How does metaphysical grounding interact with the truth-functions? I argue that the answer varies according to whether one has a worldly conception or a conceptual conception of grounding. I then put forward a logic of worldly grounding and give it an adequate semantic characterisation.

It is shown that the standard definitions of truth-functionality, though useful for their purposes, ignore some aspects of the usual informal characterisations of truth-functionality. An alternative definition is given that results in a stronger notion which pays attention to those aspects.

Benjamin Schnieder has argued that several traditional definitions of truth-functionality fail to capture a central intuition informal characterizations of the notion often capture. The intuition is that the truth-value of a sentence that employs a truth-functional operator depends upon the truth-values of the sentences upon which the operator operates. Schnieder proposes an alternative definition of truth-functionality that is designed to accommodate this intuition. We argue that one traditional definition of ‘truth-functionality’ is (...) immune from the counterexamples that Schnieder proposes and is preferable to Schnieder’s alternative. (shrink)

Many-valued logics are standardly defined by logical matrices. They are truth-functional. In this paper non truth-functional many-valued semantics are presented, in a philosophical and mathematical perspective.

Michael Lynch’s Truth as One and Many is a contribution to the large body of philosophical literature on the nature of truth. Within that genre, advocates of truth-as-correspondence, advocates of truth-as-coherence, and the like, all hold that truth has a single underlying metaphysical nature, but they sharply disagree as to what this nature is. Lynch argues that many of these views make good sense of truth attributions for a limited stretch of discourse, but he (...) adds that each of the contenders also faces a ‘scope problem’, in that none of them makes sense of all legitimate truth attributions. Some deflationists make similar observations – usually less concessive – and conclude that truth simply has no substantial, underlying nature. There is nothing for the philosopher to articulate and defend. Lynch agrees with deflationists that the realm of discourse is too diverse for any single ‘first-level’ property, such as correspondence or coherence, to fully characterize it, but against deflationism, he holds that truth does have a nature. He presents a functionalist theory. The definition is this: x is true if, and only if, x has a property that plays the truth-role. According to Lynch, the truth-role is given in terms of what he calls the ‘core truisms’ of the folk concept of truth. These are platitudes that hold of the everyday notion, and are such that anyone who denies all of them can be accused of changing the subject – of not talking about truth. A given property T plays the truth-role for a batch of propositions, just in case, for any proposition P in the batch: P is T if, and only if, where P is believed, things are as they are believed to be; other things being equal, it is a worthy goal …. (shrink)

Two of the main intuitions that underlie the phenomenon of vagueness are the truth-functional and the penumbral intuitions. After presenting and contrasting them, I will put forward Tappenden's gappy approach to vagueness (which takes into account the truth-functional intuition). I will contrast Tappenden'sview with another of the theories of vagueness that see it as a semantic phenomenon: Supervaluationism (which takes into account the penumbral intuition). Then I will analyze some objections to Tappenden's approach and some objections to Supervaluationism. (...) Finally, I will present my own worries about Tappenden's account. (shrink)

Although it is eommon to attribute to Wittgenstein in the Tractatus a treatment of general propositions as equivalent to eonjunctions and disjunctions of instance propositions, the evidence for this is not perfeetly clear. This article considers Wittgenstein’s comments in 5.521, which can be read as rejecting such a treatment. It argues that properly situating the Tractatus historically allows for a revised reading of 5.521 and other parts of the Tractatus relevant to Wittgenstein’s theory of generality. The result is that 5.521 (...) does not conflict with the view that general propositions are truth-functions of instance propositions. Common problems with such a view are to some extent obviated by the fact that Wittgenstein, following Russell and Moore, was not concerned with a syntactically defined language, but with propositions conceived as independent of a fixed language. (shrink)

