We develop a bottom-up approach to truth-value semantics for classical logic of partial terms based on equality and apply it to prove the conservativity of the addition of partial description and selection functions, independently of any strictness assumption.

Presupposition, vagueness, and oddness can lead to some sentences failing to have a clear truth value. The homogeneity property of plural predication with definite descriptions may also create truth-value gaps: The books are written in Dutch is true if all relevant books are in Dutch, false if none of them are, and neither true nor false if, say, half of the books are written in Dutch. We study the projection property of homogeneity by deploying methods of (...) general interest to identify truth-value gaps. Method A consists in collecting both truth judgments and, independently, falsity judgments. The second method, employed in experiment series B and C, is based on one-shot ternary judgments: completely true vs. completely false vs. neither. After a calibration of these methods, we use them to demonstrate that homogeneity projects out of negation, the scope of universal sentences and the scope of non-monotonic quantifiers such as exactly two, to some extent. We assess our results in light of different theoretical approaches to homogeneity—approaches based on presuppositions, scalar implicatures, and something like supervaluations, respectively. We identify free parameters in these theories and assess various variants of them based on our results. Our experimental paradigms may be of broader significance insofar as they can be applied to other phenomena which result in the failure of a sentence to have a definite truth value. (shrink)

Kuhn's alleged taxonomic interpretation of incommensurability is grounded on an ill defined notion of untranslatability and is hence radically incomplete. To supplement it, I reconstruct Kuhn's taxonomic interpretation on the basis of a logical-semantic theory of taxonomy, a semantic theory of truth-value, and a truth-value conditional theory of cross-language communication. According to the reconstruction, two scientific languages are incommensurable when core sentences of one language, which have truth values when considered within its own context, lack (...) truth values when considered within the context of the other due to the unmatchable taxonomic structures underlying them. So constructed, Kuhn's mature interpretation of incommensurability does not depend upon the notion of truth-preserving translatability, but rather depends on the notion of truth-value-status-preserving cross-language communication. The reconstruction makes Kuhn's notion of incommensurability a well grounded, tenable and integrated notion.Author Keywords: Incommensurability; Thomas Kuhn; Taxonomic structures; Lexicons; Truth-value; Untranslatability; Cross-language communication. (shrink)

We introduce a general recipe for presenting non-classical logics in a modular and uniform way as labelled deduction systems. Our recipe is based on a labelling mechanism where labels are general entities that are present, in one way or another, in all logics, namely truth-values. More specifically, the main idea underlying our approach is the use of algebras of truth-values, whose operators reflect the semantics we have in mind, as the labelling algebras of our labelled deduction systems. The (...) “truth-values as labels” approach allows us to give generalized systems for multiple-valued logics within the same formalism: since we can take multiple-valued logics as meaning not only finitely or infinitely many-valued logics but also power-set logics, i.e. logics for which the denotation of a formula can be seen as a set of worlds, our recipe allows us to capture also logics such as modal, intuitionistic and relevance logics, thus providing a first step towards the fibring of these logics with many-valued ones. (shrink)

This paper sheds light on the relationship between the logic of generalized truth values and the logic of bilattices. It suggests a definite solution to the problem of axiomatizing the truth and falsity consequence relations, \ and \ , considered in a language without implication and determined via the truth and falsity orderings on the trilattice SIXTEEN 3 . The solution is based on the fact that a certain algebra isomorphic to SIXTEEN 3 generates the variety of (...) commutative and distributive bilattices with conflation. (shrink)

After summarizing approaches taken previously to the problem of the determination of truth-Value of statements constituting literary works (ryle, Russell, Quine, Strawson, Davidson, Et. Al.) the author argues for a treatment of the problem based upon the linguistic status of such statements as translocutions and their relation to the context in which they occur.

