if φ; ψ is, inunopinionated contexts. Putting things together, it follows that indicatives andsubjunctives are Strawson equivalent. Where both are defined, each entails theother.

(Cll) φ; ψ φ 99K ψ Collapse

The present framework validates collapse. Any context which is unopinionatedabout φ supports 3φ. Yet, at any context which supports 3φ, both φ; ψ andφ 99K ψ collapse into the strict conditional. (36)-(38) suggest that this is correctprediction. Crucially, however, while indicatives and subjunctives are Strawsonequivalent, they differ in meaning. This difference in meaning (at the level ofpresupposition) allows the two forms of conditional to have distinct logics. SinceStrawson equivalence is non-transitive, we can accept collapse while maintainingthat indicatives and subjunctives validate notably different inference patterns.

5.3 Collapse

If we restrict our attention to core set of contexts licensing both indicatives andsubjunctives, collapse has prima facie appeal. In contexts which are unopinion-ated about both their antecedents and consequents, indicatives and subjunctivesappear equivalent. For example, suppose that Andi is playing Roulette. Wedon’t know how Andi will bet (or whether he’ll win), but we do know that theball will land on red (the game’s been rigged). Under these conditions, (39)-(40)are seemingly interchangeable:

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(39) If Andi bets on red, he’ll win.

(40) If Andi were to be on red, he’d win.

This phenomenon is not restricted to future-orientated conditionals. Supposethat the roulette wheel is spun and the ball indeed lands on red. Assuming weremain ignorant about how Andi bet and whether he won, (41)-(42) would beequally good ways of summarizing the information available to us:

(41) If Andi bet on red, he won.

(42) If Andi had bet on red, he’d have won.

Things are more complicated in contexts which are opinionated about the relevantconditional consequents. Consider the Adams pairs formed by (43.a-b) and (43.a-b) (in a context in which it is established that Hamlet had some author orother):

(43) a. If Shakespeare didn’t write Hamlet, someone did.

b. If Shakespeare hadn’t written Hamlet, someone would have.

(44) a. If Shakespeare didn’t write Hamlet, then no-one did.

b. If Shakespeare hadn’t written Hamlet, then no-one would have.

Pairs of conditionals like these present an apparent problem for collapse. Aspeaker who is willing to assent to (43.a) need not be willing to assent to (43.b).Similarly, a speaker willing to assent to (44.b) need not be willing to assent to(44.a). Yet, by collapse, (43.a-b) are Strawson equivalent, and, likewise, (44.a-b).

However, we should not be over-hasty in closing the case against collapse. Indeed,given the subjunctive/indicative distinction proposed above, there is reason tothink that our judgements about Adams pairs might be capable of being explainedby appeal to a couple of simple pragmatic principles (this can be seen as animplementation of an idea originating in Kartunnen and Peters (1979)).

Consider (44.a-b) first. Note, crucially, that the consequent (i.e., that no-one

wrote Hamlet) is incompatible with the context. But, where σ ¬ψ, it follows

that σ Sup(φ) → ψ only if σ ¬φ. Hence, neither conditional will becompatible with the context unless its antecedent is incompatible with it.

Since (44.a) is indicative, it is defined only at contexts compatible with itsantecedent. Hence, in contexts incorporating the information that someonewrote Hamlet, it is predicted to be unsupported, if defined. In contrast, (44.b)is capable of being both defined and supported in such a context, as long as thecontext also incorporates the negation of its antecedent.

Plausibly, there is a conversational preference for asserting sentences compatiblewith the context. If what is established at a context would entail the denial of an

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utterance, but there is some way of adding information to the context to makeit compatible, we can expect the utterance to convey that the speaker intendsthe context to incorporate that information (presupposition accommodation canbe treated as an analogue of this phenomenon). This idea can be implementedwithin the present dynamic framework.

Suppose that (i) ψi, . . . ψj ¬φ, but (ii) there is some ψk ∈L0 such that

ψi, . . . ψj , ψk 6 ¬φ (and 6 φ). Then for all σ, an utterance of φ at σ[ψi], . . . [ψj ]will convey that the speaker takes the context to incorporate the informationconveyed by (one of) the weakest such φk(s). In this case, an utterance of (44.b)will be predicted to convey that the speaker takes the context to incorporate theinformation that Shakespeare wrote Hamlet. However, in any such context, (44.a)would be undefined. Accordingly, assent to (44.b) will carry no commitment toassent to (44.a). While the two are Strawson co-entailing, they convey differentinformation pragmatically, as a result of their differing presuppositions.