Although it is eommon to attribute to Wittgenstein in the Tractatus a treatment of general propositions as equivalent to eonjunctions and disjunctions of instance propositions, the evidence for this is not perfeetly clear. This article considers Wittgenstein’s comments in 5.521, which can be read as rejecting such a treatment. It argues that properly situating the Tractatus historically allows for a revised reading of 5.521 and other parts of the Tractatus relevant to Wittgenstein’s theory of generality. The result is that 5.521 (...) does not conflict with the view that general propositions are truth-functions of instance propositions. Common problems with such a view are to some extent obviated by the fact that Wittgenstein, following Russell and Moore, was not concerned with a syntactically defined language, but with propositions conceived as independent of a fixed language. (shrink)

Fibring has been shown to be useful for combining logics endowed withtruth-functional semantics. However, the techniques used so far are unableto cope with fibring of logics endowed with non-truth-functional semanticsas, for example, paraconsistent logics. The first main contribution of thepaper is the development of a suitable abstract notion of logic, that mayalso encompass systems with non-truth-functional connectives, and wherefibring can still be dealt with. Furthermore, it is shown that thisextended notion of fibring preserves completeness under certain reasonableconditions. This (...) completeness transfer result, the second main contributionof the paper, generalizes the one established in Zanardo et al. (2001) butis obtained using new techniques that explore the properties of a suitablemeta-logic (conditional equational logic) where the (possibly)non-truth-functional valuations are specified. The modal paraconsistentlogic of da Costa and Carnielli (1988) is studied in the context of this novel notionof fibring and its completeness is so established. (shrink)

Quine argued that any context allowing substitution of logical equivalents and coextensive terms is truth functional. We argue that Quine's proof for this claim is flawed.

We explore the technical details and historical evolution of Charles Peirce's articulation of a truth table in 1893, against the background of his investigation into the truth-functional analysis of propositions involving implication. In 1997, John Shosky discovered, on the verso of a page of the typed transcript of Bertrand Russell's 1912 lecture on ?The Philosophy of Logical Atomism? truth table matrices. The matrix for negation is Russell's, alongside of which is the matrix for material implication in the (...) hand of Ludwig Wittgenstein. It is shown that an unpublished manuscript identified as composed by Peirce in 1893 includes a truth table matrix that is equivalent to the matrix for material implication discovered by John Shosky. An unpublished manuscript by Peirce identified as having been composed in 1883?1884 in connection with the composition of Peirce's ?On the Algebra of Logic: A Contribution to the Philosophy of Notation? that appeared in the American Journal of Mathematics in 1885 includes an example of an indirect truth table for the conditional. (shrink)

The material interpretation of conditionals is commonly recognized as involving some paradoxical results. I here argue that the truth functional approach to natural language is the reason for the inadequacy of this material interpretation, since the truth or falsity of some pair of statements ‘p’ and ‘q’ cannot per se be decisive for the truth or falsity of a conditional relation ‘if p then q’. This inadequacy also affects the ability of the overall formal system to establish (...) whether or not arguments involving conditionals are valid. I also demonstrate that the Paradox of Indicative Conditionals does not actually involve a paradox, but instead contains some paralogistic elements that make it appear to be a paradox. The discussion of the paradox in this paper further reveals that the material interpretation of conditionals adversely affects the treatment of disjunctions. -/- Much has been said about these matters in the literature that point in the same direction. However, there seems to be some reluctance against fully complying with the arguments against the truth functional account of conditionals, since many of the alternative accounts rely on the material conditional, or at least on an understanding of the conditional as a function of antecedent and consequent in a similar sense as the material conditional. My argument against truth functionality indicates that it may in general involve similar problems to treat conditionals as such functions, whether one deals with theories of truth, assertability or probability. (shrink)

Motivated by aspects of reasoning in theories of physics, Robin Giles defined a characterization of infinite valued Łukasiewicz logic in terms of a game that combines Lorenzen-style dialogue rules for logical connectives with a scheme for betting on results of dispersive experiments for evaluating atomic propositions. We analyze this game and provide conditions on payoff functions that allow us to extract many-valued truth functions from dialogue rules of a quite general form. Besides finite and infinite valued Łukasiewicz logics, also (...) Meyer and Slaney’s Abelian logic and Cancellative Hoop Logic turn out to be characterizable in this manner. (shrink)

Peirce's example puts another nail in the coffin of the truth-functionality thesis. Conditionals are not truth-functional.