Since the famous debate between Russell (Mind 14: 479–493, 1905, Mind 66: 385–389, 1957) and Strawson (Mind 59: 320–344, 1950; Introduction to logical theory, 1952; Theoria, 30: 96–118, 1964) linguistic intuitions about truth values have been considered notoriously unreliable as a guide to the semantics of definite descriptions. As a result, most existing semantic analyses of definites leave a large number of intuitions unexplained. In this paper, I explore the nature of the relationship between truth value intuitions (...) and non-referring definites. Inspired by comments in Strawson (Introduction to logical theory, 1964), I argue that given certain systematic considerations, one can provide a structured explanation of conflicting intuitions. I show that the intuitions of falsity, which proponents of a Russellian analysis often appeal to, result from evaluating sentences in relation to specific questions in context. This is shown by developing a method for predicting when sentences containing non-referring definites elicit intuitions of falsity. My proposed analysis draws importantly on Roberts (in: Yoon & Kathol (eds.) OSU working papers in Linguistics: vol. 49: Papers in Semantics 1998; in: Horn & Ward (eds.) Handbook of pragmatics, 2004) and recent research in the semantics and pragmatics of focus. (shrink)

This paper examines the concept of ‘areti’ as encountered in the Aristotelian ethical system in order to establish its relationship to the modern concept of virtue as well as to that of moral truth, that is, to identify its truth-value. I intend to show that the Aristotelian ‘areti’ as a developed state of character and as an advanced stage of ethical understanding entails moral truth. ‘Areti’ as a good-in-itself possesses an intrinsic value which reflects moral (...) truth, and as a means for the accomplishment of ‘eudaimonia’ (ultimate happiness) it possesses an instrumental value. I also wish to argue that this position calls for a realist as well as an objectivist (or nonrelativist) approach in Aristotle. To that effect, I examine the elements of ‘areti’ that relate it to truth, and then I use reference to some of theAristotelian virtues, such as ‘andreia’ (courage), ‘philia’ (friendship), ‘dikaiosyne’ (justice), and ‘megalopsychia’ (magnanimity), in order to examine the way moral truth functions. This examination will also try to show that Aristotle’s aretaic approach does not suffer from the ills of virtue ethics theories. (shrink)

The first section (§1) of this essay defends reliance on truth values against those who, on nominalistic grounds, would uniformly substitute a truth predicate. I rehearse some practical, Carnapian advantages of working with truth values in logic. In the second section (§2), after introducing the key idea of auxiliary parameters (§2.1), I look at several cases in which logics involve, as part of their semantics, an extra auxiliary parameter to which truth is relativized, a parameter that (...) caters to special kinds of sentences. In many cases, this facility is said to produce truth values for sentences that on the face of it seem neither true nor false. Often enough, in this situation appeal is made to the method of supervaluations, which operate by “quantifying out” auxiliary parameters, and thereby produce something like a truth value. Logics of this kind exhibit striking differences. I first consider the role that Tarski gives to supervaluation in first order logic (§2.2), and then, after an interlude that asks whether neither-true-nor-false is itself a truth value (§2.3), I consider sentences with non-denoting terms (§2.4), vague sentences (§2.5), ambiguous sentences (§2.6), paradoxical sentences (§2.7), and future-tensed sentences in indeterministic tense logic (§2.8). I conclude my survey with a look at alethic modal logic considered as a cousin (§2.9), and finish with a few sentences of “advice to supervaluationists” (2.10), advice that is largely negative. The case for supervaluations as a road to truth is strong only when the auxiliary parameter that is “quantified out” is in fact irrelevant to the sentences of interest—as in Tarski’s definition of truth for classical logic. In all other cases, the best policy when reporting the results of supervaluation is to use only explicit phrases such as “settled true” or “determinately true,” never dropping the qualification. (shrink)

∗Thanks to J. C. Beall, Alex Byrne, Jason Decker, Tyler Doggett, Paul Elbourne, Adam Elga, Warren Goldfarb, Delia Graff, Richard Heck, Charles Parsons, Mark Richard, Susanna Siegel, Jason Stanley, Judith Thomson, Carol Voeller, Brian Weatherson, Ralph Wedgwood, Steve Yablo, Cheryl Zoll, and an anonymous referee for valuable comments and discussions. Versions of this material were presented in my seminar at MIT in the Fall of 2000, and at the University of Maryland, Baltimore County. Parts of this paper also derive from (...) my comments on a paper of Scott Soames at the ‘Liars and Heaps’ conference at the University of Connecticut in the Fall of 2002. I am grateful for the help of these audiences, and especially to Prof. Soames. (shrink)

