Proof and Truth. An anti-realist perspective

Abstract

In the first chapter, we discuss Dummett’s idea that the notion of truth arises from the one of the correctness of an assertion. We argue that, in a first-order language, the need of defining truth in terms of the notion of satisfaction, which is yielded by the presence of quantifiers, is structurally analogous to the need of a notion of truth as distinct from the one of correctness of an assertion. In the light of the analogy between predicates in Frege and open formulas in Tarksi, we concentrate on the semantic status of predicates. We register a dual attitude of Dummett towards Frege’s ascription of reference to predicates. On the one it is needed to endow quantifiers with their appropriate meaning. On the other hand, the introduction of concepts, as semantic correlate of predicates, smuggles a realist flavor in the overall semantic picture. In concluding the excursus (and with it the chapter), we stress Dummett’s will of developing a semantic picture free from this realist trait.

In the second chapter, we present the idea of a proof-theoretic semantics. As for Frege true sentences denotes the truth-value True, so here closed' (i.e. categorical) valid argumentations denotes proofs. If a language contains implication-like operators, in order to characterize the condition of validity of a closed argumentation one has to introduce a notion of validity applying to ‘open’ (i.e. hypothetical) argumentations. We argue that this problem is analogous to the one posed by quantifiers in Frege-Tarksi’s style semantics. That is, implication forces one to introduce notion of validity for argumentations, which is more substantial than the correctness of the assertion of their conclusions. As we saw, the corresponding claim in the truth-based approach—that quantifiers require one to introduce a notion of truth, which is more substantial than the one of an assertion being correct—was the source of realism. Hence, Dummett proposes to reduce the semantic contribution of open argumentations to the one of their ‘closed instances’. In the truth-based perspective, this would correspond to the denial of the need of introducing concepts as the semantic correlates of predicates. We argue that Dummett’s fear, that an irreducible notion of function (represented by the need of ascribing validity to open argumentations) would lead to realism, turns out to be ill-founded.

In the third chapter, we discuss the role played by the notion of truth in the anti-realist account. The notion of truth is what the anti-realist needs to cope with the so-called paradox of deduction. The analysis of the paradox yields to distinguishing between the truth of a sentence and the truth of a sentence being recognized. In terms of these conceptual couple, we reconsider the relationship between truth and assertion in an anti-realist perspective. Grounds are provided for a thesis (which was already advanced in chapter two), according to which the notion of the assertion of a sentence being correct is primarily connected only with the canonical means of establishing a sentence. The possibility of establishing a sentence by indirect means is conceptually dependent on the practice of establishing logical relationship of dependence among sentences. That is, the notion of a closed valid non-canonical argumentation is of any theoretical relevance only in presence of a notion of validity applying to open argumentations.

In the fourth chapter, we discuss the possibility of characterizing in the proof-theoretic-semantics a notion of refutation. We develop an original characterization of refutations starting from an informal inductive specification of the condition of refutations of logically complex sentences. A sub-structural logic, called dual-intuitionistic logic, stands to this notion in the same relationship in which intuitionistic logic stands to the notion of proof so far consdered. All notions developed in chapter 2 have their corresponding (dual) ones in the framework developed. In particular, the distinctions canonical/non-canonical and closed/open argumentations. In the refutation based perspective, elimination rules have priority over introductions and the (only) assumption over the (many) conclusions.

In the conclusions, we indicate the ingredients that an anti-realist approach to meaning should incorporate, in order to avoid the difficulties we registered. The core of an alternative view is a different conception of the relationship between categorical and hypothetical notions, in which the validity of open argumentations is not reduced to that of their instances, but rather it is directly defined. As a limit case, one would get a notion of validity applying to closed argumentations.

Im ersten Kapitel wird Dummetts These zum Ursprung des Wahrheitsbegriffes aus der Richtigkeit der Behauptung dargelegt und bezugnehmend auf Freges Zuschreibung einer Bedeutung zu Prädikaten diskutiert.

Im zweiten Kapitel wird eine Analogie zwischen dem wahrheitstheoretischen und dem sogenannten beweistheoretischen Ansatz entwickelt. In der wahrheitstheoretischen Semantik wird für die Erklärung der Quantoren ein Wahrheitbegriff, der den Begriff der Richtigkeit einer Behauptung erweitert, gebraucht. Dem wird im beweistheoretischen Ansatz ein Gültigkeitsbegriff für hypothetische Argumentationen gegenübergestellt, der benötigt wird, um die Gültigkeit kategorischer Argumentationen zu definieren - sofern ein Operator wie Implikation in der Sprache vorhanden ist.

Im dritten Kapitel wird nun die Analogie zwischen den beiden Perspektiven weiterentwickelt, mit Schwerpunkt auf dem Zusammenhang zwischen Wahrheit und hypothetischer Gültigkeit.

Im vierten Kapitel wird eine neue Charakterisierung des Begriffs der Widerlegung als Spiegelbild des bisher diskutierten Beweisbegriffs entworfen. Dieser Begriff der Widerlegung teilt Vorteile und Nachteile mit dem des Beweises.

Abschließend wird im letzten Kapitel eine mögliche Weiterentwicklung der beweistheoretischen Semantik aufgezeigt, wobei Dummetts bemerkte Spannung zwischen der Richtigkeit einer Behauptung und Wahrheit lösbar wird und der neu definierte Widerlegungsbegriff eingebracht wird.