From Fine and Kamp in the 70’s—through Osherson and Smith in the 80’s, Williamson, Kamp and Partee in the 90’s and Keefe in the 00’s—up to Sauerland in the present decade, the objection continues to be run that fuzzy logic based theories of vagueness are incompatible with ordinary usage of compound propositions in the presence of borderline cases. These arguments against fuzzy theories have been rebutted several times but evidently not put to rest. I attempt to do so in this (...) paper. (shrink)

The aim of the present paper is to introduce a system, where the epistemic state of an agent is represented truth-functionally. In order to obtain this system, we propose a four-valued logic, that we call the logic of rational agent, where the fact of knowing something is formalized at the level of valuations, without the explicit use of epistemic knowledge operator. On the basis of this semantics, a sound and complete system with two distinct truth-functional negations is provided. (...) These negations allow us to express the statements about knowing or not knowing something at the syntactic level. Moreover, such a system is applied to the analysis of knowability paradox. In particular, we show that the paradox is not derivable in terms of the logic of rational agent. (shrink)

What is the fundamental insight behind truth-functionality ? When is a logic interpretable by way of a truth-functional semantics? To address such questions in a satisfactory way, a formal definition of truth-functionality from the point of view of abstract logics is clearly called for. As a matter of fact, such a definition has been available at least since the 70s, though to this day it still remains not very widely well-known. A clear distinction can be (...) drawn between logics characterizable through: (1) genuinely finite-valued truth-tabular semantics; (2) no finite-valued but only an infinite-valued truthtabular semantics; (3) no truth-tabular semantics at all. Any of those logics, however, can in principle be characterized through non-truth-functional valuation semantics, at least as soon as their associated consequence relations respect the usual tarskian postulates. So, paradoxical as that might seem at first, it turns out that truth-functional logics may be adequately characterized by non-truth-functional semantics . Now, what feature of a given logic would guarantee it to dwell in class (1) or in class (2), irrespective of its circumstantial semantic characterization? (shrink)

The expressive truth functions of two-valued logic have all been characterized, as have the expressive unary truth functions of finitely-many-valued logic. This paper introduces some techniques for identifying expressive functions in three-valued logics.

This paper aims to explain how the Tractatus attempts to unify logic by deriving the truth-functionality of logical necessity from the thesis that a proposition shows its sense. I first interpret the Tractarian notion of showing as the displaying of what is intrinsic to an expression (or a symbol). Then I argue that, according to the Tractatus, the thesis that a proposition shows its sense implies the determinacy of sense, the possibility of the complete elimination of non-primitive symbols, (...) the analyticity thesis and the strong analyticity thesis. The picture theory emerges as what provides the only acceptable account of an elementary proposition, subject to the constraint that a proposition must show its sense. The picture theory and the analyticity thesis then entail the contingency thesis (that an elementary proposition is contingent) and the independence thesis (that elementary propositions are mutually logically independent) which, together with the strong analyticity thesis, imply that all logical propositions are tautologies. (shrink)

I discuss Frege's argument - later called the slingshot - that if a construction is extensional and preserves logical equivalence then it is truth-functional. I consider some simple apparent counterexamples and conclude that they are not sentence-embedding in the required way.

The expressive truth functions of two-valued logic have all been identified. This paper begins the task of identifying the expressive truth functions of n-valued logic by characterizing the unary ones. These functions have distinctive algebraic, semantic, and closure-theoretic properties.

Although the truth value (falsity) of "Henry knows that (dogs live in trees and beavers chew wood)" remains unchanged no matter what sentence is substituted in it for "beavers chew wood", we want not to regard the second as a truth functional component (tfc) of the first. Many definitions of "tfc" (e.g., Quine's) fail to insure satisfaction of the following principle: if p is a component of r which is in turn a component of q, then p is (...) a tfc of q if and only if 1) p is also a tfc of r, and 2) r is also a tfc of q. (shrink)