Scots Philosophical AssociationUniversity of St. Andrews

A Plea for Model TheoryAuthor(s): Chris MortensenSource: The Philosophical Quarterly, Vol. 31, No. 123 (Apr., 1981), pp. 152-157Published by: Oxford University Press on behalf of the Scots Philosophical Association and theUniversity of St. AndrewsStable URL: http://www.jstor.org/stable/2218722 .

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152 CHRIS MORTENSEN

is inadequate, but Russell's later account of the concept of existence is sufficient. Of course, in "On Denoting" Russell criticised and rejected the Principles-Fregean view of denoting. But it must be stressed that it follows from what has been said above that the anti-Fregean passages are not gratuitous. If the need for the theory of descriptions is seen primarily as arising from ontological considerations, then these passages provide the indispensable basis of this need. If they fail, then the need dissolves.

University of Hong Kong

A PLEA FOR MODEL THEORY

BY CHRIS MORTENSEN

I. INTRODUCTION

Historically, necessity posed a problem. How is it that some truths somehow have to be true, independent of changes (real or imagined) in the world? The positivist answer to this problem was that the source of such truths lies in language. In a phrase, necessity is analyticity. Accounts of analyticity varied: truth by virtue of meaning, semantical rules, convention and so on. The common core to such views however, was the reduction of necessity to a phenomenon dependent on the language used. The positivist position on analyticity has been unfashionable since Quine's criticisms of it, but Graham Priest has persuasively defended a view somewhat like that of the positivists, with the interesting twist that he claims it to be consistent with, and indeed required by, a broadly Quinean epistemology.' Priest claims that there are analytic truths more or less coincident with the class so named by the positivists, and that their analytic truth arises from con- vention. But he differs from the positivists, in so far as their position is held to involve a reductive account of necessity, at least if this is understood as truth "come what may", proof against all revision. Here his Quineanism emerges, for he denies that there are any such necessities. This paper argues that Priest's views need modifying in certain important respects.

The source of analyticity, for Priest, lies in the activity of inferring, which is a practice governed by rules or norms which distinguish valid from invalid inferences. To each valid inference there corresponds a true con- ditional 'If A, then, logically, B', or 'That A entails that B', which Priest calls a logical conditional. The practice of inferring is prior to the linguistic object, the conditional, in that justification of the truth of the latter ulti- mately rests on the norms governing the former, and not vice versa. Such true logical conditionals, and anything validly deducible from them, are called "analytic".

According to Priest, the truth of logical conditionals is conventional because the validity of inferences is conventional, and the latter is conven- tional because the norms which govern inferring are determined by the relevant social practice. Allowing that practices and their accompanying norms can change, Priest rejects the suggestion that a whole practice of

1Graham Priest, "Two Dogmas of Quineanism", Philosophical Quarterly, 29 (1979), 289-301.

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A PLEA FOR MODEL THEORY 153

inferring might be mistaken, and holds that "external" evaluation of the norm implicit in a practice is impossible. Whole social practices judged according to their own norms are correct, but there are no meta-norms lying behind practices. The best we could say of different practices is that neither was more correct than the other.

Surely, one might feel, a social practice of inferring might be mistaken?2 Just as we might want to say that a whole society's moral standards could be mistaken, or that a scientific theory or paradigm could be false, even if universally adopted. Priest's answer to this has a familiar ring. The mean- ings of sentences are so closely related to the infeience relations between them that, if there were a different practice-and-norms, this would show that the meanings of the terms involved in it were different. Hence there cannot be a counterexample to a practice-backed inference norm A/. .B. A counter- example would be a case where A would be true and B false, and that is prevented by the fact that the norm also ensures the meanings of A and B sufficiently to make it impossible for B to be false when A is true. Thus there can be no genuine inter-practice disagreements about norms of validity. Priest agrees with Quine that logic must be built into our translation manual (p. 296). He thus seems to commit himself to a strong meaning-variance thesis: any difference in validity norms entails a difference in meaning of the terms whose deductive relations are governed by the norms. Otherwise, if meanings (whatever they are) could stay invariant while practice-and- norms changed, then we might find ourselves in the peculiar situation where according to one set of norms it is impossible for A to be true and B false, and according to another set it is possible, even actual.