I present an account of truth values for classical logic, intuitionistic logic, and the modal logic s5, in which truth values are not a fundamental category from which the logic is defined, but rather, an idealisation of more fundamental logical features in the proof theory for each system. The result is not a new set of semantic structures, but a new understanding of how the existing semantic structures may be understood in terms of a more fundamental notion of (...) logical consequence. (shrink)

We explore a possibility of generalization of classical truth values by distinguishing between their ontological and epistemic aspects and combining these aspects within a joint semantical framework. The outcome is four generalized classical truth values implemented by Cartesian product of two sets of classical truth values, where each generalized value comprises both ontological and epistemic components. This allows one to define two unary twin connectives that can be called “semi-classical negations”. Each of these negations deals only (...) with one of the above mentioned components, and they may be of use for a logical reconstruction of argumentative reasoning. (shrink)

After some generalities about connections between functions and relations in Sections 1 and 2 recalls the possibility of taking the semantic values of ‐ary Boolean connectives as ‐ary relations among truth‐values rather than as ‐ary truth functions. Section 3, the bulk of the paper, looks at correlates of these truth‐value relations as applied to formulas, and explores in a preliminary way how their properties are related to the properties of “logical relations” among formulas such as equivalence, (...) implication (entailment) and contrariety (logical incompatibility), concentrating for illustrative purposes on binary logical relations such as those just listed. To avoid an excess of footnotes, some points have been deferred to an Appendix as “Longer Notes”. (shrink)

Central to any form of Deflationism concerning truth (hereafter ‘DT’) is the claim that truth has no substantial theoretical role to play. For this reason, DT faces the following immediate challenge: if truth can play no substantial theoretical role then how can we model various prevalent kinds of indeterminacy—such as the indeterminacy exhibited by vague predicates, future contingents, liar sentences, truth-teller sentences, incomplete stipulations, cases of presupposition failure, and such-like? It is too hasty to assume that (...) these phenomena are all to be modelled via some epistemic conception of indeterminacy whereby indeterminacy is just some special species of ignorance which arises because of our limited powers of discrimination. Some non-epistemic model is called for—at least for certain species of indeterminacy. On what is perhaps the most enduring and popular non-epistemic model, indeterminacy gives rise to truth-value gaps. But is DT compatible with the possibility of truth-value gaps? Compatibilism says Yes; Incompatibilism says No. The broad goal of this paper is to defend a form of Incompatibilism. If DT is to make sense of various kinds of indeterminacy then truth-value gaps cannot be invoked to do so. The particular goals of this paper are: (i) To set forth a new form of Compatibilism which can address an argument against truth-value gaps given by Williamson (1994, pp. 187-192). (ii) To offer a new argument against truth-value gaps using principles entailed by DT, thereby undermining Compatibilism. (shrink)

In this paper, some qualitative scientific laws are treated in a way that is analogous to the method by which Karl Menger has clarified the nature of quantitative laws such as Boyle's law about ideal gases. The qualitative analogue of the number-valued fluents, such as temperature, are fluents whose domains consist of physical objects while their values are T and F (true and false).

There is strong disagreement about whether indicative conditionals have truth values. In this paper, I present a new argument for the conclusion that indicative conditionals have truth values based on the claim that some true statements entail indicative conditionals. I then address four arguments that conclude that indicative conditionals lack truth values, showing them to be inadequate. Finally, I present further benefits to having a worldly view of conditionals, which supports the assignment of truth values to (...) indicative conditionals. I conclude that certain types of account of indicative conditionals, which have been ignored in the literature partly on the basis of assigning truth values to indicative conditionals, deserve consideration. (shrink)

