The Ultimate Problem of Classical Logic

Classical and Nonclassical Logics, 2020

Typically, a logic consists of a formal or informal language together with a deductive system and/or a modeltheoretic semantics. The language has components that correspond to a part of a natural language like English or Greek. The deductive system is to capture, codify, or simply record arguments that are valid for the given language, and the semantics is to capture, codify, or record the meanings, or truth-conditions for at least part of the language. The following sections provide the basics of a typical logic, sometimes called "classical elementary logic" or "classical first-order logic". Section 2 develops a formal language, with a rigorous syntax and grammar. The formal language is a recursively defined collection of strings on a fixed alphabet. As such, it has no meaning, or perhaps better, the meaning of its formulas is given by the deductive system and the semantics. Some of the symbols have counterparts in ordinary language. We define an argument to be a non-empty collection of sentences in the formal language, one of which is designated to be the conclusion. The other sentences (if any) in an argument are its premises. Section 3 sets up a deductive system for the language, in the spirit of natural deduction. An argument is derivable if there is a deduction from some or all of its premises to its conclusion. Section 4 provides a model-theoretic semantics. An argument is valid if there is no interpretation (in the semantics) in which its premises are all true and its conclusion false. This reflects the longstanding view that a valid argument is truth-preserving. In Section 5, we turn to relationships between the deductive system and the semantics, and in particular, the relationship between derivability and validity. We show that an argument is derivable only if it is valid. This pleasant feature, called soundness, entails that no deduction takes one from true premises to a false conclusion. Thus, deductions preserve truth. Then we establish a converse, called completeness, that an argument is valid only if it is derivable. This establishes that the deductive system is rich enough to provide a deduction for every valid argument. So there are enough deductions: all and only valid arguments are derivable. We briefly indicate other features of the logic, some of which are corollaries to soundness and completeness.

The Computer Journal, 1992

No doubt every reader of this journal is aware that computer science is becoming infiltrated by a strange breed of people called logicians, who try to convince computer people that their arcane symbolism and obscure terminology are just what is needed to solve the software crisis, the hardware crisis, and any other difficulties that the computer world finds itself facing. Unfortunately the symbolism and the jargon can be very off-putting to anyone who has not already become immersed in formal logic; I have often met people who work with computers and are aware of how important logic is claimed to be by its devotees, and who feel that they really ought one day to make an effort to penetrate its mysteries, but who have not known how to set about doing so. This article and its sequel ('Logic as a Formal Method') are intended as a fairly gentle initiation into what logic is about and what it has to offer computer scientists. They are inevitably very sketchy and incomplete -more like the brochures that can be picked up at a travel agent's than a proper guide-book -but it is to be hoped that some, at least, of my readers will come away with a clearer picture of what lies in store for them if they decide to follow up the more detailed references.

We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given signature; if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation; in an appendix, we further extend this representation to lattices with implications. Finally, we discuss adding to our logic a conditional obeying only introduction and elimination rules, interpreted as a modality using a family of accessibility relations.

2013

iii 0 Vers une théorie des preuves pour la logique classique v 0.1 Catégories des preuves.............................. vi

2012

Clarke's book titled Logic printed by Longman Green (London) in the year 1909 has been extensively used. Logical arguments can be divided into two major parts: Hypothetical Syllogism and Categorical Syllogism. The word 'syllogism' means an argument or form of reasoning in which two statements or premises are made and a logical conclusion is drawn from them. Here we shall see the first division i.e., hypothetical syllogism and when you are confident you can move to categorical syllogism. The word 'hypothetical' is derived from 'hypothesis' which means a statement that begins with a hypothetical clause like 'if' which shows a possibility or probability of an action. The word 'categorical' means without qualifications or conditions; absolute; positive; directed; explicit; said of a statement or theory.

Journal of Philosophical Logic, 1978

2019

In this article, we will present a number of technical results concerning Classical Logic, ST and related systems. Our main contribution consists in offering a novel identity criterion for logics in general and, therefore, for Classical Logic. In partic- ular, we will firstly generalize the ST phenomenon, thereby obtaining a recursively defined hierarchy of strict-tolerant systems. Secondly, we will prove that the logics in this hierarchy are progressively more classical, although not entirely classical. We will claim that a logic is to be identified with an infinite sequence of conse- quence relations holding between increasingly complex relata: formulae, inferences, metainferences, and so on. As a result, the present proposal allows not only to differentiate Classical Logic from ST, but also from other systems sharing with it their valid metainferences. Finally, we show how these results have interesting con- sequences for some topics in the philosophical logic literature, among them for the debate around Logical Pluralism. The reason being that the discussion concerning this topic is usually carried out employing a rivalry criterion for logics that will need to be modified in light of the present investigation, according to which two logics can be non-identical even if they share the same valid inferences.

Riassumiamo brevemente la nostra attività di ricerca nel campo delle logiche non-classiche iniziata negli anni '90. In particolare, descriviamo la nostra ricerca riguardante l'applicazione delle logiche non-classiche alla rappresentazione della conoscenza e lo sviluppo di metodi di prova per logiche non-monotone e condizionali.

2010

The first edition of this book appeared in 2001. It comprised about 250 pages and covered only propositional logic. It was a good but limited introduction to non-classical logic. Its most distinctive features were its concise and clear coverage of many important systems of non-classical propositional logic,

Synthese 199, 13067–13094, 2021

In Sect. 1 it is argued that systems of logic are exceptional, but not a priori necessary. Logics are exceptional because they can neither be demonstrated as valid nor be confirmed by observation without entering a circle, and their motivation based on intuition is unreliable. On the other hand, logics do not express a priori necessities of thinking because alternative non-classical logics have been developed. Section 2 reflects the controversies about four major kinds of non-classical logics-multi-valued, intuitionistic, paraconsistent and quantum logics. Its purpose is to show that there is no particular domain or reason that demands the use of a non-classical logic; the particular reasons given for the non-classical logic can also be handled within classical logic. The result of Sect. 2 is substantiated in Sect. 3, where it is shown (referring to other work) that all four kinds of non-classical logics can be translated into classical logic in a meaning-preserving way. Based on this fact a justification of classical logic is developed in Sect. 4 that is based on its representational optimality. It is pointed out that not many but a few non-classical logics can be likewise representationally optimal. However, the situation is not symmetric: classical logic has ceteris paribus advantages as a unifying metalogic, while non-classical logics can have local simplicity advantages.