In

Socratic Epistemology: Explorations of Knowledge-Seeking by Questioning

', Cambridge: Cambridge University Press. * Ilkka Niiniluoto (1999), ''Critical Scientific Realism'', Oxford: Oxford University Press. {{Philosophy of language Mathematical logic Model theory Philosophy of language Semantics Theories of deduction Theories of truth

logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premi ...

, the semantics of logic or formal semantics is the study of the semantics, or interpretations, of formal
Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements (forms, in Ancient Greek). They may refer to:
Dress code and events
* Formal wear, attire for formal events
* Semi-formal attire ...

and (idealizations of) natural language
In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...

s usually trying to capture the pre-theoretic notion of entailment
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one ...

.
Overview

The truth conditions of various sentences we may encounter inargument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialect ...

s will depend upon their meaning, and so 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
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...

, an idealised sentence suitable for logical manipulation.
Until the advent of modern logic, Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of p ...

's ''Organon
The ''Organon'' ( grc, Ὄργανον, meaning "instrument, tool, organ") is the standard collection of Aristotle's six works on logical analysis and dialectic. The name ''Organon'' was given by Aristotle's followers, the Peripatetics.
The si ...

'', especially ''De Interpretatione
''De Interpretatione'' or ''On Interpretation'' (Greek: Περὶ Ἑρμηνείας, ''Peri Hermeneias'') is the second text from Aristotle's ''Organon'' and is among the earliest surviving philosophical works in the Western tradition to deal ...

'', 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
In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, ...

, attempting to find calculi in the spirit of Aristotle's syllogisms, but with the generality of modern logics based on the quantifier.
The main modern approaches to semantics for formal languages are the following:
* The archetype of ''model-theoretic semantics'' is Alfred Tarski
Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...

's semantic theory of truth, based on his T-schema, and is one of the founding concepts of model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...

. 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
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

is given by a mapping from terms to a universe of individual
An individual is that which exists as a distinct entity. Individuality (or self-hood) is the state or quality of being an individual; particularly (in the case of humans) of being a person unique from other people and possessing one's own need ...

s, 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
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 and André ...

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
Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian- British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He is conside ...

'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 J. Michael Dunn, Nuel Belnap, and Hugues 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 semantics'' or ''game-theoretical semantics'' made a resurgence 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 Hartry 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.
See also

* Algebraic semantics *Formal semantics (natural language)
Formal semantics is the study of grammatical meaning in natural languages using formal tools from logic and theoretical computer science. It is an interdisciplinary field, sometimes regarded as a subfield of both linguistics and philosophy of lan ...

References

* Jaakko Hintikka (2007),Socratic Epistemology: Explorations of Knowledge-Seeking by Questioning

', Cambridge: Cambridge University Press. * Ilkka Niiniluoto (1999), ''Critical Scientific Realism'', Oxford: Oxford University Press. {{Philosophy of language Mathematical logic Model theory Philosophy of language Semantics Theories of deduction Theories of truth