verification, falsification, and the logic of enquiry

P E T E R M I L N E

V E R I F I C A T I O N , F A L S I F I C A T I O N , A N D T H E

L O G I C O F E N Q U I R Y

ABSTRACT. Our starting point is Michael Luntley's falsificationist semantics for the logical connectives and quantifiers: the details of his account are criticised but we provide an alternative falsificationist semantics that yields intuitionist logic, as Luntley surmises such a semantics ought. Next an account of the logical connectives and quantifiers that combines verificationist and falsificationist perspectives is proposed and evaluated. While the logic is again intuitionist there is, somewhat surprisingly, an unavoidable asymmetry between the verification and falsification conditions for negation, the conditional, and the universal quantifier. Lastly we are lead to a novel characterization of realism.

1. L U N T L E Y AND F A L S I F I C A T I O N I S T S E M A N T I C S

In his recent book (Luntley 1988) Michael Luntley develops an experienceable logic based on a falsificationist analysis of the logical connectives. He asserts that this logic is intuitionist but does not prove the assertion. He does however make it plausible by considering which rules of inference an experiential logic might be expected to validate and by showing why the law of excluded middle should fail. In point of fact the propositional logic that emerges is intuitionist on a reasonable falsificationist understanding of the connectives, but the particular falsificationist account Luntley gives comes completely unstuck, as I shall show. The exact nature of Luntley's falsificationist account of the quantifiers is not easy to determine, as we shall see, but again it turns out that an experienceable predicate calculus is intuitionist. It is with the formal details of Luntley's account that I deal in this section, not the philosophical thesis it is intended to support.

To make an assertion is to exclude some possible experience; knowledge of an assertion makes a difference to what one experiences, or, put another way, there can be no knowledge that is compatible with all experience. This is Luntley's manifestation constraint. It applies to knowledge of an assertion, not knowledge of what it means. 1 Thus associated with every meaningful sentence there is a class of falsifying sentences, sentences which report what is excluded by the original sentence. Although the class of its falsifiers does not determine the

Erkenntnis 34: 23-54, 1991. © 1991 Kluwer Academic Publishers. Printed in the Netherlands.

2 4 P E T E R M I L N E

sense of a sentence - conceptually, sense is prior - knowledge of the falsifiers is sufficient for knowledge of its sense. 2 For any decidable sentence A the class fA of sentences that it excludes is {TA} and similarly the class f~A is {A}. Obviously A v B is falsified just in case both A and B are, A ^ B just in case one or other of A and B are. This gives us the conditions: fA vB = fA ("l fB and fA^B = fA U lB. More problematic is the case of the conditional, A ---> B. Here Luntley takes as his model the intuitionist account of the conditional in terms of a construction (or proof) performed on constructions. Luntley's analogue is a falsification concerning falsifications: fA---,B = {fA ~ fB}. This is intended to capture the idea that when one knows that A ~ B one knows that assertion of A leads to assertion of B, hence A ---> B is falsified if it is shown that falsification of B does not lead to falsification of A. Having thus defined the falsification conditions for conditional sentences negation can, again following the example of intuitionist logic, be defined in terms of some experientially determinable contradiction: -~A = A ---> 2 . According to Luntley, 1 represents the exclusion of the very possibility of experience and consequently _1_ is falsified by the assertability of any one decidable sentence. Hence f± = {{p}} where p is an effectively decidable sentence. With these definitions of falsifier classes - that's my name for the f ' s , Luntley does not supply a generic term - Luntley defines entailment and theoremhood in a somewhat convoluted way. (In the course of defining theoremhood Luntley claims that since negation is defined in terms of the conditional all theorems of the logic are of the form A ---> B. If this is true then the rule of /x-introduction is not sound, which Luntley certainly does not intend.) 3

What ought to be immediately obvious is that there is something dreadfully wrong with Luntley's conception of a falsifier class. Where A and B are distinct decidable sentences A v B is unfalsifiable because there is no single sentence common to fA and lB. But of course to falsify A v B one needs to falsify both and there need be no one experience nor any one decidable sentence that accomplishes that. What must go into the falsifier class of a sentence is not just atomic sentences but any sentence that succeeds in falsifying the sentence in question. So fAvB contains neither --hA nor -nB but does contain ~ A / , - lB. In order for the right set-theoretic relations to obtain between falsifier classes fA must contain all sentences that entail ~A. But then the initial enterprise, the definition of the logical constants and entailment in terms of falsifier classes, is rendered circular.

V E R I F I C A T I O N , FALSIFICATION 25

Perhaps I misunderstand Luntley when he says that where A is decidable ' f a comes to simply -hA and fnA c o m e s to A' . No matter. Problems ar ise when we turn our attention to conditionals. Let A, B, C and D be distinct atomic sentences and let E be the sentence (A ~ B) v (C ~ D). Then E is unfalsifiable, for fE = fA---~B n fc--->D = {fA qS fB} O { fc ~S fD} = 0 .4

What has gone wrong? Firstly, Luntley seems to forget that while there may be no logical connections between atomic sentences there may be semantic relations so that in any given case he cannot give a purely formal account of which atomic sentences and/or negations thereof belong to fA. The thought that falsification is not simply a logical mat ter is p rompted by Luntley's regarding a sentence's r o l e -

the inferential relations it enters into - as subordinate to its sense. Secondly, he has failed to take a leaf out of the intuitionist logician's book. The intuitionist logician feels no obligation to tell us which atomic sentences are proved, nor how any that are proved are proved. What the logician looks at is the connexion between proofs of compound sentences and proofs of the sentences out of which they are com- pounded. Similarly then, in developing an experiential propositional logic based on a falsificationist account of connectives we need say nothing exact about what constitutes falsification of an atomic sentence.

Our alternative falsificationist account of the logical constants mirrors the intuitionist's verificationist account. First the easy cases:

A v B is falsified when, and only when, both A and B are falsified; A A B is falsified when, and only when, A or B (or both) is falsified.

The falsification of ± is the precondition for the possibility of coherent experience so:

Z is falsified.

What are we to say of A --+ B? "Assert ion of A leads to assertion of B". Certainly then, A --~ B is falsified if B is falsified and A is not, but we want to say more than that. We want to say something along the lines of: if ever it could be the case that B is falsified when A is not then A --~ B is falsified. That is perhaps an unduly realist and/or modal way of expressing the matter . A better analogue of the intuitionist account would be to say:

26 P E T E R M I L N E

A ~ B is falsified when, and only when, it is shown to be false that any falsification of B leads to, yields, or generates, a falsification of A.

That seems to be pretty much what Luntley wants to say about the conditional in experiential logic. 5 Given the definition of negation and the stipulation regarding _k we see that:

--hA is falsified when, and only when, it is shown to be false that every falsification is a falsification of A.

Since these definitions of the logical constants are "mirror images" of the verificationist definitions of the intuitionist it comes as no surprise that the falsificationist logic of experience is intuitionist, a fact whose proof we now sketch.

Consider X, the lonely knowing subject, and let L be some propositional language. What she knows at any given moment - her "knowledge situation" at that time - permits her to falsify various sentences of L. Thus with each possible knowledge situation t we associate a function ut from the atomic sentences of L to {0, 1} defined by the condition: ut(A) = 0 if, and only if, A is falsified relative to X's knowledge situation at t. X has arrived at her present knowledge situation through acquiring evidence and building on her past knowledge situations. And no doubt she will continue to acquire knowledge. We model X's knowledge situations using elements of a partially ordered set T, the order mimicking temporal order. The use of a partial order is justified on the following anti-realist considerations. Points in the ordering prior to any given point represent knowledge situations that are possible predecessors of that situation, where the notion of possibility used is epistemic. Similarly, points after any given point represent epistemically possible extensions of the given knowledge situation. On an anti-realist perspective the only constraints placed on the past and future are given by X's present knowledge situation. In fact the past and future are only determinate for X to the extent that they are made so by her present knowledge situation. The notion of possibility is therefore grounded in what she currently knows. Let UT= (T, <~, {ut: t ~ T}), ~< being the partial order on T. Since what is once falsified remains so, the ut's satisfy the condition:

(i) for any atomic sentence A, Vs >I t[u,(A) >i ut(A)].

Also:

V E R I F I C A T I O N ~ F A L S I F I C A T I O N 27

(ii) Vt ut(A_) = O.

We define the family of functions {v~: t E T} as follows:

(iii) for atomic A, v~(A) = u~(A); (iv) if A = B v C then v~(A) = max{vt(B), v¢(C)}; (v) if A = B/x C then vt(A) = min{vt(B), vt(C)}; (vi) if A = B ~ C then vt(A) = 0 if 3s <~ t(vs(B) = 1 &

vs(C) = 0), otherwise v~(A) = 1.

