Yablo's paradox, a coinductive language and its semantics

The paper describes properties of Yablo sequences over growing domains of finite arithmetical models and over partial models of Kripke truth theory. We show that for any partial fixed-point model and for the Strong Kleene, Weak Kleene and Supervaluation valuation schema, all Yablo sentences Y (n) are neither true nor false under these schema or equivalently: the truth-value of all Yablo sentences Y (n) in fixed-point partial models under any of the above valuation scheme, is indeterminate. Furthermore, we show that under the logic of sufficiently large finite models (logic of potential infinity) all the Yablo sentences are false in the limit. The main philosophical conclusion is that a finitist, not accepting the notion of actual infinity may adopt the sl-theory of arithmetics as an adequate explication of the concept of potential infinity and simply give an answer to the Yablo paradox that in the limit (i.e. under the sl-semantics on the FM-domain of a given arithmetical model) all the Yablo sentences are false.

Lecture Notes in Computer Science, 2004

We define new, both model-theoretical and fixpoint-based, characterizations of the well-founded semantics for logic programs in the general setting of bilattices. This work lights the role of the CWA, used in the well-founded semantics as a carrier of falsehood, and shows that the definition of that semantics does not require any separation of positive and negative information nor any program transformation.

En este artículo describiremos brevemente el sistema que en lógica es conocido como lógica IF (lógica amigable con la independencia) y que fue introducido por Hintikka y Sandu en 1989. Es conocido que esta lógica tiene enunciados que son indeterminados. Tras esto, mostraremos cómo resolver la indeterminación de sus enunciados aplicando el teorema Minimax de von Neumann. Este artículo se basa en gran medida en [Sevenster y Sandu ], [Mann, Sandu, y Sevenster (2011)], [Sandu (2012)], [Sandu (en prensa)], and [Barbero and Sandu (en prensa)].

We review three pairwise similar paradoxes, the modest liar paradox, McGee’s paradox and Yablo’s paradox, which imply the ω- inconsistency. We show that is caused by the fact that co-inductive def- initions of formulae are possible because of the existence of the truth predicate.

Annals of Pure and Applied Logic, 1992

Blass, A., A game semantics for linear logic, Annals of Pure and Applied Logic 56 (1992) 183-220. We present a game (or dialogue) semantics in the style of Lorenzen (1959) for Girard's linear logic (1987). Lorenzen suggested that the (constructive) meaning of a proposition 91 should be specified by telling how to conduct a debate between a proponent P who asserts p and an opponent 0 who denies q. Thus propositions are interpreted as games, connectives (almost) as operations on games, and validity as existence of a winning strategy for P. (The qualifier 'almost' will be discussed later when more details have been presented.) We propose that the connectives of linear logic can be naturally interpreted as the operations on games introduced for entirely different purposes by Blass (1972). We show that affine logic, i.e., linear logic plus the rule of weakening, is sound for this interpretation. We also obtain a completeness theorem for the additive fragment of affine logic, but we show that completeness fails for the multiplicative fragment. On the other hand, for the multiplicative fragment, we obtain a simple characterization of game-semantical validity in terms of classical tautologies. An analysis of the failure of completeness for the multiplicative fragment leads to the conclusion that the game interpretation of the connective @ is weaker than the interpretation implicit in Girard's proof rules; we discuss the differences between the two interpretations and their relative advantages and disadvantages. Finally, we discuss how Godel's Dialectica interpretation (1958), which was connected to linear logic by de Paiva (1989) fits with game semantics.

Lan Wen, a Chinese mathematician, recently published his monograph to solve the semantic paradoxes, which provides the insight of the corresponding relation between the paradox argument and the proof of the unsolvability of certain equation systems in Boolean Algebra. In this essay, after introducing the main idea of his proposal, we present the Tarski Undefinability of Truth (TUT) theorem in algebraic semantics, then under a certain translation, the conclusion of which can be understood as a prerequisite of Wen's proposal and his proposal can also be expressed by the Paracomplete proposal. Finally, we indicate that Wen's proposal is by no means retaining the full classical logic to resolve semantic paradoxes, and we argue that, although Wen provides an insightful analysis of the algebraic structure of paradoxes, it is still insufficient from both the view of solving semantic paradoxes and the theory of truth.

Advances in Modal Logic, 2020

In "A Problem in Possible-World Semantics," David Kaplan presented a consistent and intelligible modal principle that cannot be validated by any possible world frame (in the terminology of modal logic, any neighborhood frame). However, Kaplan's problem is tempered by the fact that his principle is stated in a language with propositional quantification, so possible world semantics for the basic modal language without propositional quantifiers is not directly affected, and the fact that on careful inspection his principle does not target the world part of possible world semantics—the atomicity of the algebra of propositions—but rather the idea of propositional quantification over a complete Boolean algebra of propositions. By contrast, in this paper we present a simple and intelligible modal principle, without propositional quantifiers, that cannot be validated by any possible world frame precisely because of their assumption of atomicity (i.e., the principle also cannot be validated by any atomic Boolean algebra expansion). It follows from a theorem of David Lewis that our logic is as simple as possible in terms of modal nesting depth (two). We prove the consistency of the logic using a generalization of possible world semantics known as possibility semantics. We also prove the completeness of the logic (and two other relevant logics) with respect to possibility semantics. Finally, we observe that the logic we identify naturally arises in the study of Peano Arithmetic.

2016

The traditional representation of an epistemic scenario as a model covers only complete descriptions that specify truth values of all assertions. However, many, perhaps most, epistemic scenarios are not complete and allow partial or asymmetric knowledge. Syntactic Epistemic Logic, SEL, suggests viewing an epistemic situation as a set of syntactic conditions rather than as a model, thus also capturing incomplete descriptions. This helps to extend the scope of Epistemic Game Theory. In addition, SEL closes the conceptual and technical gap, identified by R. Aumann, between the syntactic character of game descriptions and the semantic method of analyzing games.

Between Logic and Reality, 2011

A new version of Game-Theoretical Semantics (GTS) is put forward where game rules are extended to the non-logical constants of sentences. The resulting theory, together with a refinement of our criteria of identity for functions, provide the technical basis for a game-based conception of linguistic meaning and interpretation.

Annals of Pure and Applied Logic, 1997

I present a semantics for the language of first order additive-multiplicative linear logic, i.e. the language of classical first order logic with two sorts of disjunction and conjunction. The semantics allows us to capture intuitions often associated with linear logic or constructivism such as sen-tences=games, sentences=resources or sentences=problems, where "truth" means existence of an effective winning (resource-using, problem-solving) strategy. The paper introduces a decidable first order logic ET in the above language and gives a proof of its soundness and completeness (in the full language) with respect to this semantics. Allowing noneffective strategies in the latter is shown to lead to classical logic. The semantics presented here is very similar to Blass's game semantics (A.Blass, "A game semantics for linear logic", APAL, 56). Although there is no straightforward reduction between the two corresponding notions of validity, my completeness proof can likely be adapted to the logic induced by Blass's semantics to show its decidability (via equality to ET), which was a major problem left open in Blass's paper. The reader needs to be familiar with classical (but not necessarily linear) logic and arithmetic.