Non-classical logic

Last updated

Non-classical logics (and sometimes alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is commonly the case, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth. [1]

Contents

Philosophical logic is understood to encompass and focus on non-classical logics, although the term has other meanings as well. [2] In addition, some parts of theoretical computer science can be thought of as using non-classical reasoning, although this varies according to the subject area. For example, the basic boolean functions (e.g. AND, OR, NOT, etc) in computer science are very much classical in nature, as is clearly the case given that they can be fully described by classical truth tables. However, in contrast, some computerized proof methods may not use classical logic in the reasoning process.

Examples of non-classical logics

There are many kinds of non-classical logic, which include:

Classification of non-classical logics according to specific authors

In Deviant Logic (1974) Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics. [4] The proposed classification is non-exclusive; a logic may be both a deviation and an extension of classical logic. [5] A few other authors have adopted the main distinction between deviation and extension in non-classical logics. [6] [7] [8] John P. Burgess uses a similar classification but calls the two main classes anti-classical and extra-classical. [9] Although some systems of classification for non-classical logic have been proposed, such as those of Haack and Burgess as described above for example, many people who study non-classical logic ignore these classification systems. As such, none of the classification systems in this section should be treated as standard.

In an extension, new and different logical constants are added, for instance the "" in modal logic, which stands for "necessarily." [6] In extensions of a logic,

(See also Conservative extension.)

In a deviation, the usual logical constants are used, but are given a different meaning than usual. Only a subset of the theorems from the classical logic hold. A typical example is intuitionistic logic, where the law of excluded middle does not hold. [8] [9]

Additionally, one can identify a variations (or variants), where the content of the system remains the same, while the notation may change substantially. For instance many-sorted predicate logic is considered a just variation of predicate logic. [6]

This classification ignores however semantic equivalences. For instance, Gödel showed that all theorems from intuitionistic logic have an equivalent theorem in the classical modal logic S4. The result has been generalized to superintuitionistic logics and extensions of S4. [10]

The theory of abstract algebraic logic has also provided means to classify logics, with most results having been obtained for propositional logics. The current algebraic hierarchy of propositional logics has five levels, defined in terms of properties of their Leibniz operator: protoalgebraic, (finitely) equivalential, and (finitely) algebraizable. [11]

See also

Related Research Articles

Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.

In logic, the semantic principleof bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false. A logic satisfying this principle is called a two-valued logic or bivalent logic.

<span class="mw-page-title-main">Saul Kripke</span> American philosopher and logician (1940–2022)

Saul Aaron Kripke was an American analytic philosopher and logician. He was Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and emeritus professor at Princeton University. Kripke is considered one of the most important philosophers of the latter half of the 20th century. Since the 1960s, he has been a central figure in a number of fields related to mathematical and modal logic, philosophy of language and mathematics, metaphysics, epistemology, and recursion theory.

Classical logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.

<span class="mw-page-title-main">Negation</span> Logical operation

In logic, negation, also called the logical not or logical complement, is an operation that takes a proposition to another proposition "not ", standing for " is not true", written , or . It is interpreted intuitively as being true when is false, and false when is true. Negation is thus a unary logical connective. It may be applied as an operation on notions, propositions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition is the proposition whose proofs are the refutations of .

Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic.

Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

In logic, false or untrue is the state of possessing negative truth value and is a nullary logical connective. In a truth-functional system of propositional logic, it is one of two postulated truth values, along with its negation, truth. Usual notations of the false are 0, O, and the up tack symbol .

In logic, a three-valued logic is any of several many-valued logic systems in which there are three truth values indicating true, false, and some third value. This is contrasted with the more commonly known bivalent logics which provide only for true and false.

In programming language theory and proof theory, the Curry–Howard correspondence is the direct relationship between computer programs and mathematical proofs.

Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical logic in a wider sense as the study of the scope and nature of logic in general. In this sense, philosophical logic can be seen as identical to the philosophy of logic, which includes additional topics like how to define logic or a discussion of the fundamental concepts of logic. The current article treats philosophical logic in the narrow sense, in which it forms one field of inquiry within the philosophy of logic.

A paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" systems of logic, which reject the principle of explosion.

In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic; thus, consistent superintuitionistic logics are called intermediate logics.

Kripke semantics is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Joyal. It was first conceived for modal logics, and later adapted to intuitionistic logic and other non-classical systems. The development of Kripke semantics was a breakthrough in the theory of non-classical logics, because the model theory of such logics was almost non-existent before Kripke.

Deviant logic is a type of logic incompatible with classical logic. Philosopher Susan Haack uses the term deviant logic to describe certain non-classical systems of logic. In these logics:

Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.

In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables.

Philosophy of logic is the area of philosophy that studies the scope and nature of logic. It investigates the philosophical problems raised by logic, such as the presuppositions often implicitly at work in theories of logic and in their application. This involves questions about how logic is to be defined and how different logical systems are connected to each other. It includes the study of the nature of the fundamental concepts used by logic and the relation of logic to other disciplines. According to a common characterisation, philosophical logic is the part of the philosophy of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. But other theorists draw the distinction between the philosophy of logic and philosophical logic differently or not at all. Metalogic is closely related to the philosophy of logic as the discipline investigating the properties of formal logical systems, like consistency and completeness.

<span class="mw-page-title-main">Logic</span> Study of correct reasoning

Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or logical truths. It studies how conclusions follow from premises due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics.

References

  1. ↑ Logic for philosophy, Theodore Sider
  2. ↑ Burgess, John P. (2009). Philosophical Logic. Princeton University Press. pp. vii–viii. ISBN   978-0-691-13789-6.
  3. ↑ da Costa, Newton C. A.; Krause, Décio (1994), "Schrödinger Logics", Studia Logica, 53 (4): 533, doi:10.1007/BF01057649
  4. ↑ Haack, Susan (1974). Deviant Logic: Some philosophical issues. Cambridge University Press. p. 4. ISBN   0-521-20500-X. LCCN   74-76949.
  5. ↑ Haack, Susan (1978). Philosophy of Logics. Cambridge University Press. p. 204. ISBN   0-521-29329-4.
  6. 1 2 3 Gamut, L. T. F. (1991). Logic, language, and meaning, Volume 1: Introduction to Logic. University of Chicago Press. pp. 156–157. ISBN   978-0-226-28085-1.
  7. ↑ Akama, Seiki (1997). Logic, language, and computation. Springer. p. 3. ISBN   978-0-7923-4376-9.
  8. 1 2 Hanna, Robert (2006). Rationality and logic. MIT Press. pp. 40–41. ISBN   978-0-262-08349-2.
  9. 1 2 Burgess, John P. (2009). Philosophical logic. Princeton University Press. pp. 1–2. ISBN   978-0-691-13789-6.
  10. ↑ Gabbay, Dov M.; Maksimova, Larisa (2005). Interpolation and definability: modal and intuitionistic logics. Clarendon Press. p. 61. ISBN   978-0-19-851174-8.
  11. ↑ Pigozzi, D. (2001). "Abstract algebraic logic". In Hazewinkel, M. (ed.). Encyclopaedia of mathematics: Supplement Volume III. Springer. pp. 2–13. ISBN   978-1-4020-0198-7. Also online: "Abstract algebraic logic", Encyclopedia of Mathematics , EMS Press, 2001 [1994]

Further reading

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.