The idea behind clause (vi) is that X is in a position at time t to falsify A ~ B if either B is falsified while A is not at t or if it is compatible with what she knows at t that she could have been in some previous knowledge situation s in which this was the case. Either way X knows that falsification of B does not lead directly to falsification of A.

We say that a sentence A survives in "T if Vt E T vt(A) = 1.6 Let us say that A survives (tout court) if V T V u [A survives in "T], where 'Vu' abbreviates 'for all families {ut: t ~ T}'. A survives if and only if A is a theorem of intuitionist propositional logic. We can, however, make a stronger claim. First let us say that the inference from ~1 to A survives in "T when Vt E T [if VB E E vt(B) = 1 then vt(A) = 1]. Let us say that A survives on the basis of 2£ when V T V u [the inference from ~ to A survives in "T]. The idea here is that A survives on the basis of E when any falsification of A would lead to or yield a falsification of at least one of the members of E. Our stronger claim is this: A survives on the basis of ~ if and only if there is a proof of the inference from ~ to A in intuitionist propositional logic. The proof of this claim is straightforward: if we reverse the ordering on T but keep the ut and vt's as before we obtain the structure T, and intuitionist propositional logic is sound and complete with respect to the class of all such structures. 7

The extension of experiential logic to predicate calculus requires a little thought. There appears to be no straightforward analogy with the intuitionist case. The intuitionist may construct new objects in the course of performing constructions, hence can only be assured now that all objects have some property if he has to hand some general method of construction that shows of any present and possible future object that it has the property in question. In Kripke semantics the analogue is that at each possible knowledge situation t there is a set nomt of names of objects that exist in that situation, these sets satisfying the constraint that they can only grow in the course of time: VtVs [if t ~< s then nomt C_ nom,]; and vt(VxFx ) = 1 only if Vs >~ t [Vz E

28 PETER MILNE

noms vs(Fz) = 1]. The falsificationist reversal of the ordering of the indices in these conditions would lead to the sets of names diminishing as time passes. This diminution can be motivated. The verificationist increases the set of names as he finds more and more terms that do refer. The falsificationist throws out terms as she discovers that they fail to refer. It is tempting to think that in consequence earlier falsifications of VxFx may no longer count, since they may have been due to erroneously treating as a referring term a name now known not to refer. But such a thought is insufficiently anti-realist. Another quick glance at intuitionism makes this evident.

The intuitionist mathematician proves a generalization VnA(n) about the natural numbers when he has a method of proof, a construction, that, for any n yields a proof of A(n). More generally, for a domain other than numbers, the verificationist is entitled to assert VxA(x) when he possesses a general method which, for every name x that may in due course be verified as referring to an object in the domain, yields a verification of A(x). Most generally, he can assert VxA(x) if the method works for any te rm x that may in the fullness of time be found to refer. That x refers can be checked by verifying such assertions as x A' = '^x or x ̂ ' is an object ' . So he is entitled to assert VxA(x) when assertion of x ^ ' = ' ^x leads to assertion of A(x) for all terms x. Returning to the falsificationist we see that she can falsify VxA(x) in her present knowledge situation if it is compatible with what she presently knows that for some name x either now or in the past A(x) was falsified without leading to the falsification of x ̂ ' = 'Ax or x ̂ ' is an object ' . The existential quantifier is more simply accommodated: since the stock of names that X does not know do not refer is diminishing (or at least not increasing) as t ime passes a past falsification of 3xA(x) cannot be impugned, for to falsify 3xA(x) at some time she has to falsify A(x) for all the names she still thinks at that t ime might refer.

We could expand our definition of the UT's by associating with each element t of T a non-empty set nomt of names whose claim to refer to something X has not yet falsified. However , if we consider only predicate logic with identity we can avoid that. When we have identity we can define the nomt's as follows: z E nomt if, and only if, Ut(z = z) = 1. Given the general constraints on the ut's it follows that the nomt's cannot increase. Now that we are considering the internal structure of sentences we place a restriction on the u,'s, namely that


u,(A) = 1 only if all the names occurring in A belong to nomt. From what was said above the falsification condition for the universal quantifier amounts to:

vt(VxFx) = 0 if 3s <~ t [3z E nora, v,(Fz) = 0], and is 1 otherwise.

We add this clause to the definition of the v[s:

(vii) v,(VxFx) = 0 if 3z v~(z = z --+ Fz) = 0, and is 1 otherwise.

Similarly with the existential quantifier we move from:

v~(3xFx) = 0 if [Vz E nomtvt(Fz) = 0], and is 1 otherwise.

t o :

(viii) vf(3xFx) = 0 if Vz vt(z = z A FZ) = 0, and is 1 otherwise. 8

Notice that the quantifiers "flip over": the falsification conditions for the universal quantifier appeal to an existential generalisation and vice versa. This is just the quantifier analogue of what happens with conjunction and disjunction, a flip-over which I passed over in silence (as does Luntley). Contrary to classical and intuitionist definitions of the connectives one does not use ' and ' to define 'A ' , one uses 'or ' .

Adapting the definitions of 'survival ' and of 'survival on the basis o f ' to the modified definition of the UT's we find that the falsificationist characterization of the quantifiers yields intuitionist predicate calculus. In order not to end up with an intuitionist version of free logic - although there would be no great sin in that - we have to amend the definitions to take account of the fact that not all names refer at every point t in T. Thus we say that a sentence A survives in "T if Vt E T [v~(A) = 1 when all the names in A belong to nomt]. Similarly, the inference from ~ to A survives in "T just in case Vt ~ T [if all the names in E and in A belong to nom~ and if VB E Ev,(B) = 1 then v~(A) = 1]. We also restrict attention to finite sets of premises. The reason for that is that since the quantifiers are defined substitutionally the natural definition of the consequence relation is not compact. A variety of tricks, familiar from the literature on completeness theorems using the substitutional interpretation of the quantifiers, could be employed to get around that limitation but since it affects nothing of present interest I choose just to ignore the problem. 9 The important result is this: where

3 0 P E T E R M I L N E

is a finite set of p remises A survives on the basis of E if, and only if, the in ference f rom E to A is de r ivab le in in tu i t ionis t p r ed i ca t e calculus. The p r o o f o f tha t asser t ion is much as in the case of p ropos i -

t iona l logic. 1° In this discussion of fals if icat ionist p r ed i ca t e logic I have said no th ing

of Lun t l ey ' s own account of these mat te r s . I t is t ime to do so. The mos t s t r ik ing d i f fe rence b e t w e e n ou r a p p r o a c h e s is tha t Lun t l ey only cons iders res t r i c ted quant i f iers - he dea ls only wi th express ions of the fo rm V x ( A ( x ) ~ B(x) ) , where the p r e d i c a t e A 'def ines the d o m a i n ' . In giving the falsif icat ion condi t ions for V x ( A ( x ) ~ B(x) ) the next - and crucial - d i f ference arises. Lun t l ey wri tes of objects not of names . I

quo te his account : ~

We only consider universal quantification where the domain is specified so that we are not concerned with accounting for simply VxC(x), but Vx(A(x) -9 B(x)) where 'A' is the predicate which defines the domain. Then, if for this domain VxB(x) survives, this is because we can show that if a named object a is B we can show that an arbitrary object is B, i.e., B(y). Thus, if B(y) survives, VxB(x). What then is it to be able to show that if B(a) survives then B(y)? The construction required is the construction which shows that fB(y) C_fB(,), and the construction which is required to show this for the domain defined by 'A' is re(,) C fA(~). For then, if the falsification conditions for a being A include the falsifying conditions for its being B because of a relation between the concepts of 'A' and 'B', then any arbitrary object y will also be such that its falsification conditions for B(y) are a subset of the falsification conditions for A(y). That is, considering Vx(A(x) ~ B(x)) we h a v e fvx(A(x)~B(x)) = ffB(.) c_fA(~) o r fVx(A(x)~B(x)) = {fB(a) ~ fA(a)}.