This explanation appears to be vindicated by the licensing of ‘Hey, wait aminute!/Hold up’-responses. It would be reasonable for a Marlovian, skepticalabout the play’s provenance, to respond to an utterance of (44.b) by saying‘Hey, wait a minute, I haven’t granted you that Shakespeare did write Hamlet ’.Given that such responses are, generally, a indicator that an utterance requiresaccommodation of the contested information (von Fintel (2004)), this is accordswith the present picture.

Consider, next, (43.a-b). In contrast to the former case, their consequent (i.e.,someone wrote Hamlet) is supported at the context. Hence, each conditionalwill be trivially supported as long as the context is compatible its antecedent.Where σ ψ, σ Sup(φ)→ ψ if σ 6 ¬φ.13

Plausibly, there is also a conversational preference for asserting non-trivialities.If an utterance would be trivial given what is established at a context, but thereis some way of adding information to a context to make it non-trivial (whileremaining compatible), we can expect the utterance to convey that that thespeaker intends the context to incorporate that information.

Suppose that (i) ψi, . . . ψj φ, but (ii) there is some ψk ∈L0 such that

ψi, . . . ψj , ψk 6 φ (and 6 ¬φ).14 Then for all σ, an utterance of φ at σ[ψi], . . . [ψj ]will convey that the speaker takes the context to incorporate the informationconveyed by (one of the) the weakest such φk(s). That is, if, (i) given what isestablished in the context, φ is trivially supported, but (ii) there is some way ofeliminatively updating the context which would lead φ to cease to be triviallysupported, an utterance of φ in the context will convey that the speaker takesthe present discourse to be strengthened in this way.

Since (43.a) is indicative, it is defined only at contexts compatible with itsantecedent; hence, where it is established that someone wrote Hamlet, it will

13Assuming that σ[Sup(φ)] is defined.14And for all σ s.t. σ[ψi], . . . , [ψj ][φ] is defined, σ[ψi], . . . , [ψj ][ψk][φ] is defined.

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be supported if defined. Furthermore, there is no way of strengthening such acontext which will make it non-trivially supported. Nevertheless, assent to aconditional such as (43.a) can serve a communicative purpose, since it conveysthat the speaker does not take it to be established that Shakespeare did notwrite Hamlet.

In contrast, (43.b) is capable of being defined in contexts incompatible withits antecedent. Furthermore, even if such a context supports its consequent,the subjunctive will not be trivially supported (since the minimal revision tothat context which supports its antecedent need not do so). The weakestupdate to a context which will result in such a context is an update with thenegation of the subjunctive’s antecedent (Since ψ,3φ (Sup(φ) → ψ), but

ψ,3φ,¬φ 6 (Sup(φ) → ψ) and 6 ¬(Sup(φ) → ψ) . Hence, like (44.b), anutterance of (43.b) can be expected to convey that the speaker takes it to beestablished that Shakespeare wrote Hamlet. Yet, in such a context, (44.b) willbe unsupported; if it is established that Shakespeare wrote Hamlet, the minimalrevision to that context in which he didn’t is presumably one in which Hamletwent unwritten.15

5.4 Supposition and Mood

An appealingly simple hypothesis is that the morphological marking found inindicative/subjunctive-antecedents has precisely the same semantic contributionin the complement clause of imperatives headed by ‘suppose’. This wouldsuccinctly explain the contrast between (45.a-b). The presuppositions of (45.b),but not (45.a), are predicted to be compatible with counterfactual use:

(45) a. The butler didn’t do it. ??Suppose he did. Then there’d be bloodon the candlestick.

b. The butler didn’t do it. Suppose he had. Then there’d be blood onthe candlestick.

The simple hypothesis predicts that the effect of supposition on ‘downstream’conditionals and additional suppositions is independent of its morphologicalmarking. This appears to be borne out, as (46)-(47) demonstrate:

(46) a. Suppose the Mets outscored the Cubs.

b. . . .Then, if the Cubs had scored 12, the Mets would have scored 13.

15The same pragmatic mechanism will explain the fact that, in unopinionated context, anagent can utter (‡.b), but not (‡.a), in support of the claim that that Jones took arsenic(Anderson (1951),von Fintel (1997),Stalnaker (1975)).

