Truth and Falsehood [electronic resource] : An Inquiry into Generalized Logical Values /
Shramko, Yaroslav.author.
Wansing, Heinrich.author.
The book presents a thoroughly elaborated logical theory of generalized truth values understood as subsets of some established set of (basic) truth entities. After elucidating the importance of the very notion of a truth value in logic and philosophy, the authors examine some possible ways of generalizing this notion. The useful four-valued logic of first-degree entailment by Nuel Belnap and Michael Dunn and the notion of a bilattice (a lattice of truth values with two ordering relations) constitute the basis for further generalizations. By doing so the authors elaborate the idea of a multilattice and, most notably, a trilattice of truth values – a specific algebraic structure with an information ordering and two distinct logical orderings, one for truth and another for falsity. Each logical order not only induces its own logical vocabulary, but also determines its own entailment relation. Both semantic ans syntactic ways of formalizing these relations by constructing various logical calculi are considered.
Description based upon print version of record.
Truth and Falsehood; Preface; Contents; 1 Truth Values; Abstract; 1.1...The Idea of Truth Values; 1.2...Truth Values and the Functional Analysis of Language; 1.3...The Categorial Status of Truth and Falsehood; 1.4...The Ontological Background of Truth Values; 1.5...Logic as the Science of Logical Values; 1.6...Logical Structures; 1.7...Truth Values, Truth Degrees, and Vague Concepts; 2 Truth Values and the Slingshot Argument; Abstract; 2.1...An Argument in Favor of Truth Values; 2.2...Reconstructing the Slingshot Arguments; 2.2.1 Church's Slingshot; 2.2.2 Gödel's Slingshot; 2.2.3 Davidson's Slingshot
2.3...The Slingshot Argument and Non-Fregean Logic2.4...Non-Fregean Logic and Definite Descriptions non-Fregean logic; 2.5...Non-Fregean Logic and lambda -Expressions; 2.6...Non-Fregean Logic and Indefinite Descriptions; 2.7...Concluding Remarks; 3 Generalized Truth Values: From FOUR2 to SIXTEEN3; Abstract; 3.1...Truth Values as Structured Entities; 3.2...Generalized Valuations, Four-Valued Logic and Bilattices; 3.3...Taking Generalization Seriously: From Isolated Computers to Computer Networks; 3.4...Generalized Truth Values and Multilattices; 3.5...The Trilattice of 16 Truth Values
3.6...Another Example of a Trilattice: Truth Values in Constructive Logics4 Generalized Truth Values: SIXTEEN3 and Beyond; Abstract; 4.1...Entailment Relations on SIXTEEN3; 4.2...First-Degree Systems for SIXTEEN3; 4.2.1 The Languages {\fancyscriptbold{L}t,\; \fancyscriptbold{L}f and Systems {{\bf FDE}}_{\bi{t}}^{\bi{t}}, {{\bf FDE}}_{\bi{f}}^{\bi{f}}; 4.2.2 The Language {\fancyscriptbold{L}}_{\varvec{tf}} for let and lef; 4.3...First-Degree Everywhere; 4.4...Hyper-Contradictions and Generalizations of Priest's Logic; 4.5...An Approach to a Generalization of Kleene's Logic: A Tetralattice
4.6...Uncertainty Versus Lack of Information5 Axiom Systems for Trilattice Logics; Abstract; 5.1...Truth Value Lattices and the Implication Connective; 5.2...From First-Degree Proof Systems to Proof Systems with Modus Ponens; 5.3...Odintsov's Axiomatization of Truth Entailment and Falsity Entailment in SIXTEEN3; 5.3.1 First-Degree Calculi; 5.3.2 Systems with Modus Ponens as the Sole Rule of Inference; 5.4...Discussion; 6 Sequent Systems for Trilattice Logics; Abstract; 6.1...Standard Sequent Systems for Logics Related to SIXTEEN3; 6.2...Alternative Sequent Calculi; 6.3...Extensions
6.4...Sequent Calculi for Truth Entailment and Falsity Entailment in SIXTEEN37 Intuitionistic Trilattice Logics; Abstract; 7.1...Introduction; 7.2...Sequent Calculus I16; 7.3...Kripke Completeness for I16; 7.4...Tableau Calculus IT16; 7.5...Kripke Completeness for IT16; 8 Generalized Truth Values and Many-Valued Logics: Harmonious Many-Valued Logics; Abstract; 8.1...Many-Valued Propositional Logics Generalized; 8.2...Designateddesignated truth valueantidesignated truth value and Antidesignated Values; 8.3...Some Separated Finitely-Valued Logics; 8.4...A Harmonious Logic Inspired by the Logic of SIXTEEN3
8.5...Harmony ad Infinitum
Includes bibliographical references and index.
The book presents a thoroughly elaborated logical theory of generalized truth values understood as subsets of some established set of (basic) truth entities. After elucidating the importance of the very notion of a truth value in logic and philosophy, the authors examine some possible ways of generalizing this notion. The useful four-valued logic of first-degree entailment by Nuel Belnap and Michael Dunn and the notion of a bilattice (a lattice of truth values with two ordering relations) constitute the basis for further generalizations. By doing so the authors elaborate the idea of a multilattice and, most notably, a trilattice of truth values – a specific algebraic structure with an information ordering and two distinct logical orderings, one for truth and another for falsity. Each logical order not only induces its own logical vocabulary, but also determines its own entailment relation. Both semantic ans syntactic ways of formalizing these relations by constructing various logical calculi are considered.
Logic.
Mathematical logic.
Logic.
Mathematical Logic and Foundations.
Mathematical Logic and Formal Languages.
Trends in logic