This passage calls for a n u m b e r of comment s . O f l i t t le i m p o r t a n c e pe rhaps save tha t it he lps close the gap b e t w e e n Lun t l ey ' s account and mine , it is to be n o t e d tha t Lun t l ey does not p r ec lude tak ing A ( x ) to be x A' = ' " x , a p r ed i ca t e we m a y suppose to be sat isf ied by all ob jec t s in the domain . M o r e i m p o r t a n t l y the d o m a i n of quant i f ica t ion is t aken to be a d o m a i n of ob jec t s with no ind ica t ion how, o r even whe the r , tha t d o m a i n m a y change in view of falsif icat ions ca r r i ed ou t by the knowing agent . (The ana logy with the in tui t ionis t m a t h e m a t i c i a n who m a y c rea te new objec t s has not been pushed far enough . ) Pe rha ps Lun t l ey ' s an t i - rea l i s t concep t ion of an ob jec t m a y save the day? Such

hopes are in vain: 12

The notion of some particular a is nothing more than what is characterised by knowing what sentences containing the name survive. [ . . .] Not only is the notion of an object nothing more than this; it is this.


Thus ob jec thood is de te rmined by survival, it is not something that is falsified.

I have to admit that Lunt ley ' s account of the existential quantifier is less clearly objectual . He says: 13

Clearly, 3xA(x) is an assertion ruling out the possibility that for any term n of the relevant domain, A(n) does not hold. Characterised in terms of sets of falsifiers this comes to

f3xA(x~ =fA(nl~ nfA(n2~ n - - • n fA(nk~

SO that as long as some instance A(ni) makes an effective constraint on experience and survives we are entitled to say that 3xA(x) survives.

But as this tells us nothing about whether the domain may grow in the course of time we are little the bet ter for it. Unless we know that the domain does not increase we have no guarantee that an existential general izat ion once falsified always remains so. Let us be charitable. Let us suppose that the domain does not grow in the course of time. Let us then ask: Is the predicate logic that derives f rom Lunt ley ' s character izat ion of the falsification condit ions for the universal quantifier intuitionistic? The answer, it seems to me, is no.

If Lunt ley does not permit A(x) to be x " ' = 'Ax in Vx(A(x) ~ B(x)) or denies that it is satisfied by all objects in the domain , then Vx(x = x) is not a t heo rem of falsificationist logic. If he both permits the an tecedent and allows its universal satisfaction then we may look at an arbi t rary sentence VxF(x). Now, though I do not claim to be able to r emove all ambigui ty f rom Lunt ley ' s account of the universal quantifier, I believe it best translates into a Kr ipkean formulat ion thus:

(vii °) vt(VxFx ) = 0 if :Iz E noms [3s <- t vs(Fz) = 0], and is 1 otherwise.

Informally , this says that X can falsify VxFx if it is compat ible with her present knowledge situation that some name x that to date she has not yet found to be non-referr ing occurs in the sentence F(x) which she can falsify. With (vii °) we are in no posit ion to guaran tee that a universal generalisat ion once falsified remains so.

Defining survival relative to UT's incorporat ing (vii °) ra ther than (vii) leads to the mis taken conclusion that falsificationist predicate calculus is not intuitionist. The sentence Vx(Fx v Ga) --+ (VxFx v Ga) survives

32 PETER MILNE

when Fx is V-free but is not a theorem of intuitionist logic. 14 Here ' s the p roof that it survives:

Suppose that vt(Vx(Fx v Ga) ~ (VxFx v Ga)) = O. Then 3s <<- t [v , (Vx(Fx v Ga)) = 1 and v , (VxFx v Ga) = 0]. vs(Vx(Fx v Ga)) = 1 iff Vz ~ noms Vt' <~s [vt,(Fz v Ga) = 1] iff Vz E noms Vt' <- s [v,,(Fz) = 1 or vt, (Ga) = 1] iff Vt' <~ s [ v / ( G a ) = 1 or Vz E noms v,, (Fz) = 1] only if vs(Ga) = 1 or Vz E noms vs(Fz) = 1. vs(VxFx v Ga) = 0 iff v , (Ga) = 0 and v , (VxFx) = 0 iff v , (Ga) = 0 and 3 z E n o m , 3t ' <- s v~,(Fz) = 0 iff vs(Ga) = 0 and 3 z ~ norn, v~(Fz) = 0 since for V-flee A if t ' ~< s and vt ,(A) = 0 then v~(A) = O. Hence for all t E T, vt(Vx(Fx v Ga) --~ (VxFx v Ga)) = 1.

To see that Vx(Fx v Ga) ~ (VxFx v Ga), Fx V-flee, ought not survive on a p roper falsificationist account of the quantifiers let us reinstate (vii) in place of (vii°). We can construct a counter -example as follows:

Fa, Fb, and Ga are all a tomic sentences; T = {1, 2}; ~< is the usual numerical order on T; ul(a -- a) = Ul(b = b) = Ul(Fa) = u l (Ga) =1, u~(Fb) : 0; uz(a = a) = uz(Fa) = 1, uz(b = b) = u2(Ga) = uz(Fb) = O. Given these stipulations, Vz Vz(Z = z ~ ( F z v G a ) ) = 1, hence v2 (Vx(Fx v Ga)) = 1; on the o ther hand, since vl(b = b) = i and vl(Fb) = O, Vz(VxFx) = 0 , and as v2(Ga) = 0, we obtain va(VxFx v Ga) = O.

If I have correctly por t rayed Lunt ley ' s concept ion of the quantifiers he is, surprisingly, guilty of insufficient anti-realism. The anti-realist construes the assertion of VxFx as saying: for any name z whatsoever assertion of z A' is an object ' leads to assertion of Fz. That assertion is falsified if there is any name z for which falsification of Fz does not lead to o r yield a falsification of z A' is an object ' . I t matters not at all that this last clause, namely z ^" is an object ' , may itself have been falsified.

To summarise this section, we may say that a falsificationist semantics for the logical connectives and quantifiers can be provided which possesses the desirable features that it complies with Lunt ley ' s guidelines and that intuitionist predicate calculus is sound and complete with respect to it. Exactly how close this semantics is to Lunt ley ' s own is difficult to discern. Formal ly we obtain our semantics by the simple technique of reversing the order ing of points in the models for intuitionist logic furnished by the o r thodox Kripke semantics.


2. COMBINING VERIFICATIONIST AND FALSIFICATIONIST SEMANTICS

By and large anti-realists have concentrated on verification. Luntley shifts to falsification. Michael Dum m et t has on occasion suggested the viability - and even the superiority - of falsification (although he has also expressed doubts on whether the resulting logic is intuitionist). 15 In both cases the resulting logic is intuitionist. This may suggest that an anti-realist should opt neither for an exclusively verificationist nor an exclusively falsificationist semantics for the connectives and quantifiers. Fur thermore , a sentence's role comprises inferential connexions with both verifiers and falsifiers. This suggests that some combination is probably nearer the mark. But combining the falsificationist and verificationist perspectives is not easy for although each separately yields intuitionist logic it is not obvious how to marry the two. Consider this. The stock of terms that are thought might refer is reduced when an objecthood claim is falsified while the stock of names known to refer is increased when such a claim is verified. Now a universal generalisation VxAx is verified when it is shown that any verification of x A' is an object ' leads to verification of Ax. The generalisation is falsified when it is shown that not every falsification of A x leads to falsification of x A' is an object ' . These conditions are not obviously contradictory. In fact they are jointly consistent in the sense that a universal generalisation may be verified and falsified at one and the same time! This compatibility is made manifest when one directly combines Kripkean accounts of verification and falsification, as I shall demonstrate shortly. Before that, however, we may note that the logical difficulties are by no means confined to the quantifiers. They begin in propositional logic. If we combine verificationist and falsificationist accounts of the conditional we get a similarly unpalatable consequence. The conditional A ~ B is verified when it is shown that every verification of A leads to a verification of B; it is falsified when it is shown that some falsification of B does not lead to a falsification of A. These conditions are again jointly consistent: it is possible simultaneously to verify A --, B and to falsify it! Not surprisingly, given the definition of negation in terms of the conditional and absurdity it is also possible simultaneously to verify and falsify -TA. And of course, if sentences of these forms can be simultaneously verified and falsified so can sentences of the other forms - conjunctions, disjunctions and existential generalizations. The difference is this: any sentence formed from atomic formulas using only


conjunct ion, disjunction and existential genera l iza t ion cannot be simul- taneous ly verified and falsified.

In o rde r to demons t r a t e these facts formal ly we extend the " T ' s to incorpora te a family {*ut: t E T} of mappings on the a tomic sentences of L. Intui t ively, *ut(A) = 1 if, and only if, A is verified at t relat ive to X ' s knowledge si tuat ion there and since what is once verified remains SO;

( i ' ) for any a tomic sentence A, Vs/> t [*us(A) 1> *u~(A)].