"Plato on Truth-Value and Truth-Aptness" examines Plato’s conception of truth-value and truth-aptness. The examination focuses on Philebus 36c3-50e4 where Socrates argues that pleasures can be true and false and more precisely that there are various kinds of true and false pleasures. The Philebus passage is the only one in Plato’s corpus where various kinds of truth, falsity, and truth-aptness are examined in close proximity and in relation to one another. Hence it is (...) an especially valuable and, with respect to the topics treated in this paper, neglected site. Socrates distinguishes four kinds of true and false pleasure, which I examine sequentially. In doing so, I argue that Plato distinguishes two kinds of representational truth-value and truth-aptness, propositional and non-propositional respectively, and two kinds of ontic truth-value and truth-aptness, absolute and gradable respectively. On the basis of a key passage at Philebus 42c5-7 the discussion concludes, suggestively but aporetically, with a consideration of how these various kinds of truth-value and truth-aptness are related to one another. (shrink)

I present an account of truth values for classical logic, intuitionistic logic, and the modal logic S5, in which truth values are not a fundamental category from which the logic is defined, but rather, an idealisation of more fundamental logical features in the proof theory for each system. The result is not a new set of semantic structures, but a new understanding of how the existing semantic structures may be understood in terms of a more fundamental notion of (...) logical consequence. (shrink)

Truth-value gaps have received little attention from a foundational perspective, a fact which has rightfully opened up gap theories to charges of vacuousness. This paper develops an account of the foundations of gap-like behavior which has some hope of avoiding such charges. I begin by reviewing and sharpening a powerful argument of Dummett’s to constrain the options that gap theorists have to make sense of their views. I then show that within these strictures, we can give an account (...) of gaps by drawing on elements of a broadly Stalnakerian framework for assertion and using gaps to track an amalgamation of assertoric effects. The discussion reveals that we may need special resources in our theories of assertion to posit gaps, that gaps may be unusable in characterizing the structure of mental states, and that gaps may have heterogeneous linguistic sources that result in equally heterogeneous projective and inferential behavior. (shrink)

Truth values are, properly understood, merely proxies for the various relations that can hold between language and the world. Once truth values are understood in this way, consideration of the Liar paradox and the revenge problem shows that our language is indefinitely extensible, as is the class of truth values that statements of our language can take – in short, there is a proper class of such truth values. As a result, important and unexpected connections emerge (...) between the semantic paradoxes and the set-theoretic paradoxes. (shrink)

Philosophers like to talk about propositions. There are many reasons for this. Perhaps the most common is that philosophers are sometimes more interested in the content of a thought or utterance than in the particular sentence or utterance that might express it on some occasion. Propositions are offered as these contents.

In some presentations of classical and intuitionistic logics, the objectlanguage is assumed to contain (two) truth-value constants: ⊤ (verum) and ⊥ (falsum), that are, respectively, true and false under every bivalent valuation. We are interested to define and study analogical constants ‡, 1 ≤ i ≤ n, that in an arbitrary multi-valued logic over truth-values V = {v1,..., vn} have the truth-value vi under every (multi-valued) valuation. As is well known, the absence or presence of (...) such constants has a significant deductive impact on the logics studied. We define such constants proof-theoretically via their associated I/E-rules in a natural-deduction proof system. In particular, we propose a generalization of the notions of contradiction and explosiveness of a logic to the context of multi-valued logics. (shrink)

For a very long time truth-value gaps were under a cloud of suspicion because they were considered logically unmanageable. So Frege declared that:“as regards concepts we have a requirement of sharp delimitation; if this were not satisfied it would be impossible to set forth logical laws about them”.Three-valued logic promised to dispel the cloud but in the eyes of many it had promised more than it could deliver. So in response to Reichenbach's plea for a three-valued quantum logic (...) Russell replied:“This is one of a number of questions as to which I am prevented from accepting a certain view by difficulties in carrying it out, but am prepared to alter my opinion if technical skill supplies an answer to my difficulties”.‘By now technical skill has supplied many alternative systems of three-valued logic, but few answers and numerous additional difficulties.’. (shrink)

Philosophers like to talk about propositions. There are many reasons for this. Perhaps the most common is that philosophers are sometimes more interested in the content of a thought or utterance than in the particular sentence or utterance that might express it on some occasion. Propositions are offered as these contents.