(‡) a. ??If Jones took arsenic, he’s showing exactly the symptoms he’s in fact showing.

b. If Jones had taken arsenic, he’d be showing exactly the symptoms he’s in factshowing.

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(47) a. The Mets outscored the Cubs.

b. . . . So, if the Cubs had scored 12, the Mets would have scored 13.

In the discourse context generated by (46.a), (46.b) is judged true. Despitelacking an additional layer of past-tensed marking, the information conveyedby the complement clause of the former appears required to be preserved whenevaluating the latter. In contrast, the same subjunctive can naturally be judgedfalse if it occurs in a discourse context following (47.a) instead.

The explanatory power of the simple hypothesis, along with its relative elegance,give us substantial reason to accept it. However, if we are to do so, we will requiresome explanation of why the inferences in (9)-(11) appear easier to reject whenthe supposition is stripped of past-tense morphology. If additional past-tensemorphology merely has an effect on presuppositions, we would expect (48)-(50)and (9)-(11) to be equally good.

(48) Suppose Ada is drinking red wine. (Then) if she were eating fish, she’dbe eating fish and drinking red wine.

(49) Suppose Claude is in London or Paris. (Then) if he weren’t in London,he’d be in Paris.

(50) If Lori were married to Kyle, then if she were married to Lyle, she’d bea bigamist. Suppose she’s married to Lyle. (Then) if she were marriedto Kyle, she’d be a bigamist.

To account for this contrast, I propose, we need to recognize the additional effectthat morphological marking can have on discourse structure. A discourse is notmere collection of utterances. Understanding a discourse requires understandingthe relations between distinct utterances. Grammatical mood can play a role inguiding this process. In particular, the shift between ‘indicative’/‘subjunctive’morphology can often indicate that two utterances are being presented ascontrasting.

(51) If Bob comes to the party, we’ll drink wine.a. ...If Mary were to come, we’d do shots.b. ...If Mary comes, we’ll do shots.

Whereas, in its discourse context, (51.a) is most naturally heard as introducing anincompatible alternative to the possibility Bob attending and us all drinking wine,this reading is notably less prominent for (51.b). The most natural interpretationof the latter (but not of the former) implies that, if both Mary and Bob come,we’ll drink wine and do shots.

If, as suggested in §4.2, contrast can trigger withdrawal of suppositions, thiswould provide an explanation of why the inferences in (48)-(50) are degraded.Withdrawing the downstream effect of supposition prior to evaluating the final

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subjunctive will result in the inference no longer being valid. Clearly, muchmore needs to be done to investigate the relation between supposition, moodand discourse structure. The brief discussion in this section has aimed merelyto demonstrate an approach which can allow us to preserve a simple, univocalaccount of both supposition and ‘indicative’/‘subjunctive’ morphology.

6 Accounting for the Data

The analysis in Definition 10 correctly predicts the invalidity of (Pres), (DA)and (CT) for subjunctives (and the corresponding validity for indicatives, via(Eqv)). Yet it also predicts that each inference pattern can be made valid byembedding the rightmost premise under Sup(·). That is, (PresSup), (DASup)and (CTSup) are each valid.

(PresSup) Sup(φ) Sup(ψ)→ (ψ ∧ φ)

(DASup) Sup(φ ∨ ψ) Sup(¬φ)→ ψ

(CTSup) Sup(φ)→ (Sup(ψ)→ χ), Sup(ψ) Sup(φ)→ χ

Stated in schematic terms, the validity of the inferences is attributable to theeffect of supposition on the output selection function. When the rightmostpremise is embedded under supposition, any counterfactual revisions induced byconditional in the conclusion must preserve the information supposed. This issufficient to explain the contrast between (6)-(8) and (9)-(11).

As an example, consider the proof of (PresSup). We need to show that Sup(ψ)→(ψ ∧ φ) will be supported by any state updated with Sup(φ). Take an arbitrarypoint of evaluation, σ. Update with Sup(φ) will return a φ-supporting informationstate, and a selection function, f +φ

σ , which preserves the information conveyedby φ. The conditional, Sup(ψ) → (ψ ∧ φ), will be supported at that point iff,after update with its antecedent, the consequent leaves the information stateunchanged. Update with Sup(ψ) will return an information state which supportsboth φ (in virtue of the revised input selection function) and ψ (in virtue of theeffect of supposition). Clearly, any such information state will be left unchangedby update with ψ ∧ φ. Hence, the inference is valid.16

Crucially, the deduction theorem is valid for →; that is, φ ψ iff φ → ψ.Since (PresSup), (DASup) and (CTSup) are valid, the framework therefore alsovalidates each of (Pres⇒), (DA⇒) and (CT⇒) for ;. Hence, it simultaneouslyexplains the contrast between (6)-(8) and (12)-(14).