Impor t an t ly the *u~'s satisfy the condit ion:

(ii ') VA [*u~(A) <~ u~(A)]

i.e. an a tomic sentence is verified at t only if it is not falsified there (and so not previously falsified). G iven the *ut's we define *v~'s as follows:

(iii ') for a tomic A, *vt(A) = *u~(A); ( iv ' ) if A = B v C then *vt(A) = max{*vt(B), *v~(C)}; (v ' ) if A = B/~ C then *vt(A) = min{*vt(B), *vt(C)}; (vi ' ) if A = B ---> C then *lJt(A ) = 1 if Vs t> t [*vs(B) = 0 or

*vs(C) = 1], o therwise vt(A) = O.

In incorpora t ing the quantif iers we associate two sets of names with every point in T: a n o n - e m p t y inner domain innom~ of names that can be verified as referr ing relat ive to X ' s knowledge si tuat ion at t, and an ou te r domain outnorn~ of names whose claim to refer cannot be falsified relat ive to that same background . As with nom~ above innom, and outnomt are definable: innom, = {z: *ut(z = z) = 1} and outnomt = {z: u~(z = z) = 1}. G iven these definitions it follows that the innom~'s and outnomt's have the right proper t ies :

VsVt [if s ~< t then innoms C_ innomt]; VsVt [if s ~> t then outnoms C_ outnom,]; Vt [innomt C_ outnom~].

A p a r t f rom their intuitive naturalness , these condit ions ensure that , once verified, an existential general isa t ion remains so, and, l ikewise, once falsified it stays that way. W e place a fur ther restr ict ion on the ut's and *u~'s, name ly that u~(A) = 1 only if all the names occurr ing in A be long to outnomt and *ut(A) = 1 only if all the names occurr ing in

V E R I F I C A T I O N , F A L S I F I C A T I O N 35

A belong to innom~. There is no need to alter the definition of the vt's although there is an implicit change in (v) and (vi) from nomt to outnomt. We add these clauses to the definition of the *vt's:

(vii') *vt(VxFx) = 1 if Vz*vt(z = z -+ Fz) = 1, and is 0 otherwise. (viii') *vt(3xFx) = 1 if 3z *vt(z = z /x Fz) = 1, and is 0 otherwise.

These conditions together with the falsification conditions for the connectives, (i) - (vi), and quantifiers, (vii) and (viii), given above, consti- tute the natural extension of the Kripke models containing verification and falsification conditions. We shall use "T*u as the name for such an extended Kripke model.

We now construct a model in which a universal generalisation, a conditional, and a negation are all simultaneously verified and falsified. Fa, Fb, Ga and Gb are all atomic sentences; a and b are the only names in L. T = {1, 2}; ~< is the usual numerical order on T. ul(a = a ) = u l ( b = b ) = u 2 ( a = a ) = l; u 2 ( b = b ) = O ; * u l ( a = a ) = * u 2 ( a = a ) = l ; * u l ( b = b ) = *u2(b=b)=O. u l ( F b ) = u l ( G a ) = u 2 ( G a ) = l; Ul(Fb) = u~(Gb) = u2(Fa) = u2(Fb) = u2(Gb) = 0; *u2(aa) = 1; *Ul(Fa) = * U l ( F b ) = *Ul (Ga) = * u l ( G b ) = *uz(Fa) = *ua(Fb) =

*u2(Gb) = 0. Then V z ( V x G x ) = O, *va(VxGx) = 1; v z ( F b ~ Gb) = O, *v2(Fb -+ Gb) = 1; v2(-nFb) = O, *v2(-nFb) = 1.

A simple proof by induction on the length of formulas suffices to demonstrate that, given the conditions governing atomic sentences, no structure "T . . exists in which at any point t any sentence A containing only conjunctions, disjunctions and existential generalizations satisfies the conditions *vt(A) = 1 and vt(A) = O.

It is undoubtedly tempting at thi s point to think that a little tinkering with the verification and falsification conditions will restore things to good order. 16 In fact our tinkering is highly constrained, so much so that there appears to be only one plausible candidate for the set of verification and falsification conditions. The justification of that claim is not difficult but is involved, requiring several steps and a bit of stage- setting.

Firstly, we spell out the intuitive constraints on verification and falsification that must be met by any putative set of verification and falsification conditions. (1) Relative to X 's knowledge situation at any time sentences may be verified, falsified, or undecided; these possibilities are jointly exhaustive, and mutually exclusive. (2) Verifications and


falsifications are conclusive, they are not to be overturned by subse- quent decisions. This stipulation merely elucidates the notions of verification and falsification with which we are currently occupied: verification and falsification are not defeasible.

Secondly, we say that the inference from £ to A transmits verification if whenever all the premises are verified relative to a knowledge situation the conclusion is also verified there. The inference retransmits falsification if whenever the conclusion is falsified relative to a knowledge situation at least one of the premises is also falsified there. In what follows immediately we confine our attention to propositional logic, hence we have no need at this point to incorporate clauses accommodating names lacking reference into the definitions of transmission and retransmission. The central idea is this: any logic whose deducibility relation is sound with respect to both the transmission of verification and the retransmission of falsification and that can accom- modate a standard conditional is a fragment of intuitionist logic. That is, any attempt to stipulate verification and falsification conditions (in accordance with 1 ) and 2) above) for the logical connectives, including a conditional, in terms of which rules of inference and hence a deducibility relation are to be characterized as both transmitting verification and retransmitting falsification yields intuitionist logic. In making this claim I am taking it for granted that the verification and falsification conditions for conjunction and disjunction previously introduced are correct. The bearing of this point will become apparent below.

Any candidate '--+' for being a conditional must satisfy each of the following conditions:

(a) V A V B [A, A --, B B]; (b) VEVAVB [if £, A F- B then E t- A --+ B]. 17

In justification it could be said that these conditions are inherent in our linguistic and investigative practices, especially in hypothetical reasoning. (We shall come back to that.) These conditions do not define '--+', for it is to be characterized semantically in terms of verification and falsification conditions. Rather, (a) and (b) are adequacy conditions on the resulting deducibility relation. One further, utterly uncontroversial, adequacy condition: whatever the resulting logic any sentence is deduc- ible from a set of sentences to which it belongs. (This assumption is


certainly sound with respect both to transmission of verification and retransmission of falsification.)

Suppose that there is a propositional logic, sound with respect to both the transmission of verification and the retransmission of falsification, that accommodates a conditional satisfying (a). We have immediately that if A and A --+ B are verified so is B. Hence, when A is verified and B is not A --+ B is not. (For the sake of convenience I have suppressed the qualification 'relative to X ' s knowledge situation' which attaches to each use of 'verified' in the foregoing sentences. Below I shall refer explicitly to a knowledge situation only where necessary to avoid confusion.) We also have that if B is falsified then so is one or other or both of A and A ~ B, hence when B is falsified and A is not A--+ B is falsified. Perhaps surprisingly we have at this point enough information to show us that the logic is not classical.

The negation of A, we recall, is defined as A--~ ±. Now L is always falsified, so -~A is falsified when A is verified or undecided. That being the case A v -hA is undecided when A is, hence the law of excluded middle is not a theorem of our logic since not sound with respect to the transmission of verification. Thus if, as I am assuming, the verification and falsification conditions for disjunction are correct, the only way to maintain that the logic is classical is to deny the existence of undecided sentences, a denial so unreasonable as to merit no consideration. The denial is utterly unreasonable in light of the project at hand, which is the construction of what might be called a 'logic of enquiry' , or, as Luntley would have it, an 'experienceable logic', that is, a logic read off f rom our investigative practices.

Back to the conditional. We have enough information now to set out some strong necessary conditions on its verification and a weak sufficient condition for its falsification. If X is to verify A --~ B relative to her present knowledge situation then she must not enter into any knowledge situation, past, present, or future, which permits falsification of A--~ B. In particular then, she must not enter into a knowledge situation relative to which B is falsified but A is not. She must also not enter into a future knowledge situation relative to which A is verified and B is not, for in that case A ~ B ceases to be verified. Hence:

*vt(A --~B) = 1 only if [Vs/> tVr <~ s (vr(A) = 0 or vr(B) = 1) and Vs >1 t (*v,(A) = 0 or *v,(B) = 1)]-


As yet only taken to embody a necessary condition we may note that this constraint is already more stringent than what passed for a sufficient condition for verification on our original Kripkean scheme. As for falsification, if X has been in a knowledge situation relative to which B was falsified and A was not then that past falsification sticks. Hence:

vt(A ---+ B) = 0 if 3s <~ t [v , (A) = 1 & vs(B) = 0].

