Handbook of Philosophical Logic: Volume III: Alternatives to Classical Logic
Dov M. Gabbay, Franz Guenthner
Springer Science & Business Media, Jun 29, 2013 - Philosophy - 531 pages
This volume presents a number of systems of logic which can be considered as alternatives to classical logic. The notion of what counts as an alternative is a somewhat problematic one. There are extreme views on the matter of what is the 'correct' logical system and whether one logical system (e. g. classical logic) can represent (or contain) all the others. The choice of the systems presented in this volume was guided by the following criteria for including a logic as an alternative: (i) the departure from classical logic in accepting or rejecting certain theorems of classical logic following intuitions arising from significant application areas and/or from human reasoning; (ii) the alternative logic is well-established and well-understood mathematically and is widely applied in other disciplines such as mathematics, physics, computer science, philosophy, psychology, or linguistics. A number of other alternatives had to be omitted for the present volume (e. g. recent attempts to formulate so-called 'non-monotonic' reason ing systems). Perhaps these can be included in future extensions of the Handbook of Philosophical Logic. Chapter 1 deals with partial logics, that is, systems where sentences do not always have to be either true or false, and where terms do not always have to denote. These systems are thus, in general, geared towards reasoning in partially specified models. Logics of this type have arisen mainly from philo sophical and linguistic considerations; various applications in theoretical computer science have also been envisaged.
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algebraic Anderson and Belnap argument assertion assignment atomic attack axiomatized axioms basic calculus canonical classical logic complete condition connectives consequence relation consider construction containing Deduction Theorem defined definition denoting derivation dialogue disjunction domain Dummett Dunn element equivalent example extension false finite formal free logic function Gabbay Gentzen given hence Heyting hypothesis implication induction intermediate logics interpretation introduced intuitionistic logic Kripke model language lattice Lemma many-valued logic Math mathematical matrix Meyer modal logic modus ponens monotonic monotonic functions natural deduction natural numbers negation node non-denoting singular terms notion obtained orthomodular partial logic Prawitz predicate predicate logic principle problem proof proposition provable quantifiers recursive relevance logic Routley rules Section semantics sense sentence sequence set of formulas singular terms structure subset supervaluational Symbolic Logic Tºp Troelstra true truth truth-value valid variables