Second, note that where |φ| is incoherent17, update with Sup(φ) will return an

16Note that, where ψ ∧ π is non-idempotent, the entire conditional will be undefined atσ[Sup(φ)]

17φ is incoherent iff there is no c s.t. c|φ| = c

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absurd state (if defined). 18 Epistemic contradictions are incoherent in preciselythis sense. Thus, when embedded under supposition they will return absurdity,and when embedded in the antecedent of a conditional, they will require theconsequent to be evaluated at an absurd state. Assuming, in both cases, thisyields infelicity, we account for the markedness of (17)-(19).

Finally, we can account for observations regarding counterfactual use in §2.3.The only fixed point of sequential update with |φ|�|¬φ| is ∅.

(52) Sup(φ); (Sup(¬φ)→ ψ)

(53) Sup(φ)→ (Sup(¬φ)→ ψ)

Yet after update with Sup(φ), update with Sup(ψ) will return an informationstate which is a fixed point of |φ|; |¬φ|. Hence, both (52)-(53) (the translations of(23)-(24) respectively) require evaluation of their mostly deeply nested consequentat an absurd state.

7 Conclusion

Suppositional theories of conditionals have some form of pre-theoretical plau-sibility. For example, they vindicate the intuition behind the Ramsey test,that:

“If two people are arguing ‘If p, then q?’ [. . . ], they are adding phypothetically to their stock of knowledge and arguing on that basisabout q”(Ramsey (1931, 247)).

However, extant theories have also tended to be lacking, both in empirical supportand in formal specification of the alleged connection between instructions tosuppose and conditionals.

The present paper has considered a range of data satisfying the former desider-atum. It has then shown how, by developing a theory satisfying the latterdesideratum, this data can be explained. On the resulting account, naturallanguage conditionals decompose into a strict conditional and an embeddedinstruction to suppose. The effect of the latter is restricted to the conditional’sconsequent, rather than changing the discourse context (as unembedded sup-positions do). Thus, in a slogan: conditional antecedents are sentence levelsuppositions; supposition is a discourse level conditional antecedent.

18Update with Sup(φ) will be undefined at a proper σ in those cases where σ[(φ)] isnon-absurd –since in this case success and minimality cannot be jointly satisfied.

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A Appendix

Where LA={A,A′, . . . } is a set of sentential constants, we define the language Las follows:

Definition 11. L=L0∪ L1∪ L2, where:

L0= � if φ ∈LA, then φ ∈L0;

� if φ ∈L0, then ¬φ,2φ ∈L0;

� if φ, ψ ∈L0, then φ ∧ ψ, φ ∨ ψ ∈L0;

� nothing else is a member of L0.

L1= � if φ ∈L0, then Sup(φ) ∈L1;


L2= � if φ ∈L0∪L1 and ψ ∈L0∪L2, then φ→ ψ ∈L2;


An update semantic model is a tuple M=〈WM , | · |M 〉. A world w,w′, · · · ∈WM

is a function {0,1}LA . An information state c, c′, · · · ∈ P(WM ) is a set of worlds.With indexation to a model elided for legibility, |·| is a US-interpretation functiondefined on L0, s.t.:

Definition 12.

i. c|A| = {w ∈ c|w(A) = 1};

ii. c|¬φ| = c \ c|φ|;

iii. c|φ ∧ ψ| = c|φ||ψ|;

iv. c|φ ∨ ψ| = c|φ| ∪ c|ψ|;

v. c|2φ| =

{c, if c|φ| = c;

∅, otherwise..