We have got this far only on the basis of (a) above. From (b) we have that B ~-A--~ B. And so verification of B suffices for verification of A - + B, falsification of B is necessary for falsification of A ~ B. In consequence, if A is verified and B is undecided then A --+ B too is undecided. We now have a number of cases in which the status of A --* B relative to X ' s knowledge situation - verified, falsified, or undecided - depends only on the status of A and B relative to that same knowledge situation. We represent these in Figure 1.

A

B --+ V U

V V U

U V *

F V *

Fig. 1.

Now since A FA we have, by (b), that ~-A--+A. Consequently, at least for some sentences A and B, A ~ B is verified when A and B are either both undecided or both falsified. And, assuming we are fight about conjunction, C/x D ~ C, where ' ~ ' represents transmission of verification and retransmission of falsification, so by (b) ~ (C/x D) ~ C, showing that for at least some sentences A and B, A ~ B is verified when A is falsified and B is undecided. But we have no reason to believe that Figure 1 can be completed by uniform substitution of V for *. Indeed our necessary condition for verification and sufficient condition for falsification give us every reason to suppose this is not the case. But just to see what does go wrong, suppose, for the sake of argument, that the status of A --+ B relative to X ' s knowledge situation is determined by the status of A and B relative thereto. In short, suppose that the conditional is functional in the same fashion as conjunction and disjunction. Then we would obtain Figure 2.


A

B V U

V V U U V V F V V

F

F F V

Fig. 2.

Considering the case when A and B are both undecided relative to X ' s current knowledge situation, we find that A --+ B is verified although if ever A were verified relative to some later knowledge situation and B to remain undecided A ~ B would then become undecided, contrary to condition 2) above. Similarly, if A were verified and B falsified A --* B would then become falsified. Since both A ~ B and B ~ A are verified relative to X ' s current knowledge situation, condition 2) entails that A cannot be verified without B 's also being verified and vice versa, nor can one be falsified without the other also being falsified. But A and B are arbitrary currently undecided sentences. So all undecided sentences are verified if one is, all are falsified if one is. If X ever decides a previously undecided sentence she becomes omniscient! Avoiding that conclusion one would be forced to suppose that undecided sentences ever remain so. That is, the only undecided sentences are undecidable.

We may sum up the last paragraph like this: if it is possible, as it ought to be, to resolve sometime indecision in the light of information subsequently gained, then the conditional of the logic of enquiry must be non-functional. It is however only non-functional to the extent per- mitted by Figure 1. This leaves us with the task of filling out the non- functional verification and falsification conditions for the conditional.

Falsification of A--* B relative to a knowledge situation entails that B is falsified relative to that knowledge situation, hence v t (A ~ B) = 0 entails vt (B) = 0. On the other hand, since *vt((A A B) --+A) = 1, for any A and B, v~(B) = 0 is not sufficient for falsification of A --* B. Thus our previous sufficient condition would seem to set the limits for falsification:

(ix) v t (A - + B ) = 0 if, and only if, 3s <~ t [v , (A) = 1 & v , (B) = 0].

Similarly, as consideration of (A A B) ~ A together with the sufficiency


of *vt(B) = 1 for *v,(A ~ B) = 1 makes plausible, our necessary condition sets the limits on verification:

(ix') *vt(A---+B) = 1 if, and only if, [Vs>~tVr<~s (vr(A) = 0 or vr(B) = 1) and Vs >t t (*v,(A) = 0 or *v,(B) = 1)].

As is readily seen, these conditions do comply with conditions 1) and 2). Intuitionist logic is sound and complete with respect to both the transmission of verification and the retransmission of falsification when these new conditions are adopted. The proof of this fact is made easier by noticing that (i) the falsification conditions have not changed, and so (ii) the falsification conditions are independent of the verification conditions (but not vice versa). We already know that the falsification conditions are sound and complete with respect to intuitionist logic. Any falsificationist model UT can be extended to a model "T.u by adding *ut's that are uniformly zero at each point t in T. We can use (ii) to prove that any inference that transmits verification relative to all knowledge situations also retransmits falsity relative to all knowledge situations. This can be done by showing that given some structure "T , , in which at some point all the members of E are unfalsified and A is falsified we can build a new structure " ' T ' , , , - I shall call it an end-model - in which at some point all the members of E are verified and A is not. To construct the end-model, let t be a point in T at which all the members of E are unfalsified and A is falsified. Let T ' = {s E T : s ~< t}. For all s in T ' , u" = u,. For all s in T ' other than t itself, *u's = *u,; *u't = ut. As is readily checked, it turns out that in this structure all the members of 2 are verified at t and A is not. Hence any inference that is sound under transmission of verification is also sound under retransmission of falsification. The last step is to show that the intuitionist introduction and elimination rules are sound with respect to transmission of verification. The only tricky case is --,-introduction and here it helps to use the fact that, as we know, the rules are sound with respect to retransmission of falsification.

We can extend these results to the quantifiers. The verification and falsification conditions look the same as before but in the case of verification of a universal generalization the condition depends on the modified verification condition for the conditional. The same sorts of procedure as just outlined permit the extension of the results.

One comment is in order here. This determination of the verification and falsification conditions for the conditional can scarcely be called


purely semantic. Intuitions about which inference patterns should turn out to be valid carried much of the weight. I suspect that our semantic intuitions concerning verification and falsification are just not strong enough to determine, unaided, verification and falsification conditions for the logical connectives, at least not for the conditional. 18 On the other hand, the adequacy conditions on the resulting deducibility relation are too weak to uniquely determine by themselves the nature of the conditional. It is the interplay of semantic intuitions and inferential practice that leads to a fully elaborated semantics. Given the multiplicity of conceptions of the conditional one could hardly expect otherwise.

3 . AN E X C U R S U S ON T H E C O N D I T I O N A L

Stated informally, the verificationist notion of the conditional holds that A --~ B is verified if verification of A leads to verification of B. X would certainly know that that was not the case if she had ever verified A without verifying B. However , if we take such a failure to count as sufficient for the falsification of A --* B we would no longer be able to guarantee the soundness of hypothetical reasoning as it is represented by adequacy condition (b) above. It is compatible with the antecedent of (b) that X should have verified A at one time and only later verified

and B later. On the informal construal presently under consideration this possibility is not compatible with (b)'s consequent, for it would not be the case that every verification of ~ leads to a verification of A --* B - the latter sentence could in fact be falsified.

What if we insist on this informal, purely semantic, characterization. of the conditional which says that A --* B is verified if every verification of A leads to a verification of B and falsified if some falsification of B does not lead to a falsification of A? Read literally this informal gloss yields a very strong conception of the conditional. Bearing in mind conditions (1) and (2), we might arrive at these stipulations:

(x) v,(A ~ B) = 0 if, and only if, [3s <~ t (v ,(A) = 1 & v,(B) = 0) or 3s <~ t(*vs(A) = 1 & *v,(B) = 0)].

(x ') *vt(A -+ B) = 1 if, and only if, [Vs >1 t Vr <~ s (vr(A) = 0 or vr(B) = 1) and Vs/> tVr <~ s (*v~(A) = 0 or *v~(B) = 1)].

They are motivated by the idea that if, given X ' s current knowledge situation, it could have been the case that A was verified and B was not then X can see that verification of A does not lead to verification

42 PETER MILNE

of B, and hence can falsify A--+ B. This accounts for the extra case added on to the falsification conditions. If she is to verify A--+ B it must not be possible for her to enter into a knowledge situation in which A ~ B is falsified, and so the verification conditions have to be tightened up in corresponding fashion. Clearly (1) and (2) are satisfied with respect to these new stipulations.

The logic that results when these conditions are adopted is certainly not intuitionist for instances of the following theorems of intuitionist logic all fail to obtain:

A ---> (B ---> A); A ~ ((A ~ B) ---> B); (d --~ B) ---> (A ---> (d ~ B)); ( A --> B ) --> ( ( B --> C ) --> ( A ---> C ) ) ; (B --~ C) ~ ((A ~ B) ~ (A ---> C)); (A ----> (B ----> C)) ----> (B ---~ ( A ----~ C) ); ((A/x B) ---> C)) ---> (A ~ (B ~ C)); (m ~ B) ---* ( ~ B ---> 7A) ; (A --~ ~B ) ---> (B --~ -hA).

On the other hand the following are verifiable at all knowledge situations, falsifiable at none:

( A /x (A----> B))---> B; (A---->(A---> B))---~(A---> B); A----~ 7 7 A ; ~A--->(A----> B); (A ~ (B ~ C)) ---> ((A/x B) ~ C); ((A B) /, (B C)) (A ---, C); ( (d ---> B) /x (d --~ C)) ---> (A --> (B A C)); ((A ~ (B v C))/x ~C)---> (A ~ B).

