The Logic of Truth

The distinction has occasioned a long controversy between Carnap and W.V. Quine. The latter distinguishes two sorts of analytic statements: the logical truths, characterized by their remaining true under all reinterpretations of the descriptive terms; and the statements, which reduce to logical truths with the help of definitions or by substitution of synonyms for synonyms. In “Two Dogmas of Empiricism”, he directs his criticism mainly against the latter arguing that the explications so far provided move in a circle, since, in the end, the concept of synonymy is explained in terms of analyticity. He reaches the conclusion that the only form of synonymy not exhibiting this deficiency is that introduced by explicit conventions. But according to him, this procedure is of no avail, because the philosophically interesting content of the general term “analytic in L” (with variable L) cannot be grasped with its help. Carnap’s suggestion of defining a new basic concept “analytic in L0” does not solve the problem since we could just as well designate this with a neutral word, say “K”, thus avoiding the illusion of an explanation. The attempt to define synonymy with the help of semantic rules fails in a similar way: it accomplishes nothing more than to place certain statements under the rubric “Semantic rules” — a notion which is no clearer than the explicandum. Carnap replies that his definition of “analytically true” is given with regard to artificial or formalized languages for which we can determine syntactic and semantic rules in such a way that the concept of truth, in Tarski’s sense, can be applied to them. Analytic statements other than logical truths will thus be distinguished by means of so-called meaning postulates which serve to make the truth-value of such statements independent of empirical matters. I share Carnap’s astonishment that Quine finds the term “semantic rule” unintelligible while formulating no complaint against similar notions such as that of a syntactically well-formed formula, or that of an axiom etc. It appears that he cannot be satisfied with rules, because he expects a theory which would give us an empirical, i.e., a behavioristic, criterion of analyticity for natural languages. His chief dissatisfaction seems to come from the fact that no clear explicandum has been put forward — in contrast, e.g., with the notion of truth, where we have a sufficiently clear pre-theoretical concept. Thus, according to him, there is nothing that could be clarified by an explication.

The concept of truth in African philosophy is an intrinsic reality whose structural and ontological manifestation exhibits the ontology and physical manifestation of falsehood. This is provable by the method of IP (Indirect Proof) as in modem logic and can be called shadow concept or inclusive disjunction equivalent to that in modern logic. In Igbo thought, truth is equivalent to the concept of permanence while the concept of falsehood is equivalent to change or impermanence, although there is a sense in which falsehood implies possibility of truth, in which the dynamic of falsehood is a dialectical change, a progression to the unfolding truth, the absolute perfect spirit underlying every history, be it that of people, science, philosophy and objects. Hence truth telling cultures have the attribute of a scientific and critical high social order while a lie telling culture will be consistent with an underdeveloped, chaotic social order. Truth telling is a sustainable culture and epistemology in Igbo society, a major ethnic group in Nigeria and West Africa, and therefore should be advocated for Igbo sustainable development goals.

Michael Kremer defines fixed-point logics of truth based on Saul Kripke’s fixed point semantics for languages expressing their own truth concepts. Kremer axiomatizes the strong Kleene fixed-point logic of truth and the weak Kleene fixed-point logic of truth, but leaves the axiomatizability question open for the supervaluation fixed-point logic of truth and its variants. We show that the principal supervaluation fixed point logic of truth, when thought of as consequence relation, is highly complex: it is not even analytic. We also consider variants, engendered by a stronger notion of ‘fixed point’, and by variant supervaluation schemes. A ‘logic’ is often thought of, not as a consequence relation, but as a set of sentences – the sentences true on each interpretation. We axiomatize the supervaluation fixed-point logics so conceived.