A US+-selection function f ∈ FM = (P(WM ) × P(WM )P(WM )) → P(WM ) isa function from an information state and a function from information statesto information states, to an information state. + is a function from L0 to afunction from US+-selection functions to US+-selection functions, subject to thefollowing constraint:

Definition 13. f+φ(c, |ψ|) = f(c, |φ|�|ψ|)

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An enriched context σ, σ′, · · · ∈ P(WM )×FM is a pair 〈cσ, fσ〉 of an informationstate and a selection function. σ is proper (adequate) if f σ is proper (adequate).[·]M is an enriched interpretation function induced by | · |M (as above, indexationto a model is elided).

Definition 14.

i. σ φ iff cσ = cσ[φ].

ii. ψi, . . . ψj φ iff, for all σ, σ[ψi], . . . [ψj ] φ (if σ[ψi], . . . [ψj ][φ] isdefined).

Definition 15.

For all φ ∈L0:

i. σ[φ] = 〈cσ|φ|, fσ〉.ii. σ[Sup(φ)] = 〈fσ〈cσ, |φ|〉, f+φσ 〉.

For all φ ∈L0∪L1, ψ ∈L0∪L2:

iii. σ[φ→ ψ] =

{σ, if σ[φ] ψ


First, note that I/E is valid for Sup(φ)→ ψ:

Observation 1. Let σ�f=〈cσ, f〉. Where φ ∈L0, (σ[φ])�f = (σ�f )[φ]. Hence,

σ iff σ�f φ.

Observation 2. fσ〈cσ, |φ|�|φ|〉 = fσ〈fσ〈cσ, |φ|〉|φ|〉 = fσ〈cσ, |φ|〉, for all properσ.

Fact 1. Sup(φ ∧ ψ)→ χ Sup(φ)→ (Sup(ψ)→ χ)

Proof. Observe that Sup(φ ∧ ψ) → χ Sup(φ) → (Sup(ψ) → χ) iffor all proper σ: σ[Sup(φ ∧ ψ)] = σ[Sup(φ)][Sup(ψ)]. By Definition 15,σ[Sup(φ ∧ ψ)] = 〈fσ〈cσ, |φ|�|ψ|〉, f+φ∧ψσ 〉. In comparison, σ[Sup(φ)][Sup(ψ)] =〈f+φσ 〈fσ〈cσ, |φ|〉, |ψ|〉, (f+φσ )+ψ〉. First, note that, by simplification, f+φ∧ψσ =(f+φσ )+ψ. Next, note that f+φσ 〈fσ〈cσ, |φ|〉, |ψ|〉 = fσ〈, fσ〈cσ, |φ|〉, |φ|�|ψ|〉. Yet,by simplification, fσ〈fσ〈cσ, |φ|〉, |φ|�|ψ|〉 = fσ〈cσ, |φ|�(|φ|�|ψ|)〉, if there is somec′ s.t. c′|φ||ψ| 6= ∅. If there is no such c′, both σ[Sup(φ ∧ ψ) → χ] andσ[Sup(φ)→ (Sup(ψ)→ χ)] are trivially true. So suppose there is. It follows fromObservation 2, by simplification, that f+φσ 〈fσ〈cσ, |φ|〉|ψ|〉 = fσ〈cσ, |φ|�|ψ|〉.Hence, since fσ〈cσ, |φ|�|ψ|〉 = fσ〈cσ, |φ ∧ ψ|〉, it follows that σ[Sup(φ ∧ ψ)] =σ[Sup(φ)][Sup(ψ)].

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Next, note that a restricted form of MP is also valid for Sup(φ)→ ψ:

Lemma 1. For all φ ∈L0∪L2, if for all χ ∈L0, f〈c, |χ|〉 = f ′〈c, |χ|〉, then

〈c, f〉 φ iff 〈c, f ′〉 φ.

Proof. Suppose that for all χ ∈L0, f〈c, |χ|〉 = f ′〈c, |χ|〉. We need to prove, by

induction, that 〈c, f〉 φ iff 〈c, f ′〉 φ.

Base case: let φ ∈L0. It follows directly from Observation 1 that for all f, f ′,〈c, f〉 φ iff 〈c, f ′〉 φ.

Inductive hypothesis: Let φ ∈ L0∪L2. If for all χ ∈L0, f〈c, |χ|〉 = f ′〈c, |χ|〉, then

〈c, f〉 φ iff 〈c, f ′〉 φ.