Priest contrasts his approach to the study of validity with the model theoretic approach. The model theorist investigates arguments first by looking for the conditions under which sentences are true, and then by attempting to find arguments which are valid in the sense of being truth- preserving; that is, arguments such that whenever the premisses are true, so is the conclusion. On this approach, the theory of truth precedes that of validity, which emerges as a by-product. On the other hand, Priest's view is that validity is practice-determined, not truth-determined, and, as we have already seen, that truth and falsity emerge via the determination of meaning by practice-relative validity norms.

II. CONSTRAINTS ON ENTAILMENT I wish to argue that the model theoretic approach to validity should not

be relegated to the back seat Priest has reserved for it. I first introduce the locution 'It is a matter of convention that A', and establish some of its properties by noting Priest's account of conventionality:

I shall argue that the validity of a deductive inference is, in a sense I shall make precise, a conventional matter. It follows that analytic truths are, in the same sense, conventionally true. . . A rule is valid if and only if it is one of the rules "in force", i.e., one of the standards which govern people's practice of inferring. . . Thus, which rules are valid is a matter which depends on human agreement (of action). In this sense, it is conventional. (p. 296, emphasis mine)

2Such a case is interestingly described by F. B. D'Agostino, "Generative Semantics: Ethnoscience or Ethnomethodology?' in Evidence and Argumentation in Linguistics, ed. T. Perry (Berlin, 1980), especially 295-6. His example is of a society which, because of certain deeply engrained views of human nature, is unable to draw the consequences of other beliefs which manifestly go against their prejudices.

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154 CHRIS MORTENSEN

Presumably, then, its being false that A entails B also depends on human agreement in action. That is, A does not entail B iff A/.'.B is not one of the rules which govern the practice of inferring. We may sum up Priest's account thus: it is a matter of convention that A (where A is an assertion that a given argument is valid or invalid) if and only if the truth or falsity of A is determined by the presence or absence of suitable practices. Two principles seem to follow from this. The first is that conventionality distributes over 'if .. .then': if, if A then B, then if it is a matter of convention that A then it is a matter of convention that B. What can be said for this principle? It seems correct to me for the reason that if B were not a matter of convention - in Priest's terms, not (wholly) determined by a social practice - then A could not be either. Suppose that its being true that B were subject to change over time with atmospheric pressure; then so would A have to be (at least partly), independently of social practice. Priest seems to accept something like this principle since it is logical conditionals and anything deducible from them which are said to be analytic, and analyticity is to be accounted for by the convention. The second principle we need is that it is a matter of convention that A iff it is a matter of convention that -A. This follows fairly directly from our summing up of Priest's explanation of the idea of conventionality. Analytic falsehoods are as much a problem as analytic truths, and the explanation of the one needs to be on all fours with the explanation of the othel.

Let us now observe that logical conditionals imply ordinary condibionals. If A entails B, then if A, then B. Now a sufficient condition for the falsity of the latter is that A be true and that B be false. Hence, if A entails B, then -(A&-B). Let A be 'Snow is white' and B be 'Coal is not black'. It seems clear that A&,B here is not a matter of convention, but of fact. Hence, by our second principle, it is not a matter of convention that -(A&-B) either. So by our first principle, it is not a matter of convention

that A entails B. So at least some entailment claims are not a matter of convention. Equally, it is not a matter of convention that A does not entail B. Similar arguments can be given with other substitutions for A and B including various truth functions. Briefly, putative conventions constrained by fact cannot be entirely conventions.

There are several replies Priest might make to this argument. The one I suspect he would make is to say that it is a conventional matter after all that snow is white and coal is black. This is not as strange as it might look at first glance. It might be argued that the idea of conventionality (in the intended sense) ought not to be understood as a sentential operator 'It is a matter of convention that . . .', but as a predicate of sentences ''. . .' is conventional'. Thus in the example, the premiss: it is not a matter of con- vention that snow is white and coal is black, should really be the premiss: 'Snow is white and coal is black' is not conventional. That premiss is how- ever questionable, since presumably many would agree that there are con- ventions in force determining at least the meaning of 'Snow is white and coal is black'. But is this such an easy way out? If 'Snow is white' is con- ventional simply because its truth-value depends on conventions of meaning, we can say the same about any sentence. What then would be distinctive in Priest's view of validity? The distinction between fact and convention needed to make conventionalism interesting would be eliminated.