During his long career, Rudolf Carnap held various different views about the concept of truth and its philosophical significance. As a good logical empiricist, he insisted on the distinction between logical and factual statements, and employed his technical powers to give rigorous characterizations of the notions of logical, analytic, and factual truth. The development of Carnap’s views reflected his ability to quickly absorb new influences and his broad interests ranging from logic to epistemology, philosophy of language, and philosophy of science (Sections 1 and 2). Carnap was the co-founder of logical semantics with Alfred Tarski (Section 3), and therefore it is especially interesting to see how Carnap’s work was related to Tarski’s early definition of truth and to the later Tarskian model theory (Section 4). In Section 5, we discuss some difficulties in Carnap’s liberally empiricist treatment of scientific theories. The final section 6 makes some remarks on Carnap’s contemporary relevance.

Truth is a concept always present in scientific activity, but which is not frequently thematized, due to the fact that the classic definition of truth, ‘adequation of the intellect to the thing’, is still valid with different shades. Martin Heidegger first and Hans Georg Gadamer later sustained that the conception of truth as the correspondence or adequation of the intellect to the thing was established in a germinal of the Western thought, specifically in Plato’s work. This definition of truth replaces the original meaning of the Greek word a-letheia, meaning ‘unconcealment’. This article states that another conception of truth is in force, which can be useful in scientific research and practice. In particular, (1) this paper will analyse a definition of truth as ‘unconcealment’ introduced by Phenomenology and Philosophical Hermeneutics; and (2) it will also demonstrate the usefulness of this notion in the field of Psychiatry and Neuroscience. For these purposes, this paper will deal in the first place, with the problem of method and understanding in Hermeneutics and the rigour which scientific knowledge can reach. Then, the notion of truth according to the hermeneutic conception will be developed. Finally, the impact of this difference in the conception of truth for scientific practice will be analysed, placing special emphasis on Neurosciences and Psychiatry.

The starting point of the article is a passage in Tarski’s work entitled The Semantic Concept of Truth and the Foundations of Semantics (1944), in which he points out the difference between the concept of truth and other semantic concepts. According to Tarski, unlike other semantic concepts, “truth” does not indicate a relation but a property (property of sentences). Although Tarski did not elaborate on this claim in his seminal paper, I argue that this passage may be taken as a starting point for further considerations of the essence of truth. In particular, one of the most important issues is the classical concept of truth and the problem of correctly constructing the definition of truth understood as not a relation but a property.

The central element of the concept of truth in Smiglecki’s Logica (1618) is his approach to formulating definitions. Where the establishing of the truth is concerned, he always points to compliance at the level of the community (conformitas) in respect of whether the intellectual recognition of a thing or things is in accordance with its intellectual equivalent, or the principles behind the latter, where these are understood as designating the corresponding idea inherent in the intellect of God. This is a form of the classical definition of truth --- similar to that used by St. Thomas Aquinas --- with a wide scope of applicability: to the field of existence (transcendental truth), to cognition and language (logical truth), and even to moral beliefs (moral rightness). Smiglecki distinguishes three types of truth: truth assigned to being, truth assigned to cognition, and truth assigned to moral convictions. Of these, the first is identified with transcendental truth, while the second is attributed not only to propositions and sentences, but also to concepts. The truth of concepts results from compliance with things by way of representation, while the truth of propositions and sentences issues from a compliance with things involving the implementation of some form of expression or other. Logical truth pertains to propositions rather than concepts. The kind of moral truth he writes about is what we would now be more likely to call “truthfulness”. With the exception of moral truth, which he defined as compliance of a statement with someone’s internal thoughts, Smiglecki considers every kind of truth to be a conditioned state of the object of knowledge. He says (a) that the ultimate object of reference of human cognitive functioning is a real being, absolutely true by virtue of compliance with its internal principles and their idea as present in the intellect of God, and (b) that the compatibility of human cognition with a real being is the ultimate basis for the truth of human cognition itself. Just knowing the real truth of a being furnishes us with an absolute form of truth, whereas merely knowing the default character of purely fictional objects can be true at most in relative terms. This pattern certainly also holds for sentences about what does not exist, and for sentences about future events whose truth or falsity is a contingent affair. Truth is of particular importance in science (scientia), although its scope covers a wider area than the area of scientific knowledge itself. It thus reaches to the very essence of scientific excellence. On the other hand, scientific knowledge is by its very nature always true. Therefore, truth is derived from our possession of its essence.

