On Semantic Games for Łukasiewicz Logic

In Philosophical Logic, the Liar Paradox has been used to motivate the introduction of both truth value gaps and truth value gluts. Moreover, in the light of "revenge Liar" arguments, also higher-order combinations of generalized truth values have been suggested to account for so-called hyper-contradictions. In the present paper, Graham Priest's treatment of generalized truth values is scrutinized and compared with another strategy of generalizing the set of classical truth values and defining an entailment relation on the resulting sets of higher-order values. This method is based on the concept of a multilattice. If the method is applied to the set of truth values of Belnap's "useful four-valued logic", one obtains a trilattice, and, more generally, structures here called Belnap-trilattices. As in Priest's case, it is shown that the generalized truth values motivated by hyper-contradictions have no effect on the logic. Whereas Priest's construction in terms of designated truth values always results in his Logic of Paradox, the present construction in terms of truth and falsity orderings always results in First Degree Entailment. However, it is observed that applying the multilattice-approach to Priest's initial set of truth values leads to an interesting algebraic structure of a "bi-and-a-half" lattice which determines seven-valued logics different from Priest's Logic of Paradox.

In the present chapter, we discuss the possibility of generalizing the very notion of a truth value by constructing truth values as complex units which may possess a ramified inner structure. We consider some approaches to truth values as structured entities and summarize this point in the notion of a generalized truth value conceived as a subset of some basic set of initial truth values of a lower degree. We are essentially led by the idea which is at the heart of Belnap and Dunn’s useful four-valued logic, where the set \({{\mathbf{2}}} = \{T,F\}\) of classical truth values is generalized to the set \({{\mathbf{4}}} = {{\fancyscript{P}}}({{\mathbf{2}}}) = \{\varnothing,\{T\},\{F\},\{T,F\}\}.\) We argue in favor of extending this process to the set \({{\mathbf{16}}} = {{\fancyscript{P}}}({{\mathbf{4}}}).\) It turns out that this generalization is well-motivated and leads to a notion of a truth value multilattice. In particular, we proceed from the bilattice \(FOUR_{2}\) with both an information and truth-and-falsity ordering to another algebraic structure, namely the trilattice \(SIXTEEN_{3}\) with an information ordering together with a truth ordering \(and\) a (distinct) falsity ordering. We also consider another exemplification of essentially the same structure based on the set of truth values one can find in various constructive logics.

In the article [4], a new constructive set theoretic principle called Refinement was introduced and analysed. While it seemed to be significantly weaker than its alternative, the more established axiom of Fullness (a constructive version of the Powerset axiom from classical set theory), it was shown to suffice to imply many of the mathematically important consequences. In this article, we will define for each set A a set of truth values which measures the complexity of the equality relation on A. Using these sets we will show that Refinement is actually equivalent to Fullness on the basis of the other axioms of constructive Zermelo‐Fraenkel set theory (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Normal forms for fuzzy relations and their contribution to universal approximation

It is shown that the infinite-valued first-order Godel logic G° based on the set of truth values {1/k: k e w {0}} U {0} is not r.e. The logic G° is the same as that obtained from the Kripke semantics for first-order intuitionistic logic with constant domains and where the order structure of the model is linear. From this, the unaxiomatizability of Kroger's temporal logic of programs (even of the fragment without the nexttime operator O) and of the authors' temporal logic of linear discrete time with gaps follows.

We study the aggregation of individuals’ beliefs into their collective judgment on a certain proposition, called the conclusion , which is logically related to other preliminary propositions. While the literature has highlighted the crucial dependency of voting outcomes on the selection of an agenda —i.e., the set of propositions voted on—the structure of this dependency is not well understood. We introduce two novel premise-based rules, defined through the conjunctive normal form and disjunctive normal form of the conclusion. Our theorem shows that for any profile of voters’ beliefs, these rules respectively yield the highest and lowest voting outcomes, with respect to a given total order over the set of truth values, within the class of all premise-based rules. This result clarifies the extent to which a voting outcome can be manipulated through the selection of an agenda.

Hedge algebras: An algebraic approach to structure of sets of linguistic truth values

