We introduce a subclass of Kripke's fixed points in which falsehood is the preferred truth value. In all of these the truthteller evaluates to false, while the liar evaluates to undefined (or overdefined). The mathematical structure of this family of fixed points is investigated and is shown to have many nice features. It is noted that a similar class of fixed points, preferring truth, can also be studied. The notion of intrinsic is shown to relativize to these two subclasses. The (...) mathematical ideas presented here originated in investigations of so-called stable models in the semantics of logic programming. (shrink)

Two periods in the history of logic and philosophy are characterized notably by vivid interest in self-referential paradoxical sentences in general, and Liar sentences in particular: the later medieval period (roughly from the 12th to the 15th century) and the last 100 years. In this paper, I undertake a comparative taxonomy of these two traditions. I outline and discuss eight main approaches to Liar sentences in the medieval tradition, and compare them to the most influential modern approaches to such sentences. (...) I also emphasize the aspects of each tradition that find no counterpart in the other one. It is expected that such a comparison may point in new directions for future research on the paradoxes; indeed, the present analysis allows me to draw a few conclusions about the general nature of Liar sentences, and to identify aspects that would require further investigation. (shrink)

The complexity of the set of truths of arithmetic is determined for various theories of truth deriving from Kripke and from Gupta and Herzberger.

Gupta’s Rule of Revision theory of truth builds on insights to be found in Martin and Woodruff and Kripke in order to permanently deepen our understanding of truth, of paradox, and of how we work our language while our language is working us. His concept of a predicate deriving its meaning by way of a Rule of Revision ought to impact significantly on the philosophy of language. Still, fortunately, he has left me something to.

In this paper, we distinguish two versions of Curry's paradox: c-Curry, the standard conditional-Curry paradox, and v-Curry, a validity-involving version of Curry's paradox that isn’t automatically solved by solving c-curry. A unified treatment of curry paradox thus calls for a unified treatment of both c-Curry and v-Curry. If, as is often thought, c-Curry paradox is to be solved via non-classical logic, then v-Curry may require a lesson about the structure—indeed, the substructure—of the validity relation itself.

In a lengthy review article, C. Anthony Anderson criticizes the approach to property theory developed in Quality and Concept (1982). That approach is first-order, type-free, and broadly Russellian. Anderson favors Alonzo Church’s higher-order, type-theoretic, broadly Fregean approach. His worries concern the way in which the theory of intensional entities is developed. It is shown that the worries can be handled within the approach developed in the book but they remain serious obstacles for the Church approach. The discussion focuses on: (1) (...) the fine-grained/coarse-grained distinction, (2) proper names and definite descriptions, (3) the paradox of analysis and Mates’ puzzle, and (4) the logical, semantical, and intentional paradoxes. (shrink)

This paper aims to argue for two related statements: first, that formal semantics should not be conceived of as interpreting natural language expressions in a single model (a very large one representing the world as a whole, or something like that) but as interpreting them in many different models (formal counterparts, say, of little fragments of reality); second, that accepting such a conception of formal semantics yields a better comprehension of the relation between semantics and pragmatics and of the role (...) to be played by formal semantics in the general enterprise of understanding meaning. For this purpose, three kinds of arguments are given: firstly, empirical arguments showing that the many models approach is the most straightforward and natural way of giving a formal counterpart to natural language sentences. Secondly, logical arguments proving the logical impossibility of a single universal model. And thirdly, theoretical arguments to the effect that such a conception of formal semantics fits in a natural and fruitful way with pragmatic theories and facts. In passing, this conception will be shown to cast some new light on the old problems raised by liar and sorites paradoxes. (shrink)

Logic: the Basics is an accessible introduction to the core philosophy topic of standard logic. Focussing on traditional Classical Logic the book deals with topics such as mathematical preliminaries, propositional logic, monadic quantified logic, polyadic quantified logic, and English and standard ‘symbolic transitions’. With exercises and sample answers throughout this thoroughly revised new edition not only comprehensively covers the core topics at introductory level but also gives the reader an idea of how they can take their knowledge further and the (...) philosophical questions around logic. -/- Logic: the Basics is essential reading for first-year undergraduate philosophy students on standard introductory logic courses. (shrink)

§1. Introduction. When truth-theoretic paradoxes are generated, two factors seem to be at play: the behaviour that truth intuitively has; and the facts about which singular terms refer to which sentences, and so on. For example, paradoxicality might be partially attributed to the contingent fact that the singular term, "the italicized sentence on page one", refers to the sentence, The italicized sentence on page one is not true. Factors of this second kind might be represented by a ground model: an (...) interpretation of all the names, function symbols, and predicates in the potentially self-referential language under study, with the exception of the predicate "x is true". Formally, suppose that L is an uninterpreted first order language. M = 〈D, I〉 is a classical model for L if D is a nonempty set; and I is a function assigning to each name of L a member of D, to each n-place function symbol of L a function from Dn to D, and to each nonlogical n-place predicate of L a function from Dn to {t, f}. Suppose furthermore that L+ is obtained by adding a new one-place predicate T to L, and that L has a quote name ‘A’ for each sentence A of L+. We follow Gupta and Belnap [5] in defining S =df {A: A is a sentences of L+}. A classical model M = 〈D, I〉 for L is a ground model for L iff both S ⊆ D, and I(‘A’) = A for each A ∈ S. A classical ground model for L is a representation of the supposedly unproblematic fragment of L+: a representation of which terms refer to which objects, and of which objects have which nonsemantic properties. (shrink)

The paper explores applications of Kripke's theory of truth to semantics for anti-luck epistemology, that is, to subjunctive theories of knowledge. Subjunctive theories put forward modal or subjunctive conditions to rule out knowledge by mere luck as to be found in Gettier-style counterexamples to the analysis of knowledge as justified true belief. Because of the subjunctive nature of these conditions the resulting semantics turns out to be non-monotone, even if it is based on non-classical evaluation schemes such as strong Kleene (...) or FDE. This blocks the usual road to fixed-point results for Kripke's theory of truth within these semantics and consequently the paper is predominantly an exploration of fixed point results for Kripke's theory of truth within non-monotone semantics. Using the theory of quasi-inductive definitions we show that in case of the subjunctive theories of knowledge the so-called Kripke jump will have fixed points despite the non-monotonicity of the semantics: Kripke's theory of truth can be successfully applied in the framework of subjunctive theories of knowledge. (shrink)

Philip Kremer, Department of Philosophy, McMaster University Note: The following version of this paper does not contain the proofs of the stated theorems. A longer version, complete with proofs, is forthcoming. §1. Introduction. In "The truth is never simple" and its addendum, Burgess conducts a breathtakingly comprehensive survey of the complexity of the set of truths which arise when you add a truth predicate to arithmetic, and interpret that predicate according to the fixed point semantics or the revision-theoretic semantics for (...) languages expressing their own truth concepts. Burgess considers various sets that can be said to represent truth in this context, and shows that their complexity ranges from Π11 or Σ11 to Π12 or Σ12. Thus, enriching arithmetic with a truth predicate increases its complexity, which is otherwise only ∆11. (shrink)