I think that what has gone wrong with Priest's position is a too rigid reliance on deducibility relations as the determinants of meaning. Only such a view could give us the strong meaning-variance thesis. Language is an

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A PLEA FOR MODEL THEORY 155

instrument for talking about the world, so there need to be some links with the world for such talk to be possible. Deducibility relations by themselves only relate pieces of language to other pieces of language. Yet whatever meaning is, it presumably constrains the relations between language and world. So meaning is not entirely a matter of deducibility relations. In the case of theoretical predicates we might find it plausible to constitute their meanings very largely by their deducibility relations with other predicates. But in the chosen example, 'Snow is white' has quite close connections with middle sized, easily observable objects. Here it is more natural to allow that no meaning shift has occurred if those connections remain, even if some variation in the practice of inference occurred. Thus, given the semantic connections of 'Snow is white' and 'Coal is black' with the world, their truth constrains possible inference practices. Some such may be right and others wrong. So, also, cross-practice criticism is possible.

A different response to the previous argument would be to hold that deducibility relationships are conventional up to truth and falsity, in the sense that beyond the restriction that no falsehood may be validly deduced from a truth, there are no objective, inter-practice inference norms. We might draw an analogy with Poincare's conventionalism. For Poincare, choice of geometry was conventional (in a somewhat different sense from Priest's), but different geometries require a different physics, for example in the postulation of universal forces. Thus it is geometry-plus-physics which is conventional. But it is not entirely conventional. Not just any geometry- plus-physics will do; only those are acceptable whose consequences square with the facts of the matter, especially the results of observation. Hence observation is a constraint on convention, or geometry-plus-physics is con- ventional up to phenomena. So, also, a similar position might be taken here. In fact I think it is the position which ought to be taken, and it might even be argued that Priest's idea of conventionality is best understood in such a way that being conventional does not rule out being partly determined by facts other than the facts of practice. But there are aspects of Priest's position which get in the way of taking this line. As noted earlier, he prefers to hold that different inference norms will give different meanings and make counterexamples impossible. The reason for his holding this is, I suggest, that to make the present move would seem to give up inference conventions as the sole source of meaning. It would also be to give up the complete priority of validity over the truth (or falsity) of corresponding conditionals. Some of Priest's remarks also suggest that he fears that in reinstating the theory of truth as a non-derivative part of the theory of validity we run the risk of reinstating propositions which are true in all models, that is, truths whose necessity is not accountable for by practice-relative conventions and norms. I will suggest later that there is not much to fear on this score.

III. MODEL THEORY The fact that the truth of conditionals is constrained by semantic connec-

tions between language and world, and thlus by the changing nature of the world itself, is a thin end of the wedge for the reinstatement of a partly model-theoretic account of validity against Priest's rejection of it. Let us say that an argument is valid iff it is (in some sense) truth-preserving. The precise sense of truth-preservation may vary with choice of logic, so we leave it unspecified here. Saying this need not affect much of what Priest has to say: he can still hold that validity is conventional, that its norms

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156 CHRIS MORTENSEN

are grounded in practice, and that validity is prior to truth-preservation and the truth of logical conditionals. It was, however, argued above that validity is not entirely a conventional matter, that it is not entirely deter- mined by practice but is also constrained by the world. That would be enough to require the model-theoretic investigation of semantical notions like truth to be a necessary part of the investigation of validity. Now the model-theoretic idea of truth-preservation needs careful statement in order to avoid commitment to the principle Ex Falso Quodlibet (EFQ). One must avoid arguing that if, for validity, whenever premisses are true the conclusion must also be true, then, if the premisses could not be true (in the sense of being inconsistent), we automatically have a valid argument to any con- clusion. But one need not be committed to this. For example, the model theory of the various relevant logics, as developed by Routley, Meyer and others,3 gives a model-theoretic account of validity which is not committed to EFQ.