Inductive Step: Suppose that, for some φ ∈ L0∪L2, if f〈c, |χ|〉 = f ′〈c, |χ|〉for all χ ∈L0, then 〈c, f〉 φ iff 〈c, f ′〉 φ. Pick some arbitrary f, f ′ suchthat f〈c, |χ|〉 = f ′〈c, |χ|〉 for all χ ∈L0. We will prove that, for any ψ ∈L0,

(i) 〈c, f〉 ψ → φ iff 〈c, f ′〉 ψ → φ and (ii) 〈c, f〉 Sup(ψ) → φ iff

〈c, f ′〉 Sup(ψ)→ φ.

Starting with (i): note that, since ψ ∈ L0, 〈c, f〉[ψ] = 〈c|ψ|, f〉 and 〈c, f ′〉[ψ] =

〈c|ψ|, f ′〉. But, by hypothesis, 〈c|ψ|, f〉 φ iff 〈c|ψ|, f ′〉 φ. So 〈c, f〉 ψ → φ

iff 〈c, f ′〉 ψ → φ.

Moving on to (ii): first, note that 〈c, f〉 Sup(ψ)→ φ iff 〈f〈c, |ψ|〉, f+ψ〉 φ.We know, by hypothesis, that f〈c, |ψ|〉 = f ′〈c, |ψ|〉. So we need to prove that

〈f〈c, |ψ|〉, f+ψ〉 φ iff 〈f〈c, |ψ|〉, f ′+ψ〉 φ, First, note that, for all χ ∈L0,|ψ|�|χ| = |ψ ∧ χ|. Thus, for any χ ∈L0, and f ∗: f∗+ψ〈c, |χ|〉 = f∗〈c, |ψ ∧ χ|〉.But we know that f〈c, |ψ ∧ χ|〉 = f ′〈c, |ψ ∧ χ|〉. So, for all χ ∈L0, f+ψ〈c, |χ|〉 =

f ′+ψ〈c, |χ|〉. So, by the inductive hypothesis, we know that 〈f〈c, |ψ|〉, f+ψ〉 φ

iff 〈f〈c, |ψ|〉, f ′+ψ〉 φ. So 〈c, f〉 Sup(ψ)→ φ iff 〈c, f ′〉 Sup(ψ)→ φ.

By induction on the complexity of φ, it follows that for all φ ∈L0∪L2, if for allχ ∈L0, f〈c, |χ|〉 = f ′〈c, |χ|〉, then 〈c, f〉 φ iff 〈c, f ′〉 φ.

Fact 2.

i. Sup(φ)→ ψ, φ ψ, if ψ ∈ L0

ii. Sup(φ) → ψ, φ ψ, if ψ = (Sup(ψi) → ψj) ∈ L2 and φ, ψi are consis-tent.

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Proof. Let σ be an arbitrary proper point of evaluation. If either σ[Sup(φ)→ψ] = ∅ or σ[φ] = ∅, then trivially σ[Sup(φ)→ ψ][φ] ψ. Thus we need to show

that, if σ[Sup(φ)→ ψ] = σ and σ[φ] 6= ∅, then σ[Sup(φ)→ ψ][φ] ψ.

Since σ[φ] 6= ∅, it follows from success that σ[Sup(φ)] = 〈cσ|φ|, f+φσ 〉 = 〈cσ[φ], f+φσ 〉.Since σ[Sup(φ)→ ψ] = σ, we know that 〈cσ[φ], f+φσ 〉 ψ. So we need to show

that if 〈cσ[φ], f+φσ 〉 ψ, then σ[φ] = 〈cσ[φ], fσ〉 ψ.

Where φ ∈L0, this follows directly from Observation 1. So, (i.) follows immedi-ately. So, suppose instead that ψ = (Sup(ψi)→ ψj) ∈L2.

Note that by simplification, for any χ ∈L0, s.t. φ, χ are consistent, f+φσ 〈cσ[φ], |χ|〉 =fσ〈cσ[φ], |φ|�|χ|〉 = fσ〈fσ〈cσ[φ], |φ|〉, |χ|〉. But, by success, fσ〈cσ[φ], |φ|〉,= cσ[φ].

So, for any such χ ∈L0, f+φσ 〈cσ[φ], |χ|〉=fσ〈cσ[φ], |χ|〉. Thus, it follows from

Lemma 1 that, where ψi, φ are consistent: 〈cσ[φ], f+φσ 〉 (Sup(ψi) → ψj)

iff σ[φ] = 〈cσ[φ], fσ〉 (Sup(ψi) → ψj). So, for any proper σ, σ[Sup(φ) →((Sup(ψi)→ ψj))][φ] (Sup(ψi)→ ψj).

