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A Perspective on the Intuitionistic Conception of Truth* Goran Sundholm

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A Perspective on the Intuitionistic Conception of Truth* Goran Sundholm Existence, Proof and Truth-Making: A Perspective on the Intuitionistic Conception of Truth* GOran Sundholm ABSTRACT. Truth-maker analyses construe truth as existence of which can best be seen as an attempt at giving content proof, a well-known example being that offered by Wittgenstein in to the metaphors involved in the so called correspon- the Tractatus. The paper subsumes the intuitionistic view of truth as dence theory of truth, z existence of proof under the general truth-maker scheme. Two generic constraints on truth-maker analysis are noted and positioned with In general, a truth-maker analysis is given in terms respect to the writings of Michael Dummett and the Tractatus. of three items, namely, Examination of the writings of Brouwer, Heyting and Weyl indicates (1) the (category of) truth-bearers A, the specific notions of truth-maker and existence that are at issue in the intuitionistic truth-maker analysis, namely that of proof in the (2) for each object a of the category of truth-bearers, sense of proof-object (Brouwer, Heyting) and existence in the non- a category TM a of truth-makers for A, propositional sense of a judgement abstract (Weyl). Furthermore, possible anticipations in the writings of Schlick and Pf~tnder are and finally, noted. (3) the relevant notion of existence for the category TM A . Leibniz's Satz vom Grunde provides a case in point: A true proposition is one where the predicate is contained in its Michael Dummett writes: subject, or, more generally, whose consequent is contained in its antecedent, and it is therefore necessary that there should be some From an intuitionistic standpoint . an understanding of a connexion between notions of the terms, i.e. that there should be mathematical statement consists in the capacity to recognize a an objective foundation from which the reason for the proposi- proof when presented with one; and the truth of such a statement tion can be given, or, an a priori proof can be found. 3 can consist only in the existence of such a proof. 1 Here the truth-bearer is a proposition in the old- It is the purpose of the present paper to offer some fashioned sense, that is, a judgement of subject/copula/ remarks of a philosophical and historical nature on this predicate form conception of truth and its meaning-theoretical use. SisP Leibniz construes all truth, whether necessary or con- o tingent, as analytic. The proposition is true when the predicate is contained in the subject, such containment The intuitionistic scheme being made evident from a certain ground, or reason, A is true namely a possibly infinite a priori proof of the judge- ment in question, consisting of (possibly infinite) there exists a proof of A resolutions of the concepts S and P into primitive com- ponents. A truth of reason has a finite proof, whereas can profitably be construed as a special instance of the the resolution proceeds to infinity in the case of a truth general truth-maker analysis of truth: of contingent fact. The truth-maker is this a priori proof A is true and the relevant notion of existence is resolvability (in the mind of God). Modern truth-maker analyses were there exists a truth-maker for A, offered in the early parts of the present century by Topoi 13: 117-126, 1994. © 1994 KluwerAcademic Publishers. Printed in the Netherlands. 118 GORAN SUNDHOLM Husserl, Moore, and Russell, using as truth-makers, for When the truth-bearers are propositions, the above example, facts, complexes and Sachverhalte. The most constitutes a simple solution to the old Quinean chal- elaborated of these modern attempts, surely, is that lenge of providing a criterion of identity for proposi- presented by Wittgenstein in the Tractatus. 4 tions. A second general point can be made concerning . the notion of existence that is involved in a meaning- theoretically constrained truth-maker analysis. Here the The particular properties of a given truth-maker analysis meaning of a sentence A is laid down in terms of its will obviously depend on the three specific choices truth condition made as to the truth-bearers, their truth-makers, and the A is true. appropriate mode of existence for the chosen truth- makers. Clearly, in view of the rich variety of truth- Thus, on pain of a vicious regress of ever-descending maker analyses that has been offered, one cannot expect meaning-explanations, the predicate to establish too many general theses, applicable to any •.. is true truth-maker analysis, from the very formulation of the generic truth-maker scheme itself. cannot be propositional in nature. In other words, it is In the Dummett passage quoted above, the link not a propositional function formulated in the language between truth- and meaning-theoretical issues is faced from which the sentences