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. (shrink)

Can mystics intuit something of what modern physicists calculate? And if so, how? The question of the relation between the classical mysticisms and modern science is approached in Part I in terms of the multiple forms and definitions of 'truth value'. Intuition/epiphany, pragmatism, coherence, and correspondence are considered as forms of truth that have also been proposed for unitive mystical experience. Since 'correspondence' or 'representation' has been the definition at the core of modern science, it in particular (...) is approached by combining Lakoff and Johnson on the roots of all abstract knowledge in physical metaphor and Gibson on the ecological array of perception as the template from which such metaphors must be drawn. A series of similarities between the maximally inclusive metaphors underlying unitive mystical experience and cosmological physics, as abstracted from an ecological array already resonant with multiple levels of physical reality, suggests a basis for the correspondence between these otherwise distinct cognitive domains. Part II extends this approach to widely posited cross cultural values of spiritual or mystical realization, in terms of gratitude, compassion, faith, and inner freedom. When fully embodied as forms of personal conduct, these are also ultimately derivable from the openness and flow patternings of the ecological array itself. The possibility that the more abstract spiritual values of the major world mysticisms exemplify not only revelatory and pragmatic forms of truth, but are also broadly correspondent with both the principles of immediate concrete perception and modern physics, may eventually allow a contemporary re- formulation of the microcosm-macrocosm unities basic to traditional cultures. (shrink)

According to the accepted translation-failure interpretation, the problem of incommensurability involves the nature of the meaning-referential relation between scientific languages. The incommensurability thesis is that some competing scientific languages are mutually untranslatable due to the radical variance of meaning or/and reference of the terms they employ. I argue that this interpretation faces many difficulties and cannot give us a tenable, coherent, and integrated notion of incommensurability. It has to be rejected. ;On the basis of two case studies, I find that (...) the confrontations between many classical incommensurable languages are not confrontations between two untranslatable languages with different distribution of truth values, but rather the confrontations between incompatible fundamental presuppositions at the ontological level. We can always identify a truth-value gap between two incommensurable languages. Such a truth-value gap indicates a communication breakdown between the two language communities on the one hand, and is caused by the incompatible fundamental presuppositions underlying them on the other. ;I thereby identify the truth-value functional relationship between sentences, instead of the meaning-referential relationship between terms, as the dominant semantic relation between two incommensurable languages. According to my presuppositional interpretation of incommensurability, the real secret of incommensurability lies in the ontological setup of two competing presuppositional languages. When two presuppositional languages with incompatible factual commitments encounter with each other, the confrontation leads to a truth-value gap, and consequently a communication breakdown between them. Formally put, two scientific languages are incommensurable when core sentences of one language, which have truth values when considered within its own context, lack truth values when considered within the context of the other due to an ontological gap between them. ;The presuppositional interpretation makes many significant contributions to the discussion of the issue of incommensurability and the related metaphysical and epistemological issues: It confirms the existence of the phenomenon of incommensurability and makes it metaphysically and epistemologically significant. It establishes the tenability and integrity of the notion of incommensurability. It avoids many alleged unattractive epistemological and metaphysical consequences of the translation-failure interpretation. (shrink)

The purpose of the paper is to refute the fregean assumption that declarative sentences refer to truth-Values. A consequence of the assumption is that the truth-Value of a declarative sentence containing another as part remains unchanged when the part is replaced by another sentence having the same truth-Value, Provided that the part as part has only customary reference and expresses a complete thought. The refutation proceeds by demonstrating this consequence to be false.