To model the indicative and subjunctive conditional operators introduced in§5, let us introduce a new unary operator, δ(·). δ(·) introduces definednessconditions to our language (following Beaver and Krahmer (2001)). Where:

φ ∈L0, σ[δ(φ)] =

{σ, if σ φ;

undefined, otherwise.

Then [φ 99K ψ] =def [δ(3φ)]�[Sup(φ)→ ψ] and [φ; ψ] =def [δ(3¬φ)]�[Sup(φ)→ψ].

Definition 16. Let L∗ be the consequent closure of L0 under 99K, defined asfollows:

� if φ ∈L0, then φ ∈L∗;

� if φ ∈L0 and ψ ∈L∗, then φ 99K ψ ∈L∗;

� nothing else is a member of L∗

L∗ is the extension L0 with 99K which allows for arbitrarily many embeddedinstances of 99K in its consequent.

Fact 3. Where ψ ∈L∗, σ[φ→ ψ] = σ[φ 99K ψ] (where defined).

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Proof. First, note that φ→ ψ and φ 99K ψ are both tests. Thus, where defined,σ φ 99K ψ iff σ[φ 99K ψ] = σ; otherwise σ[φ 99K ψ] = ∅ (and similarly for→). It follows that σ[φ→ ψ] = σ[φ 99K ψ](where defined) iff for all σ on which

φ 99K ψ is defined, σ φ→ ψ iff σ φ 99K ψ.That is, to prove that their denotations coincide where defined, it suffices todemonstrate that they are (Strawson) co-entailing.

Suppose φ, ψ ∈L0. Let σ be an arbitrary proper point of evaluation s.t.σ[φ 99K ψ] is defined; that is, σ 3φ. It follows from conservativity that

cσ[Sup(φ)][ψ] = cσ[φ][ψ]. Thus, σ φ→ ψ iff σ φ 99K ψ. Next, note that since

φ, ψ ∈L0, we know that φ→ ψ 2(¬(φ∧¬ψ)). Hence, by Observation 1, it fol-

lows for all f, that σ φ→ ψ iff σ�f φ→ ψ. So, σ φ→ ψ iff σ�f φ 99K ψ.

Suppose, for induction, that where φ ∈L0, ψ ∈L∗, for all f : σ φ → ψ

iff σ�f φ 99K ψ. Let χ ∈L0. We will show that σ χ → (φ 99K ψ) iff

σ χ 99K (φ 99K ψ).

Let σ be an arbitrary σ s.t. σ[χ 99K (φ 99K ψ)] is defined. Thus, σ 3φ. It

follows from conservativity that cσ[χ] = cσ[Sup(χ)]. So σ[Sup(χ)] = (σ[χ])�f+φ

.

Since, for all f, σ φ → ψ iff σ�f φ → ψ it follows that σ[χ] φ → ψ

iff (σ[χ])�f+φ

φ 99K ψ. But trivially, σ[χ] φ → ψ iff σ[χ] φ 99K ψ.

Thus σ[χ] φ 99K ψ iff σ[Sup(χ)] φ 99K ψ. So σ χ → (φ 99K ψ) iff

σ χ 99K (φ 99K ψ).

Finally, we will prove the claims in §6:

Observation 3. Where φ ∈L0: σ φ iff cσ = cσ|φ|.

Lemma 2. For all φ, ψ ∈L0: c|φ||ψ| = c iff c|ψ||φ| = c.

Proof. Suppose c|φ||ψ| = c. Then, since every φ, ψ ∈L0 is eliminative,19 c|φ|=cand c|ψ| = c. So c|ψ||φ| = c.

Lemma 3. For all φ, ψ ∈L0: if c|φ| = c iff c|ψ| = c, then f〈c, |φ|〉 = f〈c, |ψ|〉.

Proof. Observe that if c|φ| = c iff c|ψ| = c, then |2φ| = |2ψ|. So f〈c, |2φ|〉 =f〈c, |2ψ|〉. But, by transparency, f〈c, |φ|〉 = f〈c, |2φ|〉 and f〈c, |ψ|〉 = f〈c, |2ψ|〉.Thus f〈c, |φ|〉 = f〈c, |ψ|〉.

Fact 4.