Type-2 operations on finite chains

I. Some Programmatic Comments. 1. Logic, Epistemology and the Unity of Science: An Encyclopedic Project in the Spirit of Neurath and Diderot Shahid Rahman and John Symons. 2. An International Encyclopedia of the Unified Sciences translated by John Symons and Ramon Alvarado) Otto Neurath. II. Game Theory and Independence Friendly Logic as a Unifying Framework. 3. Towards a Unity of the Human Behavioral Sciences Herbert Gintis. 4. Some Coloured Remarks on the Foundations of Mathematics in the 20th Century Gerhard Heinzmann. 5. Logical Versus Nonlogical Concepts: An Untenable Dualism? Jaakko Hintikka. 6. Semantic Games in Logic and Epistemology Ahti-Veikko Pietarinen. 7. IF Logic, Game-Theoretical Semantics and New Prospects for Philosophy of Science Ahti-Veikko Pietarinen and Gabriel Sandu. III. Unity and Plurality in Science and in Logic. 8. Concepts Structured through Reduction: A Structuralist Resource Illuminates the Consolidation-Long-Term Potentiation (LTP) Link John Bickle. 9. The Unity of Science and the Unity of Being: A Sketch of a Formal Approach C. Ulises Moulines. 10. Logical Pluralism and the Preservation of Warrant Greg Restall. 11. In Defence of the Dog: Response to Restall Stephen Read. 12. Normic Laws, Non-monotonic Reasoning, and the Unity of Science Gerhard Schurz. 13. The Puzzling Role of Philosophy in Life Sciences: Bases for a Joint Program for Philosophy and History of Science Juan Manuel Torres. 14. The Creative Growth of Mathematics Jean Paul Van Bendegem. 15. Quantum Logic and the Unity of Science John Woods and Kent Peacock. IV. The Logic of the Knowledge-Seeking Activities. 16. Belief Contraction, Anti-formulae and Resource Overdraft: Part II Deletion in Resource Unbounded Logics Dov Gabbay, Odinaldo Rodrigues and John Woods. 17. Reasoning about Knowledge in Linear Logic: Modalities and Complexity Mathieu Marion and Mehrnouche Sadrzadeh. 18. A Solution to Fitch's Paradox of Knowability Helge Ruckert. 19. Theories of Knowledge and Ignorance Wiebe van der Hoek, Jan Jaspars and Elias Thijsse. 20. Action-Theoretic Aspects of Theory Choice Heinrich Wansing. 21. Some Computational Constraints in Epistemic Logic Timothy Williamson. IV. Contributions from Non-Classical Logics. 22. The Need for Adaptive Logics in Epistemology Diderik Batens. 23. Logics for Qualitative Reasoning Paulo Veloso and Walter Carnielli. 24. Logic of Dynamics and Dynamics of Logic: Some Paradigm Examples Bob Coecke, David J. Moore and Sonja Smets. 25. Complementarity and Paraconsistency Newton C. A. Da Costa and Decio Krause. 26. Law, Logic, Rhetoric: a Procedural Model of Legal Argumentation Arno Lodder. 27. Essentialist Metaphysics in a Scientific Framework Ulrich Nortmann. Index.

A many‐valued sentential logic with truth values in an injective MV‐algebra is introduced and the axiomatizability of this logic is proved. The paper develops some ideas of Goguen and generalizes the results of Pavelka on the unit interval. The proof for completeness is purely algebraic. A corollary of the Completeness Theorem is that fuzzy logic on the unit interval is semantically complete if and only if the algebra of the truth values is a complete MV‐algebra. In the well‐defined fuzzy sentential logic holds the Compactness Theorem, while the Deduction Theorem and the Finiteness Theorem in general do not hold. Because of its generality and good‐behaviour, MV‐valued logic can be regarded as a mathematical basis of fuzzy reasoning.

An Algebraic Account of References in Game Semantics

In this paper, we study alethic modal operators in Łukasiewicz’ many-valued logics where truth values are certain rational numbers from the closed unit interval [0,1]. The alethic modal operators necessary and possible are added to Ł3 using Tarski’s idea. The result is a modal logic denoted by Ł\(_{3}^{\mathrm{mod}}\). A formula-equivalency between Ł n and Ł\(_{n}^{\mathrm{mod}}\) is stated for n ≥ 3. Truth value assignments, or valuations, in Łukasiewicz’ n-valued modal logic are considered. The laws of excluded middle and contradiction are considered and found that they are possible in Ł\(_{n}^{\mathrm{mod}}\). Normal modal many-valued systems based on axiom schemes (K), (T), (S4), (B) and (S5) are considered.

In earlier papers, I have sketched an approach to logical and linguistic semantics that embodies some of the same ideas on which Wittgenstein’s notion of language game is based.1 One of these ideas is that in order to appreciate the semantics of a word (or any other primitive expression of a language) we should study its function in the rule-governed human activities that serve to connect the language (or the fragment of a language) with the world. What Wittgenstein called “language games” can typically be considered such linking activities. In the languages (or parts of languages) that I will study in this chapter, certain activities of this kind are construed as games in the strict sense of the mathematical theory of games. They are called “semantical games”, and the semantics based on them is called game-theoretical semantics. Its basic ideas are explained most easily by reference to formal but interpreted first-order languages. Such a language, say L, can be assumed to have a finite number of primitive predicates that are interpreted on some given fixed domain D.

Game semantics is a rich and successful class of denotational models for programming languages. Most game models feature a rather intuitive setup, yet surprisingly difficult proofs of such basic results as associativity of composition of strategies. We seek to unify these models into a basic abstract framework for game semantics, game settings. Our main contribution is the generic construction, for any game setting, of a category of games and strategies. Furthermore, we extend the framework to deal with innocence, and prove that innocent strategies form a subcategory. We finally show that our constructions cover many concrete cases, mainly among the early models [5, 23] and the recent, sheaf-based ones [40].

Understanding Game Semantics Through Coherence Spaces