FREGE AND THE JUSTIFICATION OF LOGICAL LAWS In recent work on Frege, one of the most salient issues has been whether he was prepared to make serious use of semantic notions such as reference and truth. Those not familiar with this debate are often surprised to hear of it. Surely, they say, Frege's post-1891 writings are replete with uses of 'true' and 'refers'. But no one wants to deny that Frege makes use of such terms: Rather, what is at issue is how Frege understood them; more precisely, what is at issue is whether Frege employed them for anything like the purposes for which philosophers now employ them. What these purposes are, or should be, is itself a matter of philosophical dispute, and, although I shall discuss some aspects of this issue, my goal here is not to address it directly. My purpose here, rather - one of them, anyway - is to argue that Frege did make very serious use of semantic concepts: In particular, he offered informal mathematical arguments, making use of semantic notions, for semantic claims. For example, he argues that all of the axioms of the Begriffsschrift - the formal system in which he proves the basic laws of arithmetic - are true, that its rules of inference are truth-preserving, and that every well-formed expression in Begriffsschrift has been assigned a reference by the stipulations he makes about the references of its primitive expressions. Let me say at the outset that Frege was not Tarski and did not produce, as Tarski did, a formal semantic theory, a mathematical definition of truth. But that is not of any significance here. One does not have to provide a formal semantic theory to make serious use of semantic notions. At most, the question is whether Frege would have been prepared to offer such a theory, or whether he would have accepted the sort of theory Tarski provided (or some alternative), had he known of it. On the other hand, the issue is not whether Frege would have accepted Tarski’s theory of truth, or Godel ’ s proof that first-order logic is complete, as a piece of mathematics; it is whether he would have taken these results to have the kind of significance we (or at least some of us) would ascribe to them. Tarski’s argument in ‘The concept of truth in formalized languages’ shows that all axioms of the calculus of classes are true; the completeness theorem shows that every valid first-order schema is provable in certain formal systems.

Both the presuppositions underlying a scientific theory and the theoretical claims of which the theory consists play a twofold role at the same time. They are both settle, and they are meant to confine the way in which the terms specific to the theory may be interpreted. These two functions (descriptive and semantic) cannot be separated in any sharp way. The process of the formation of scientific theories consists of the formation of both the conceptual apparatus (the language of the theory) and the set of claims intended to account for the regularities characteristic of the examined phenomena. One of the main theses of this paper is that the hypotheses one can formulate within a given theory may not have a definite truth value, if the conceptual apparatus of the theory in question is not “veristic”, i.e. the correspondence between the concepts and the elements of structure of the phenomena to which the theory refers is not defined in a complete and unique manner. The concept of truth and the relevance of this concept for metatheoretical investigations is the central issue of the paper, both its present part and the one to appear in the next issue. the considerations the reader is going to find in this part are carried out in a rather loose way. A more formal exposition of some of the topics undertaken here is to be found in the forthcoming part.

With each imaginary city a separate but coalescent epistemological riddle, Calvino’s Invisible Cities is a meditation on the troubles of knowing the world in which we live and our place in it. The only systematicity to be found in Calvino’s ecology of knowledge is in the paradoxes; common sense is playfully evaded in favor of illogical realism and non-Euclidean quantum physics. Radical is, I argue, the a-systemic object ontology and the agential epistemology Calvino offered 50 years ago, if interpreted as a reflection on the limitations of knowledge. Logocentrism (i), the concept of logical truth (ii), progressive timelines of accumulated wisdom (iii), and the assumption of the possibility of objectivity (iv) leave instead space for silent forms of communication and uncertainty (i), the primacy of doubt (ii), and the redistribution in the world of time, agency, and cognition (iii–iv), all of which flatten hierarchies and decenter the human. They also oblige us to confront fears, desires, and assumptions in order to start knowing and perhaps enjoying the difficulty of knowledge.