are taken whose meaning is squarely from the outset, and the meaning of a sentence explained using the predicate in question. When com- is, of course, commonly formulated in terms of truth bined with the truth-maker analysis this yields the result conditions. These, on the other hand, are given in terms that the existence involved in such an analysis is not of a notion of truth that is construed according to the that of the existential quantification truth-maker analysis in question. Therefore, on the 3xe(x). hypothesis that meaning is given in terms of truth conditions, the truth-maker analysis will be subjected to It must be observed first that there is no such thing as meaning-theoretical constraints. Such constraints, then, the meaning of this quantifier independently of its provide further conceptual structure which might enable occurrence in existential propositions; on the contrary, us to formulate some general insights concerning the the meaning of the quantifier is explained as part of the truth-maker analysis of truth. existential proposition having such and such a truth- condition. Furthermore, the proper form of the quan- tification is . (3x ~ D)P(x), One remark concerning truth-maker analyses that can since the usual, model-theoretic explanation of the be made bears on the application of the notion of semantics of the quantifier suppresses the domain of identity to truth-bearers. An appropriate notion of quantification, owing to the separation of syntax and identity for truth-bearers can be obtained without effort semantics. Here when the quantified formulae express given the truth-maker analysis. An identity criterion is propositions, rather than serve as the formal objects of readily formulated in terms of the truth-makers: metamathematical study, the domain of quantification A is the same truth-bearer as B has to be explicitly included. The meaning of the exis- iff tential quantifier 3 is accordingly explained in terms of the truth-maker categories TM A and TMB coincide the conditions under which (3x ~ D)P(x) is true. On the truth-maker analysis this truth condition is An alternative, essentially equivalent choice would analysed as: be to use instead: there exists a truth-maker for (3x ~ D)P(x). A is the same truth-bearer as B iff If the existence involved here were itself to be expressed every truth-maker for A is a truth-maker for B, by means of an existential quantifier, firstly, in order and vice versa. that the quantifier be applicable, the relation of truth- EXISTENCE, PROOF AND TRUTH-MAKING 119 making between the truth-maker and the true pro- If to grasp the meaning of a sentence involves knowing when it position would have to be a propositional function. •.. is true, then this knowledge must involve knowing what it is, in general, for a sentence.., to be true. To know this is neither Secondly, an infinite regress would arise: necessary or sufficient to know, of every particular sentence, (3z e the category of truth-makers for (3x under what conditions it is true. It must, rather, involve knowing D)P(x)) how in all cases the concept of truth is related to that of meaning. To be more exact, we ought here to speak of an implicit grasp of [the relation of truth-making obtains between z the connection between the concepts of meaning and of truth, and the proposition (3x e D)P(x)], rather than of a knowledge of it. 6 since the first existential quantifier 3z has yet again to This connection between meaning and truth can take be expressed in terms of the truth-condition for an either of two forms. It is direct when meaning is given existentially quantified proposition, and so on. Thus, the in terms of truth. The connection will be mediated when notion of existence at issue here is not that of the meaning is ultimately given in terms of another notion. existential quantifier. Accordingly, also the status of the In this second case, as Dummett rightly notes, there has relation of truth-making between the truth-bearer and to be a uniform manner of deriving the notion of truth the truth-maker is sub judice and one cannot expect it from the latter notion in terms of which meaning is then to be that of a propositional function: if it were, it would ultimately explained: immediately combine with the existential quantifier to yield a proposition of the rejected, regress-infested type. we need to distinguish between a feature of an expression which is dependent upon its meaning and one that is actually part of that meaning. This distinction underlies our intuitive inclination to take the meaning as something we confer on an expression ..... If some feature of an expression is merely dependent on its meaning, then it did not have to be conferred and cannot be Phenomena related to, or originating in, this unusual altered; that is, it cannot be altered without altering something status of the relation of truth-making have been noted else to which it is responsible, and from which it derived.
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