This comes about in large measure because the unrestricted intuitionist rule for ~- in t roduc t ion , (b) above, is not sound although this weaker analogue is:

(b') V A V B [if A ~- B then F-A --+ B].

The following rule for --->-introduction is also sound:

(c) VNVAVB [if ~, A F- Z then 2 F A ~ B],

as is this rule:

(d) for all n > 0, VA1. • . V A n V B V C [if A1/x • • • /x An F- B then C--~ A b . • • , C--> A n f- C--> B ].

In point of fact (a) and (b') alone may be taken as constitutive of the


notion of a conditional. 19 So in adopting this semantic definition we would have arrived at a coherent conception of a conditional. Taking (b ' ) , (c) and (d) as sequent-calculus rules governing --> and substituting them for (b) in intuitionist logic, yields a logic that is sound with respect to both transmission of verification and retransmission of falsification where the conditional is defined by our most recent formulation of its verification and falsification conditions. But the logic is not complete relative to our verificationist-cum-falsificationist semantics ((A---~ (B v C) ) /x -n C) -+ (A--~B) is not provable (although all the other sentence schemata on the second list are). What other rules must be added I do not know. 2° Whatever the logic its status is somewhat heterodox. While it refuses one "fallacy of relevance" - A --+ (B ~ A)) - it endorses another - ~A- -+ (A ~ B). Of course the latter comes about because of the rules involving ±. But endorsing a minimal logic does not seem to be an option here. Albeit that one may find it congenial to think that under certain circumstances ± is not actually falsified - for example, if X ' s mind were ever a t abu la rasa - our semantic strategy demands that if sometimes it goes unfalsified it must be verifiable relative to at least one possible knowledge situation. Intuit- ively there is no possibility of that. 21

The non-intuitionist logic we have come across here has nothing much to do with the combination of verificationism and falsificationism. It has everything to do with the nature of the modified verification conditions, in particular, on their being both forward- and backward- looking with respect to verification of the conditional. Suppose for example that one were to take Kripke 's original semantics for intuitionist logic, modified only by substituting for (vi ') a version of our latest verification condition for the conditional that suppresses mention of the v's. That is, we adopt the condition:

*vt(A-+B) = 1 if, and only if, Vs/> t V r <~ s [*vr(A) = 0 or * v r ( B ) = 1].

Intuitionist logic is not sound with respect to these verification conditions. None of the theses on the first list above is logically valid while, on the other hand, those on the second list are and the rules (b ' ) , (c) and (d) are all sound. But the modified verification and fal,ification conditions seem to be correct if one takes to heart the informal characterization of the conditional. This raises the startling possibility that intuitionist logic has been misidentified.


I took over from Luntley the idea that A ~ B is falsified when a falsification of B does not lead to a falsification of .4. This may not be quite the analogue of the intuitionist's conditional. Luntley notes that the intuitionist has a proof of A --+ B when he has an effective method of converting any proof of A into a proof of B and a proof that that is what the method achieves, z2 The analogue for falsifiability would be that A ~ B is falsified if it is verifiably the case that there is no method for transforming a falsification of B into a falsification of A. That stipulation threatens the autonomy of a falsificationist semantics for the logical connectives. Luntley does not mention this. On general grounds he argues that where warrant or " p r o o f " is not conclusive what it means to assert a sentence is to be distinguished from the conditions in which assertion is warranted, for else one does not know what it is that is asserted in those conditions. What assertion of A ~ B claims is that falsification of B leads to falsification of A, it excludes falsification of B not leading to falsification of A.23 From the point of view of the elaboration of a logic of enquiry this division of labour is not open to us. What is of interest is exactly the conditions under which A ~ B is falsified by the enquiring agent. 24

If A ~ B is to be verified, verification of A leads to verification of B. Now if it is compatible with X ' s present knowledge situation that at some time past she could have been in a knowledge situation relative to which A could be verified but B could not, surely that suffices to show her now, in her present knowledge situation, that verification of A does not lead to verification of B, that A - + B cannot be verified relative to her present situation. So, if the logic of enquiry is intuitionist X must ignore the past to the extent that possible past failure of verification of A to lead to verification of B does not matter , unless, of course, B has been falsified. Given the verificationist's project, X is quite justified in this. She is entitled to use any theses that are presently verifiable, that is that can be verified relative to her current knowledge situation, in determining whether verification of A leads to verification of B. The assertion of a conditional sentence is to be understood not as claiming that verification of A leads to verification of B in isolation from all else that X may be entitled to assert, which involves a remark- ably strong necessitarian view of the conditional. Rather X is currently entitled to assert A ~ B if, given what else she is currently entitled to assert, verification of A leads to verification of B in any knowledge situation in which she can verify at least what she can currently verify,


that is in any possible future knowledge situation, provided, of course, that A --+ B is not falsified there.

It is tempting to apply analogous reasoning to the case of falsification. Why should A -+ B be falsified relative to X ' s current knowledge situation because relative to some possible past situation B was falsifiable and A was not? Why should the more lenient attitude taken towards verification not be extended towards falsification? The short answer is that modus ponens would not then be sound with respect to the retransmission of falsity. And when modus ponens fails, we do not have a conditional (adequacy condition (a)). Perhaps fortunately, there is also a semantic defence of the hard line. Falsification of B at t is carried out relative to the background of sentences that B has not yet falsified by t. After X has falsified some more sentences it may well be that falsification of A occurs whenever B is falsified, but the more sentences X falsifies the more likely such correlations are to arise accidentally. There is an asymmetry between verification and falsification, an asymmetry that is reflected in the two underlying ideas:

counting in too few verifications would make it too easy to falsify A ~ B;

counting in too many falsifications would make it too easy not to falsify A --+ B.

In both cases it is falsification of A--+ B that is at issue, not its verification, and that is the root of the asymmetry.

When X verifies a sentence her attitude towards it changes from uncertainty to something more positive, and she can look on it as an acquisition. When she falsifies a sentences she is entitled to some negative attitude towards it. The mistake behind the appeal for leniency is to think that similarly X acquires something. When X falsifies A it is rather as if the sentence drops out of consideration; X no longer enter- tains it. The negative attitude towards A, to which its falsification entitles her, need not be correlated with a positive attitude to any other sentence, and afor t ior i not towards --qA. In this respect enquiry differs from, for example, subjective interpretations of probabili ty in which, largely on the basis of an unexamined adherence to classical logic, a decrease in X ' s degree of belief in A is correlated with an increase in her degree of belief in its negation. On the other hand, although falsification of A need not be correlated with the verification of any


sentence, verification of A ought to be sufficient for falsification of 7 A on any understanding of negation consonant with a logic of enquiry. In light of these considerations there is no compelling reason why we should not stick to the hard line. We see that our earlier stipulation of verification and falsification conditions (ix) and (ix') is indeed coherent. To it we now return.

4. T H E A S Y M M E T R Y B E T W E E N V E R I F I C A T I O N

A N D F A L S I F I C A T I O N

We modelled our original falsification conditions on the Kripkean conditions for verification and then combined the two in the obvious and most natural way only to discover that they are incompatible. Taking a step back and imposing obvious constraints on the notions of verification and falsification and adequacy constraints on what counts as a conditional we found that the Kripke model served us well for falsification conditions but needed revision with respect to verification conditions. With regard to verification it looks as if intuitionism gives us the right logic for the wrong reasons! - taking the main activity of mathematics to be disproof rather than proof would set things to rights. But we need not go that far. Intuitionist mathematics has no independent notion of falsification, and presumably that is the reason why its notion of proof is quite coherent. This presumption is borne out by a further consideration, namely that where there are independent notions of verification and falsification - independent in that falsification of A is not definitionally equivalent to verification of ~ A - there is an unavoidable asymmetry between the verification and falsification conditions of the conditional. Unlike the cases of conjunction and disjunction they are not and cannot be dual to one another. Our first formulation of falsification conditions simply dualised the standard Kripkean semantics for intuitionist logic. That was permissible only to the extent that we tacitly presupposed a context in which verification makes no sense. Once we acknowledge both verification and falsification we have to refine our account. Under dualising necessary conditions become sufficient conditions, 'and's become 'or 's , *v's become v's, l ' s become O's, V's become 3's , ~<'s become ~>'s, and vice versa, by way of illus- tration, dualising

*v~(A A B) = 1 only if *v~(A) = 1

VERIFICATION, FALSIFICATION 47

we get

vt(A /x B) = 0 if vt(A) = O.