Model theory does not merit the criticism Priest directs at it. He claims in effect that a defect of the model-theoretic approach is that it commits one to classical logic or at least to modal logic (p. 297). In the light of the Routley-Meyer semantics, this is false. He also argues that validity is epistemically prior to truth-preservation, in that for the model-theorist to begin to determine which inferences are valid using the notion of truth- preservation, an infinite number of inferences would need to be checked in order to decide whether a given inference form was valid. This is a curious argument for a conventionalist to use. If it were really a matter of social practice which inferences were valid, then it would equally be a matter of practice which inferences were truth-preserving. In fact, Priest himself describes principles for the discovery of valid inferences which fit the model- theorist's activity well enough (p. 295). We begin with a collection of infer- ences (and, we might add, truth judgements) and the aim is to construct a theoretical account of validity (add: truth, and truth-preservation), which does justice to the data. "During the process there is the characteristic interplay between theory and data" (ibid.).

IV. MODEL THEORY IS NOT COMMITTED TO PRACTICE-INDEPENDENT VALIDITY

Let us recall the earlier distinction between 'A entails B' and 'If A then B'. It was argued that the truth of A-B (i.e., -(A&-B) ) was a non- conventional constraint on the truth of 'A entails B'. We can go further if we want to hold what seems natural, that some propositions of the form 'If A then B' are true, not as a matter of convention, and equivalent neither to 'A entails B' nor to A= B. For then the nonconventional truth of 'If A then B' is a further constraint on validities, and a further model theoretic requirement for entailments to satisfy. The reinstatement of model theory may however suggest the possibility of practice-independent validities and necessities. I will conclude by expressing agreement with Priest that none such are to be found.

Simply asserting 'If A then B' (given an intended account of its truth conditions) carries no commitment to practice-independent necessities and validities, only to a (practice-independent) conditional. The constraints on the notion of entailment that we have argued for, then, do not force practice-

3For example, in "The Semantics of Entailment, I" in Truth, Syntax and Modality, ed. H. Leblanc (Amsterdam, 1973), 199-243.

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A PLEA FOR MODEL THEORY 157

independent necessities on us: the argument so far is neutral as to whether there are any. Let me tentatively suggest another argument for the same conclusion. One of the directions of recent logical research has been into the semantical conditions under which various propositions hold and fail. One of the upshots has been a growing body of information about how to construct models to refute more and more propositions. It is, for example, no news that countermodels can be constructed to large numbers of the theorems of the very natural modal logic S5, on which David Lewis' modal realism is based. It is also a straightforward matter to construct counter- models to the laws of excluded middle and noncontradiction.4 Recent work by Errol Martin has even shown how to construct countermodels to every instance of A - A.5 In the light of these kinds of results, it seems to me that it would be a bold claim that there is any proposition that cannot be made to come out false in some structure. If one wishes to avoid making this bold claim, then the price to be paid for maintaining the existence of some necessary propositions and universal validities is to be forced to assert a preferred set of semantical structures among the describable ones and to show that it is a correct description of reality. An example might help. If one wants to assert Lj(A= DO A), and give O, 0 their usual Kripke-style meanings, then modal semantics shows that we need to claim that there is a relation R of relative possibility between worlds which is symmetric. There are Kripke-style worlds structures in which R is not symmetric, and in which D(A = D OA) is false. So to hold that D(A = D OA) is true, we would need to give some reason why the non-symmetric relational worlds structures should be ignored. It would, of course, be improper to argue that one has so fixed the meaning of C, 0 that D(A= E OA) must come out true in all worlds. If LA means 'A is true in all worlds possible relative to this one', then it is a further question what the structure of the set of worlds is. It is no part of that meaning of O that all worlds are possible relative to every other world, which is sufficient for S5 and hence for the truth of D(A= D OA). (Nor can it be claimed that WA means 'A is true in all worlds of an S5 structure of worlds' so that the necessity of Az D OA follows; because it would still need to be shown that there exists such a structure in re.) The point can be summed up as follows. If any proposition can be refuted in some structure, then an argument for a preferred class of structures (and hence for objective necessities) will not be forthcoming from model theory alone In fact, I do not believe that any argument for objective necessities is successful. Model theory merely tells us what it is for various propositions to be true. Hence, I believe, the argument of earlier sections does not threaten Priest's denial of objective necessities and validities.

Australian National University

4See e.g. R. Routley, R. K. Meyer et al, Relevant Logics and Their Rivals (Canberra, 1981), Ch.2.

5E. P. Martin, The P-W Problem, Ph.D. Dissertation, Australian National Uni- versity, 1978.

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