19φ is eliminative iff for all c|φ| ⊆c

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i. Sup(φ) Sup(ψ)→ (φ ∧ ψ)

ii. Sup(φ ∨ ψ) (¬φ)→ ψ

iii. Sup(φ)→ (Sup(ψ)→ χ), Sup(ψ) Sup(φ)→ χ

Proof. i. Sup(φ) Sup(ψ)→ (ψ ∧ φ) iff for all proper σ, σ[Sup(φ)][Sup(ψ)]φ ∧ ψ. First, note that since Sup(φ), Sup(ψ) ∈L iff φ, ψ ∈L0, we know thatφ, ψ ∈L0. Next, note that cσ[Sup(φ)][Sup(ψ)] = f+φσ 〈cσ[Sup(φ)], |ψ|〉 = fσ〈cσ[Sup(φ)], |φ|�|ψ|〉.So, by success, cσ[Sup(φ)][Sup(ψ)]|φ ∧ ψ| = cσ[Sup(φ)][Sup(ψ)]. So, by Observa-

tion 3, σ[Sup(φ)] ψ → (ψ ∧ φ). Since σ was arbitrary it follows that

Sup(φ) ψ ⇒ (φ ∧ ψ).

Proof. ii. Sup(φ∨ψ) Sup(¬φ)→ ψ iff for all proper σ, σ[Sup(φ∨ψ)][Sup(¬φ)]

ψ. Next, note that cσ[Sup(φ∨ψ)][Sup(¬φ)] = f+(φ∨ψ)σ 〈cσ[Sup(φ∨ψ)], |¬φ|〉 = fσ〈cσ[Sup(φ∨ψ)], |φ∨

ψ|�|¬φ|〉. By success, it follows that cσ[Sup(φ∨ψ)][Sup(¬φ)]|(φ∨ψ)|�|¬φ| = cσ[Sup(φ∨ψ)][Sup(¬φ)].From update semantics we know that c is a non-empty fixed point of |(φ∨ψ)|�|¬φ|only if c is a fixed point of |ψ|. But we know that ψ ∈L0. So, by Observa-

tion 3, σ[Sup(φ ∨ ψ)] Sup(¬φ) → ψ. Since σ was arbitrary, it follows that

Sup(φ ∨ ψ) Sup(¬φ)→ ψ

Proof. iii. Sup(φ)→ (Sup(ψ)→ χ), Sup(ψ) Sup(φ)→ χ iff for all proper σ,

if σ Sup(φ)→ (Sup(ψ)→ χ), then σ[Sup(ψ)][Sup(φ)] χ. For an arbitrary

proper σ, suppose σ Sup(φ)→ (Sup(ψ)→ χ). Then σ[Sup(φ)][Sup(ψ)] χ.So, it suffices to demonstrate that σ[Sup(φ)][Sup(ψ)] = σ[Sup(ψ)][Sup(φ)].

We know that σ[Sup(φ)][Sup(ψ)]=〈f+φσ 〈fσ〈cσ, |φ|〉, |ψ|〉, (f+φσ )+ψ〉 and thatσ[Sup(ψ)][Sup(φ)] = 〈f+ψσ 〈fσ〈cσ, |ψ|〉, |φ|〉, (f+ψσ )+φ〉.

First, note that, since φ, ψ ∈L0, it follows from Lemma 2, that c|φ||ψ| = c iffc|ψ||φ| = c. Accordingly, by Lemma 3, for all c, fσ〈c, |φ|�|ψ|〉=fσ〈c, |ψ|�|φ|〉.Thus (f+φσ )+ψ = (f+ψσ )+φ. Furthermore, note that, by success and sim-plification, f+φσ 〈fσ〈cσ, |φ|〉, |ψ|〉 = fσ〈cσ, |φ|�|ψ|〉 and f+ψσ 〈fσ〈cσ, |ψ|〉, |φ|〉 =fσ〈cσ, |ψ|�|φ|〉. But by Lemmas 2 and 3, fσ〈cσ, |φ|�|ψ|〉 = fσ〈cσ, |ψ|�|φ|〉. Soσ[Sup(φ)][Sup(ψ)] = σ[Sup(ψ)][Sup(φ)]. Since σ was arbitrary, it follows that

Sup(φ)→ (Sup(ψ)→ χ), Sup(ψ) Sup(φ)→ χ.

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