The paper maintains that Frege's quantification of sentence positions motivates his identification of sentences as proper names of truth-values; that this identification is fully compatible with the Context Principle; that the relation of a thought to its truth-value is the primary case of the relation of sense to meaning. The paper offers a reconstruction of Frege's defense of in pp. 33-5 of “On Sense and Meaning”O artigo sustenta que a quantificação Fregeana sobre posições de sentença motiva sua (...) identificação de sentenças como nomes próprios de valores de verdade; que esta identificação é inteiramente compatível com o Princípio do Contexto; que a relação de um pensamento com o seu valor de verdade é o caso primário da relação do sentido para com o significado. O artigo oferece uma reconstrução da defesa de Frege de nas pp. 33-5 de “Sobre o Sentido e o Significado”. (shrink)

The author recently claimed that Pr(P, Q), where Pr is a probability function and P and Q are two sentences of a formalized language L, qualifies as an estimate--made in the light of Q--of the truth-value of P in L. To substantiate his claim, the author establishes here that the two strategies lying at the opposite extremes of the spectrum of truth-value estimating strategies meet the first five of the six requirements (R1-R6) currently placed upon probability (...) functions and fail to meet the sixth one. He concludes from those two results that the value for P and Q of any function satisfying R1-R5 must rate "minimally satisfactory" and the value for P and Q of any function satisfying R1-R6 must rate "satisfactory" as an estimate--made in the light of Q--of the truth-value of P in L. (shrink)

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. (shrink)

Various four- and three-valued modal propositional logics are studied. The basic systems are modal extensions BK and BS4 of Belnap and Dunn's four-valued logic of firstdegree entailment. Three-valued extensions of BK and BS4 are considered as well. These logics are introduced semantically by means of relational models with two distinct evaluation relations, one for verification and the other for falsification. Axiom systems are defined and shown to be sound and complete with respect to the relational semantics and with respect to (...) twist structures over modal algebras. Sound and complete tableau calculi are presented as well. Moreover, a number of constructive non-modal logics with strong negation are faithfully embedded into BS4, into its three-valued extension B3S4, or into temporal BS4, BtS4. These logics include David Nelson's three-valued logic N3, the four-valued logic N4 bottom, the connexive logic C, and several extensions of bi-intuitionistic logic by strong negation. (shrink)

The question is asked whether one can consistently both be a minimalist about truth, and hold that some meaningful assertoric sentences fail to be either true or false. It is shown that one can, but the issues are delicate, and the price is high: one must either refrain from saying that the sentences lack truth values, or else one must invoke a novel non-contraposing three-valued conditional. Finally it is shown that this does not help in reconciling minimalism with (...) emotivism, where this latter is understood as involving the view that ethical sentences are neither true nor false. (shrink)

Antirealism about the past is apparently in conflict with our acceptance of a set of systematic linkages between the truth-values of differently tensed sentences made at different times. Arguments based on acceptance of these so-called truth-value links seem to show that fully accounting for our use of the past and future tenses will involve use of a notion of truth which is not epistemically constrained and is thus antirealistically unacceptable. I elaborate these difficulties through an examination (...) of work by Dummett and Wright. I agree with Wright's rejection of Dummett's proposal but go on to argue that Wright's account fares no better. Building on what I take to be the failure of Wright's account, I offer a solution for the antirealist which diagnoses the problem as stemming from an equivocation in the meanings assigned to the tensed truth predicates. I close by raising a different problem for antirealist accounts of the past, one which is, however, related to more general difficulties for antirealist theories of meaning. (shrink)

Misunderstanding occurs between speakers when they disagree about the meaning of words in use. In the case of truth-values, Frege took these to be referents of sentences which consist of classes of accepted (i.e. “true”) or rejected (i.e. “false”) sentences. From this usual depiction of truth and falsity, a general algebraic framework is proposed to systematize the use of truth-values from a dialogical point of view of logic. A special attention will be paid to two radically opposed (...) pseudo-speakers: Heraclites and Nagarjuna, according to whom truth-values respectively refer to everything or nothing. Finally, dialectical (pseudo-Hegelian) negation will be rendered as a very special function: an ontological object-forming operator, similar to the arithmetic successor-forming operator Sn+1(x) on integers and consistent within our generalized theory of truth-values as variable referents. (shrink)