In this chapter, I discuss the relativity of truth and define the concept of viewpoint-relative truth. The relativity of truth is perhaps the strongest form of epistemological relativity that can used to deduce the relativity of knowledge and reality. One of the fundamental problems of epistemology is overcoming doubt, which is why I will introduce epistemology by presenting the problem of scepticism. After this, I discuss different theories of truth. The contextuality, or the dependency on context, of truth is focal to viewpoint relativism, and I build the definition of viewpoint-dependent truth upon it. As an example of relativism that utilises contextuality, I will present MacFarlane’s theory of relativism, which represents the so-called “new relativism.” Finally, I explain what I mean by truth relativism.

Can Tarski's definition of truth be taken to be an analysis of truth – i.e., a method of defining formal truth predicates that are capable of playing the role of truth in all theoretical contexts in which that notion is needed? It is argued that although Tarski's truth predicates can play many of the roles demanded of truth, they cannot play the role of truth in theories of meaning and interpretation. Crucial to the argument is the observation that there is a conceptual connection between our ordinary notions of truth and meaning, in virtue of which statements of the truth conditions of sentences provide some information about their meanings. Since there is no corresponding connection between Tarski's notion of truth and the notion of meaning, statements of the Tarski‐truth conditions of sentences provide no information about meaning. The chapter closes with a detailed defense of Tarski's model‐theoretic analysis of logical truth and logical consequence against John Etchemendy's objections to it.

The Hilbert–Bernays Theorem establishes that for any satisfiable first-order quantificational schema S, one can write out linguistic expressions that are guaranteed to yield a true sentence of elementary arithmetic when they are substituted for the predicate letters in S. The theorem implies that if L is a consistent, fully interpreted language rich enough to express elementary arithmetic, then a schema S is valid if and only if every sentence of L that can be obtained by substituting predicates of L for predicate letters in S is true. The theorem therefore licenses us to define validity substitutionally in languages rich enough to express arithmetic. The heart of the theorem is an arithmetization of Gödel's completeness proof for first-order predicate logic. Hilbert and Bernays were the first to prove that there is such an arithmetization. Kleene established a strengthened version of it, and Kreisel, Mostowski, and Putnam refined Kleene's result. Despite the later refinements, Kleene's presentation of the arithmetization is still regarded as the standard one. It is highly compressed, however, and very difficult to read. My goals in this paper are expository: to present the basics of Kleene's arithmetization in a less compressed, more easily readable form, in a setting that highlights its relevance to issues in the philosophy of logic, especially to Quine's substitutional definition of logical truth, and to formulate the Hilbert–Bernays Theorem in a way that incorporates Kreisel's, Mostowski's, and Putnam's refinements of Kleene's result.

This volume presents a series of essays published by Charles Kahn over a period of forty years, in which he seeks to explicate the ancient Greek concept of Being. He addresses two distinct but intimately related problems, one linguistic and one historical and philosophical. The linguistic problem concerns the theory of the Greek verb einai, ‘to be’: how to replace the conventional but misleading distinction between copula and existential verb with a more adequate theoretical account. The philosophical problem is in principle quite distinct: to understand how the concept of Being became the central topic in Greek philosophy from Parmenides to Aristotle. But these two problems converge on what Kahn calls the veridical use of einai. In the earlier papers he takes that connection between the verb and the concept of truth to be the key to the central role of Being in Greek philosophy. In the later papers he interprets the veridical in terms of a more general semantic function of the verb, which comprises the notions of existence and instantiation as well as truth.

This chapter introduces a broad class of definitions—generalized schematic definitions—that have hitherto been unrecognized and uses them to set out a flexible framework for the study of truth in a broad range of situations. The idea of saturated hypotheses of a revision rule is deployed to construct a semantical system S for the interpretation of these definitions. The chapter goes on to build a general revision theory of truth T on the basis of the system S. Basic consequences of theory T for the behavior of truth in classical as well as three-valued contexts are noted. It is shown that T accounts for the abbreviation and generalization functions of truth, while Fixed-Point Theories fail to do so. It is also shown that only a theory of truth that tolerates omega-inconsistency can fully account for these functions.