The major cause of trouble is that in dualising the eminently correct

v,(A ~ B) = 0 if *vt(A) = 1 and vt(B) = O,

(i.e., A ~ B is falsified if A is verified and B falsified) we obtain the extremely regrettable

*vt(A -+ B) = 1 only if vt(A) = 0 or *vt(B) = 1.

(i.e., A -+ B is verified only if A is falsified or B verified). What makes this so regrettable is that when A is neither verified nor falsified A -+ A is not verified, contradicting even the weak adequacy condition (b ') . (Perhaps not quite so troubling, but also to be regretted, is that dualising

*vt(A --> B) = 1 only if Vs >~ t [v,(A) = 0 or va(B) = 1]

yields

v,(A ~ B ) = 0 if 3s >~ t [*v,(A) = 1 and *vs(B) = O]

which takes us back to our non-intuitionist logic.) In a context in which it makes sense to have both verification and falsification conditions the conditions for the conditional, and hence for negation and the universal quantifier, cannot be duals to one another. This is an intriguing, if somewhat disconcerting, result.

5. REALISM(S)

As they have been formulated here, it is the falsification conditions that can be stated independently, not the verification conditions. This may suggest that falsification has primacy, thereby giving weak support to Dummet t ' s suggestion. But it is only weak support for the point of the present exercise has been to demonstrate that one does not have to make a choice between the two. Our linguistic practices lead us to assent and dissent, to assertion and denial; our investigative practices lead us to verify and falsify. The same logic is embodied by both assertion and denial, by both verification and falsification. They are two sides of the same coin. If assent to a set of premises rationally compels assent to a conclusion, then denial of the conclusion rationally


compels dissent from at least one premise. And the same holds good in reverse. Anti-realism has tended to give one-sided accounts of linguistic and investigative practices. That one-sidedness has come from holding too tightly to intuitionist mathematics, the one undeniably great - but one-sided - exemplar of anti-realist practice.

So much for anti-realism. What of the partisan of classical logic whom last we saw faced with an unenviable choice between giving up his logic or embracing a stupid notion of undecidedness? The realist, as we shall call him, has a way to resist that choice. He simply denies the functionality of disjunction. The realist admits that there are cases in which A is undecided and B falsified in which A v B is undecided. But it is not so in all cases. In fact, he maintains, it is not so when we take instances of the law of excluded middle. Conditions (iv) and (iv') may have looked all right at the time but really they are not. They make the disjunction of a logic of enquiry functional when in fact it is not. (The realist cannot deny that an undecided A leads to a falsified q A for that falls out of the requirement that modus ponens holds good under retransmission of falsification, and rejection of modus ponens is no way to defend classical logic.) Can the anti-realist argue for the functionality of disjunction? She certainly can given the usual introduction and elimination rules which, since common to intuitionist and classical logic, the realist is unlikely not to concede.

Again we have adequacy constraints, this time on disjunction:

(e) VAVB[A ~-A v B and B ~-A v B];

(f) V A V B V C [if A ~- C and B F- C then A v B F- C].

From (e) we have immediately that A v B is verified when either A or B or both is, that A v B is falsified only if both A and B are. From (e) and (f) it follows that A v B and B v A are interderivable, hence the verification and falsification conditions for disjunction must be sym- metric in A and B. Recalling that A~-A, we find from (f) that A v A ~- A. If verification of a disjunction did not require verification of at least one disjunct this derivation would not be sound. Likewise, if falsification of both disjuncts did not ensure falsification of the disjunction. Hence A v B is verified just in case at least one disjunct is verified; it is falsified just in case both disjuncts are falsified. 2s

Does that show that the realist's investigative practices are incoher- ent? Not at all. But it does allow a sharper characterization of the


realist. The realist denies a tacit assumption that has run through the whole enterprise of specifying verification and falsification conditions. The realist denies componentialism, which he regards as no great loss since he does not see verification and falsification conditions as giving the meaning of sentences anyway. He says that A v ~ A is verified independently not only of the current status of A and ~ A but independently of any considerations about their status in any knowledge situation. Their content is irrelevant to this verification. But what is the realist doing here? He is bringing to bear upon the logic of enquiry conceptions external to what is given in our investigative practices. The realist accepts componentialism with respect to truth-conditions, thereby establishing that the law of excluded middle holds no matter what, and draws on that "discovery" to deny componentialism with regard to the logic of enquiry. The anti-realist, having no truck with the transcendent metaphysics behind the realist's claims, derives her logic from the verification and falsification conditions of complex sentences. The anti-realist develops a pure logic of enquiry. The realist's logic is adulterated by external considerations. What is the source of these considerations? Not enquiry itself, but an a priori imposition of a metaphysical framework on the world prior, conceptually at least, to all enquiry. The realist has no pure logic of enquiry, no experienceable logic. A metaphysical viewpoint is brought to bear in such a way that a transcendent understanding of the logical connectives is forced onto the logic of enquiry.

That, at least, is one way to view the realist. We obtain another characterization when we concentrate on negation. Our semantics for negation permit a distinction between falsification of A and verification of -hA. The latter entails the former but not vice versa. Now the following line of argument, which a realist might employ, appears to have a certain cogency. If A is falsified now it remains so, hence A cannot be verified relative to any possible future knowledge situation, and that surely suffices, even on an anti-realist understanding, for verification of ~A. Against our original Kripkean conditions (vi) and (vi') for verification and falsification of the conditional (which determine those of negation) this argument is indeed cogent. It is blocked only by the fact that verification and falsification are not mutually exclusive. It is only this anomaly that protects our original specifications from grievous damage. The damage is ruinous in that the law of excluded middle would be unfalsifiable but not necessarily verified with respect to


all possible knowledge situations. Thus falsification would obey classical logic, whereas verification would obey an intermediate logic, an extension of intuitionist logic endorsing -~A v -n -hA but not the law of excluded middle. Such a parting of the ways would put paid to the semantic project at hand, although that would trouble the realist not a whit.

When ranged against the revised conditions the argument fares less well. For failure to verify A in all future knowledge situations, while necessary, does not suffice for verification of -hA. The past must also be taken into consideration. Our revised conditions give us a very strong notion of negation. But our revised conditions are well motivated.

Although it makes little impression on the anti-realist the realist thinks there is still something to be gained from this line of thought. The realist sees the process of enquiry against the background of a more-or-less determinate reality that fixes truth-values that quite pos- sibly transcend all possibility of verification and/or falsification. This leads him to see verification of A as sufficient for A's truth, falsification of A as sufficient for its falsity. The realist is then well placed to affirm that the falsification of A is sufficient for the verification of 7 A . If A is falsified, it is false. If A is false its negation is true. Since we have established that ~ A is true on the basis of A's falsification, nothing more is needed to verify ~ A . Of course in arguing this way the realist is imposing his understanding of negation on the process of enquiry. He fashions his logic of enquiry to fit an antecedently given understanding of the connectives. This allows us to give an epistemic characterization of the realist: his investigative practices are such that he unreservedly takes a falsification of A to be a verification of A's negation.

From the literature on anti-realism there are two characterizations of realism. On one realism maintains that there are no limits to the application of classical logic. On the other realism upholds the metaphysical thesis that every sentence is determinately either true or false. As much of the literature, both for and against, explains, these two descriptions are not a perfect match. How does this new epistemic characterization compare? It is weaker than either of the standard ones. It is compatible with the idea that there are objective indeterminacies "in the world". That, I think, can only be to the good. Vagueness, whatever problems it may cause for classical logic or bivalence, does not, it seems to me, commit one to anything that might reasonably be


called anti-realism (that is the point of the 'more-or-less' above). Nor, p a c e Dummett , does the denial of truth-values to sentences containing non-referring singular terms. 26 The new characterization is also compatible with giving up classical logic for verification. As has been noted already, the thesis that falsification of A Suffices for verification of its negation leads to the law of excluded middle being unfalsifiable but only to -hA v ~ A being verified. This law is not derivable in intuitionist logic and so still transcends our investigative practices - it still represents an a priori metaphysical commitment. But worst of all from the present perspective the realist has committed himself to a mismatch between verification and falsification. By going beyond what is given in enquiry the realist obtains two logics of enquiry, one for verification, one for falsification.

