A Note on the Unprovability of Consistency in Formal Theories of Truth

Journal of Philosophical Logic - Why is it that even strong formal theories of truth fail to prove their own consistency? Although Field (Mind, 115, 459, 2006) has addressed this question for many...     

Truth and Consistency

This paper investigates relations between truth and consistency. The basic intuition is that truth implies consistency, but the reverse dependence fails. However, this simple account leads to some troubles, due to some metalogical results, in particular the Gödel-Malcev completeness theorem. Thus, a more advanced analysis is required. This is done by employing the concept of ω-consistency and ω-inconsistency. Both concepts motivate that the concept of the standard truth should be introduced as well. The results are illustrated by an interpretation of the well-known logical square and its generalization.     

Consistency,Truth and Ontology

After a brief survey of the different meanings of consistency, the study is restricted to consistency understood as non-contradiction of sets of sentences. The philosophical reasons for this requirement are discussed, both in relation to the problem of sense and the problem of truth (also with historical references). The issue of mathematical truth is then addressed, and the different conceptions of it are put in relation with consistency. The formal treatment of consistency and truth in mathematical logic is then considered, with particular attention paid to the relation between syntactic and semantic properties of sets and calculi. After the crisis of mathematical intuition and the dominance of the formalistic view, it seemed that consistency could totally replace the requirement of truth in mathematics, also in the sense that the existence of “objects” of axiomatic systems could be granted by their consistency. A rejection of this claim is presented, whose central point is a detailed analysis of the theorem that any consistent set S of sentences of first order logic has a model. A critical scrutiny shows that this model is very peculiar, being offered by the elements of the same language that is being interpreted, and the satisfiability conditions for any sentence being constituted by the mere fact of belonging to S. Though not being insignificant from a metatheoretical point of view, this theorem fails to endow consistency (even in this privileged case) with an “ontological creativity”, that is, with the capability of providing a model ontologically distinct from the language itself (which is the precondition for the classical notion of truth that is also preserved in the Tarskian semantics and model theory). A final discussion regarding the different “ontological regions” and the referential nature of truth clarifies the different aspects of the whole issue discussed.     

A Note on Typed Truth and Consistency Assertions

In the paper we investigate typed (mainly compositional) axiomatizations of the truth predicate in which the axioms of truth come with a built-in, minimal and self-sufficient machinery to talk about syntactic aspects of an arbitrary base theory. Expanding previous works of the author and building on recent works of Albert Visser and Richard Heck, we give a precise characterization of these systems by investigating the strict relationships occurring between them, arithmetized model constructions in weak arithmetical systems and suitable set existence axioms. The framework considered will give rise to some methodological remarks on the construction of truth theories and provide us with a privileged point of view to analyze the notion of truth arising from compositional principles in a typed setting.     

Truth Values and the Value of Truth

This paper explores the ways in which truth is better than falsehood, and suggests that, among other things, it depends on the kinds of proposition to which these values are attached. Ordinary singular propositions like "It is raining" seem to fit best the bivalent "scheme" of classical logic, the general proposition "It is always raining" is more appropriately rated according to how often it rains, and a "practically vague" proposition like "The lecture will start at 1" is appropriately rated according to its nearness to exactness. Implications for logic of this "rating system" are commented on.     

Logical Truth and Tarskian Logical Truth

This paper examines the question of the extensional correctness of Tarskian definitions of logical truth and logical consequence. I identify a few different informal properties which are necessary for a sentence to be an informal logical truth and look at whether they are necessary properties of Tarskian logical truths. I examine arguments by John Etchemendy and Vann McGee to the effect that some of those properties are not necessary properties of some Tarskian logical truths, and find them unconvincing. I stress the point that since the hypothesis that Tarski's definitions are extensionally correct is deeply entrenched, the burden of proof is still on the shoulders of Tarski's critics, who have not lifted the burden. This revised version was published online in June 2006 with corrections to the Cover Date.     

Deflationary Truth and the Liar

Journal of Philosophical Logic -     

真之符合论与真之等同论辨析

传统的真之符合论面临一个问题:由谁来判别一个命题是否符合事实?为从上帝之眼回归人类之眼,普特南提出内在实在论,以区别于外在实在论。这里进而提出关于真的内在符合论,以区别于传统的外在符合论;其关键在于把事实看作语言性的,并将语言性事实的观点分为外延等同论和内涵等同论。从主观性和客观性的角度看,内涵等同论和外在符合论处于两个极端,而外延等同论和内在符合论则处于中道,并且是二位一体的。在哲学史上,这两种符合论和两种等同论的要素贯穿于弗雷格、罗素、塔斯基和普特南等人的真理论之中。     

Coherence and the Truth Connection

There is an objection to coherence theories of knowledge to the effect that coherence is not connected with truth, so that when coherence leads to truth this is just a matter of luck. Coherence theories embrace falliblism, to be sure, but that does not sustain the objection. Coherence is connected with truth by principles of justified acceptance that explain the connection between coherence and truth. Coherence is connected with truth by explanatory principle, not just luck.     

Psychoanalysis and the hermeneutic turn: a critique of Narrative Truth and Historical Truth

Through a discussion of Donald Spence's Narrative Truth and Historical Truth, a critical introduction to the hermeneutic or interpretive perspective is presented. Spence's book has generally been assumed to offer a hermeneutic reformulation of psychoanalysis. However, its presuppositions are incompatible with fundamental tenets of contemporary hermeneutic thought, as expressed in the philosophies of Heidegger, Gadamer, and the later Wittgenstein. Spence's basic assumptions are classically empiricist and positivistic. His vision of human experience is essentially associationistic and Humean; it treats experience as involving two processes, the passive reception of raw sense data and a subsequent projection of meaningful interpretation. Spence advocates the gathering of brute data while denying or downplaying the epistemological value of theorizing and of interpretive understandings. These assumptions are contrasted with those of the hemeneutic philosophers. Unlike these philosophers, Spence tends to dichotomize coherence and correspondence theories of truth. As a result, he wavers between relativism (regarding therapeutic interpretations) and objectivism (regarding scientific knowledge).