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Information about Formal Semantic

Formal semantics redirects to this page. See also Formal semantics of programming languages.


The truth conditions of various sentences we may encounter in arguments will depend upon their meaning, and so conscientious logicians cannot completely avoid the need to provide some treatment of the meaning of these sentences. The semantics of logic refers to the approaches that logicians have introduced to understand and determine that part of meaning in which they are interested; the logician traditionally is not interested in the sentence as uttered but in the proposition, an idealised sentence suitable for logical manipulation.

Until the advent of modern logic, Aristotle's Organon, especially De Interpretatione, provided the basis for understanding the significance of logic. The introduction of quantification, needed to solve the problem of multiple generality, rendered impossible the kind of subject-predicate analysis that governed Aristotle's account, although there is a renewed interest in term logic, attempting to find calculi in the spirit of Aristotle's syllogistic but with the generality of modern logics based on the quantifier.

The main modern approaches to semantics for formal languages are the following:
  • Model-theoretic semantics is the archetype of Alfred Tarski's semantic theory of truth, based on his T-schema, and is one of the founding concepts of model theory. This is the most widespread approach, and is based on the idea that the meaning of the various parts of the propositions are given by the possible ways we can give a recursively specified group of interpretation functions from them to some predefined mathematical domains: an interpretation of first-order predicate logic is given by a mapping from terms to a universe of individuals, and a mapping from propositions to the truth values "true" and "false". Model-theoretic semantics provides the foundations for an approach to the theory of meaning known as Truth-conditional semantics, which was pioneered by Donald Davidson. Kripke semantics introduces innovations, but is broadly in the Tarskian mold.
  • Proof-theoretic semantics associates the meaning of propositions with the roles that they can play in inferences. Gerhard Gentzen, Dag Prawitz and Michael Dummett are generally seen as the founders of this approach; it is heavily influenced by Ludwig Wittgenstein's later philosophy, especially his aphorism "meaning is use".
  • Truth-value semantics (also commonly referred to as substitutional quantification) was advocated by Ruth Barcan Marcus for modal logics in the early 1960s and later championed by Dunn, Belnap, and Leblanc for standard first-order logic. James Garson has given some results in the areas of adequacy for intensional logics outfitted with such a semantics. The truth conditions for quantified formulas are given purely in terms of truth with no appeal to domains whatsoever (and hence its name truth-value semantics).
  • Game-theoretical semantics has made a resurgence lately mainly due to Jaakko Hintikka for logics of (finite) partially ordered quantification which were originally investigated by Leon Henkin, who studied Henkin quantifiers.
  • Probabilistic semantics originated from H. Field and has been shown equivalent to and a natural generalization of truth-value semantics. Like truth-value semantics, it is also non-referential in nature.
In theoretical computer science, formal semantics is the field concerned with the rigorous mathematical study of the meaning of programming languages and models of computation.
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proposition is the content of an assertion, that is, it is true-or-false and defined by the meaning of a particular piece of language. The proposition is independent of the of communication.
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Aristotle (Greek: Ἀριστοτέλης Aristotélēs) (384 BC – 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great.
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Organon is the name given by Aristotle's followers, the Peripatetics, to the standard collection of six of his works on logic. The works are Categories, Prior Analytics, De Interpretatione, Posterior Analytics, Sophistical Refutations
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Aristotle's work De Interpretatione (the Latin title by which it is usually known) or On Interpretation (Greek Περὶ Ἑρμηνείας or Peri Hermeneias
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The term quantification has several meanings, general and specific. Primarily it covers all those acts which quantify observations and experiences by converting them into numbers through counting and measuring. It is thus the basis for mathematics and for science.
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The problem of multiple generality names a failure in traditional logic to describe certain intuitively valid inferences. For example, it is intuitively clear that if:
Some cat is feared by every mouse
then it follows logically that:

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Traditional logic, also known as term logic, is a loose term for the logical tradition that originated with Aristotle and survived until the advent of modern predicate logic in the late nineteenth century.
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Alfred Tarski (January 14, 1902, Warsaw, Russian-ruled Poland – October 26, 1983, Berkeley, California) was a logician and mathematician who spent four decades as a professor of mathematics at the University of California, Berkeley.
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The semantic theory of truth holds that any assertion that a sentence is true can be made only as a formal requirement regarding the language in which the proposition itself is expressed.
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The T-schema (also known as Convention T) is the inductive definition that lies at the heart of any realisation of Alfred Tarski's semantic theory of truth, expressing the commutation of truth over logical operators.
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This article discusses model theory as a mathematical discipline and not the informally used term mathematical model as used in other parts of mathematics and science.

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First-order logic (FOL) is a formal deductive system used by mathematicians, philosophers, linguists, and computer scientists. It goes by many names, including: first-order predicate calculus (FOPC), the lower predicate calculus,
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As commonly used, individual refers to a person or to any specific object in a collection. In the 15th century and earlier, and also today within the fields of statistics and metaphysics, individual
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Truth-conditional semantics is an approach to semantics of natural language that sees the meaning of a sentence being the same as, or reducible to, the truth conditions of that sentence.
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Donald Herbert Davidson (March 6, 1917 – August 30, 2003) was an American philosopher, who served as Slusser Professor of Philosophy at the University of California, Berkeley, from 1981 to 2003, after having also held substantive teaching appointments at Stanford
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Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke, beginning when he was a teenager.
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Proof-theoretic semantics is an approach to the semantics of logic that attempts to locate the meaning of propositions and logical connectives not in terms of interpretations, as in Tarskian approaches to semantics, but in the role that the proposition or logical connective plays
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Gerhard Karl Erich Gentzen (November 24, 1909 – August 4, 1945) was a German mathematician and logician.

Gentzen was born in Greifswald, Germany. He was one of Hermann Weyl's students at the University of Göttingen from 1929 to 1933.
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Dag Prawitz (born 1936) is a Swedish philosopher and logician. He is best known for his work on proof theory and the foundations of natural deduction.

Main Books

  • Natural Deduction. A Proof-Theoretic Study, 1965.
  • ABC i Symbolisk Logik, 1991.

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Sir Michael Anthony Eardley Dummett F.B.A., D. Litt, (born 1925) is a leading British philosopher. He has both written on the history of analytic philosophy, and made original contributions to the subject, particularly in the areas of philosophy of mathematics, philosophy of logic,
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Ludwig Josef Johann Wittgenstein (IPA: ['luːtvɪç 'joːzɛf 'joːhan 'vɪtgənʃtaɪn]
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In the semantics of logic, truth-value semantics is an alternative to Tarskian semantics. It has been primarily championed by Ruth Barcan Marcus, H. Leblanc, and M. Dunn and N. Belnap.
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Ruth Barcan Marcus (born 1921 in Bronx, New York) is the philosopher and logician after whom the Barcan formula is named. She is a pioneering figure in the quantification of modal logic and the theory of direct reference.

Education

  • B.A.

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Game semantics (German: dialogische Logik) is an approach to the semantics of logic that grounds the concepts of truth or validity on game-theoretic concepts, such as the existence of a winning strategy for a player.
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Jaakko Hintikka (born January 12 1929) is a Finnish philosopher and logician.

Hintikka was born in Vantaa. After teaching for a number of years at Florida State University, Stanford, University of Helsinki, and the Academy of Finland, he is currently Professor of Philosophy
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Leon Henkin (19 April 1921–1 November2006) was a logician at the University of California, Berkeley. He was principally known for the "Henkin Completeness Proof": his version of the proof of the semantic completeness of standard systems of first-order logic.
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Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration.
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In western philosophy, reason has had a twofold history. On the one hand, it has been taken to be objective and so to be fixed and discoverable by dialectic, analysis or study.
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The history of logic documents the development of logic as it occurs in various cultures and traditions in history. While many cultures have employed intricate systems of reasoning, logic as an explicit analysis of the methods of reasoning received sustained development originally
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