There is one further line of argument the realist might take. It is the weakest yet. It is also, I believe, the most interesting. The realist agrees not to impose his transcendent understanding of the connectives, but to let the logic of enquiry, experienceable logic, speak for itself. How- ever, he still maintains that the limits of enquiry are set by the way the world is independently of the process of enquiry (independently of experience). He maintains that one can only verify what is true, and likewise only falsify what is false. Hence a possible course of enquiry is bounded by what is the case. In this sense of possible there is then only one possible course for any enquiry, namely the course it actually takes. So in giving a semantic account of the logic of enquiry the realist will reject our anti-realist argument for using partial orders. Epistemic possibility is, he thinks, irrelevant. The proper realist semantics for a logic of enquiry uses linear orders only. Now it is a feature of linear orders that every instance of the thesis ~ A v ~ -n A is verified at every point, falsified at none. More generally, all instances of (A--+ B ) v (B--+A) are verified at every point, falsified at none. Moreover, the law of excluded middle is not unfalsifiable. There is n o mismatch between verification and falsification. In fact adding (A-+ B) v (B--+A) to intuitionist logic as an axiom schema yields a logic that is sound and complete with respect to our semantics in linear orders. 27 The realist can therefore be characterized as maintaining the unrestricted validity of (A ~ B) v (B ~ A). This is the weakest description yet in that he will assert it even when he concedes that a falsification of A need not be equated with a verification of 7 A . (It is part and parcel of the realist's view that A -+ B may be verifiable relative to X's knowledge situation although X has no means of verifying that this is so.)


This last characterization of realism makes clear something that ought perhaps to be emphasised, namely that this outline of the logic of enquiry proceeds independently of considerations of meaning concerning the non-logical vocabulary of the language in question. How the atomic sentences of the language acquire whatever significance they do is immaterial. The logical connectives and quantifiers are defined in terms of the dependencies of the verification and falsification conditions of complex sentences containing them on those of their parts, but then that is what one would expect of a logic of enquiry. The realist, in his last guise, happily concedes that much to anti-realism, viewing it as no concession at all. Whether this is the only way to give meaning to connectives is another matter, and one that is independent of the issue at hand. Nevertheless the metaphysical differences between realism and anti-realism are reflected in the different logics that emerge.

N O T E S

i Luntley (1988, pp. 46-53). 2 Ibid., pp. 152, 257. 3 Ibid., pp. 257-266, substantiates all claims made in this paragraph save that concerning the rule of /x- introduct ion, for which see ibid., p. 114. 4 Luntley says that the members of the falsifier classes are atomic sentences (ibid, p. 153) but I am not at all sure that he can consistently do so and even if he can I do not think that he means it. Suppose that A and B are semantically unrelated atomic sentences. Which atomic sentences falsify their disjunction? 5 Ibid., p. 261. 6 Cf. Luntley, op. cit., pp. 155,265. 7 This result, due originally to Saul Kripke (although foreshadowed by work of Michael D u m m e t t and E. J. Lemmon) , is proved in many places. In giving the present account of falsification I have had in mind a hybrid proof taken from D u m m e t t (1977) and Tennant (1978). s The variable z here and below ranges over closed terms of the language L, so the quantifiers are in effect defined substitutionally. In writing VxFx and Fz I am of course taking for granted all sorts of syntactic niceties. 9 For substitutional quantification and strong completeness see D u n n and Belnap (1968) or Leblanc (1982), Part 2. 10 Again we base this assertion on a result of Kripke's. 11 Op. cit., pp. 265-266, with some changes in notation. 12 Ibid., p. 178, emphases in the original. 13 Ibid., p. 265. 14 For its non- theoremhood see Dummet t (1977, pp. 197-207). 12 See D u m m e t t (1978, p. xI), and 'Truth ' and the postscript thereto, pp. 1-24. For Dummet t ' s reservations see Dummet t (1976, p. 126).


16 There is an alternative: to maintain that al though the logic is formally the same in each case the meaning of the connectives differs, being given on the one hand by verification conditions, on the other by falsification conditions, and so it is not one and the same sentence that is verified and falsified. This line entails that neither introduction rules, nor elimination rules, nor both together, nor the whole deducibility relation of the logic suffice to give the meaning of the connectives, a consequence that would be of some importance for contemporary philosophical logic. 17 Conditions (a) and (b) are satisfied by the material conditional, the intuitionist conditional, and the conditional of many many-valued logics. (a) is also satisfied by the conditional of strong implication, and by the subjunctive conditional on many accounts, so at best (a) and (b) partially determine the logic we are to obtain. (b) comes in for some discussion below. 18 My own intuitions extend only this far: verification of A -~ B and of A entails verification of B; verification of A---~B and falsification of B entails falsification of A; verification of A and falsification of B constitutes a falsification of A ~ B. 19 Cf. Curry (1977, pp. 172-173). Causal conditionals and, on some analyses, counter- factuals, do not satisfy (b'). 2o Whether every inference that transmits verification also retransmits falsification and vice versa remains an open question. A sufficient condition for the sentences falsified at no point in any "T~u-structure coinciding exactly with the sentences verified at every point of every "T.~,-structure is an affirmative answer to this question: is it the case that whenever a disjunction is falsified at no point in any "T. ,-s tructure one of the disjuncts is also falsified at no point in any "T.u--structure? (There is an easy proof of this well known property of disjunctive theorems of intuitionist logic in our "weak" semantics that employ (ix) and (ix') but it does not transfer to the present setting where we use (x) and (x').) Whether every inference that transmits verification and retransmits falsification is sound intuitionistically is at present unknown although I have found no reason to doubt it. An observation: if E, B ~ C then E ~ B ~ C and E, A ~ B ~ A --~ C just in case at every point of every "T. , -s t ructure non-falsification of all the members of ~, together with verification of B suffice for verification of C. The semantic versions of (b') , (c) and (d) all follow from this but how is it itself to be formulated syntactically? It may be that there is no logic that fits all the verification and falsification conditions in the way desired. It is also possible that if there is such a logic it is not a proper fragment of intuitionist logic. But let us be bold. Let us conjecture that t ransmission of verification and retransmission of falsification coincide and coincide on a fragment of intuitionist logic. 2a Logicians of a paraconsistent stripe may demur but no doubt we parted company from them when we baulked at the possibility of s imultaneous verification and falsification that resulted from marriage of the two Kripkean schemes. 22 Luntley (1988, pp. 134-135). For similarly worded accounts of the intuitionist conditional see D u m m e t t (1977, pp. 12-13), and Makinson (1973, p. 73). Luntley omits 'effective' but I assume nothing rests on that omission. 23 Op. cit., pp. 136-156. 24 There is a current, s temming from Heyting, in the explication of intuitionism that says, without qualification, that A ~ B is proved when any proof of A yields a proof of B. See Sundholm (1983) and Weinstein (1983). 25 It is a triviality to show that conjunction is functional. 26 D u m m e t t (1981, p. 438).

54 PETER MILNE

27 In its axiomatic formulation the logic is known as LC. It is strictly stronger than KC, the logic obtained by adding the schema ~ A v ~ T A . See Dummett (1976), pp. li-lii. Completeness follows from soundness and a remark in Dummett, loc. cit. It can be proved directly following the completeness theorem for intuitionist logic in Tennant (1978) and noticing that LC-closed extensions of a disjunctive LC-closed set are linearly ordered by inclusion. The falsificationist order reversal, the incorporation of *u/s, and the argument concerning end-models carry over. Soundness under the verification conditions is again to be established directly.

R E F E R E N C E S

Curry, Haskell B.: 1977, Foundations of Mathematical Logic, Dover, New York. Dummett, Michael: 1976, 'What is a Theory of Meaning? (II)', in G. Evans and J.

McDowell (eds.), Truth and Meaning, Oxford University Press, Oxford. Dummett, Michael: 1977, Elements of Intuitionism, Oxford University Press, Oxford. Dummett, Michael: 1981, The Interpretation of Frege's Philosophy, Duckworth, London. Dunn, J. Michael and Nuet D. Belnap, Jr.: 1968, 'The Substitutional Interpretation of

the Quantifier', Norms 2, 177-185. Leblanc, Hugues: 1982, Existence, Truth and Provability, SUNY Press, Albany, New

York. Luntley, Michael: 1988, Language, Logic and Experience: The Case for Anti-Realism,

Open Court, La Salle. Makinson, D. C.: 1973, Topics in Modern Logic, Methuen, London. Sundholm, Grran: 1983, 'Construction, Proofs and the Meaning of Logical Constants',

J. Phil. Logic 12, 151-172. Tennant, Nell: 1978, Natural Logic, Edinburgh University Press, Edinburgh. Weinstein, Scott: 1983, 'The Intended Interpretation of Intuitionist Logic', J. Phil. Logic

12, 261-270.

Manuscript submitted on June 12, 1989 Final version received on March 14, 1990

Philosophy Department Massey University Palmerston North New Zealand

verification, falsification, and the logic of enquiry

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