Semantics of logic
Encyclopedia
In logic
, formal semantics is the study of the semantics
, or interpretation
s, of formal
and (idealizations of) natural language
s usually trying to capture the pre-theoretic notion of entailment
. (Although both linguistics
and logic lay claim to providing theories of natural language, according to Geach, logic generally ignores the "idiotism of idiom", and sees natural languages as cluttered with idiom
s of no logical interest.)
A formal language can be defined apart from any interpretation of it. This is done by designating a set of symbol
s (also called an alphabet
) and a set of formation rules (also called a formal grammar) which determine which string
s of symbols are well-formed formula
s. When transformation rules (also called rules of inference) are added, and certain sentences are accepted as axiom
s (together called a deductive system
or a deductive apparatus) a logical system is formed. An interpretation of a formal language is (roughly) an assignment of meaning
s to its symbols and truth-conditions to its sentences.
The truth conditions of various sentences we may encounter in argument
s 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:
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, formal semantics is the study of the semantics
Semantics
Semantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....
, or interpretation
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...
s, of formal
Formal language
A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
and (idealizations of) natural language
Natural language
In the philosophy of language, a natural language is any language which arises in an unpremeditated fashion as the result of the innate facility for language possessed by the human intellect. A natural language is typically used for communication, and may be spoken, signed, or written...
s usually trying to capture the pre-theoretic notion of entailment
Entailment
In logic, entailment is a relation between a set of sentences and a sentence. Let Γ be a set of one or more sentences; let S1 be the conjunction of the elements of Γ, and let S2 be a sentence: then, Γ entails S2 if and only if S1 and not-S2 are logically inconsistent...
. (Although both linguistics
Linguistics
Linguistics is the scientific study of human language. Linguistics can be broadly broken into three categories or subfields of study: language form, language meaning, and language in context....
and logic lay claim to providing theories of natural language, according to Geach, logic generally ignores the "idiotism of idiom", and sees natural languages as cluttered with idiom
Idiom
Idiom is an expression, word, or phrase that has a figurative meaning that is comprehended in regard to a common use of that expression that is separate from the literal meaning or definition of the words of which it is made...
s of no logical interest.)
A formal language can be defined apart from any interpretation of it. This is done by designating a set of symbol
Symbol
A symbol is something which represents an idea, a physical entity or a process but is distinct from it. The purpose of a symbol is to communicate meaning. For example, a red octagon may be a symbol for "STOP". On a map, a picture of a tent might represent a campsite. Numerals are symbols for...
s (also called an alphabet
Alphabet
An alphabet is a standard set of letters—basic written symbols or graphemes—each of which represents a phoneme in a spoken language, either as it exists now or as it was in the past. There are other systems, such as logographies, in which each character represents a word, morpheme, or semantic...
) and a set of formation rules (also called a formal grammar) which determine which string
String (computer science)
In formal languages, which are used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set or alphabet....
s of symbols are well-formed formula
Well-formed formula
In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word which is part of a formal language...
s. When transformation rules (also called rules of inference) are added, and certain sentences are accepted as axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s (together called a deductive system
Deductive system
A deductive system consists of the axioms and rules of inference that can be used to derive the theorems of the system....
or a deductive apparatus) a logical system is formed. An interpretation of a formal language is (roughly) an assignment of meaning
Meaning (non-linguistic)
A non-linguistic meaning is an actual or possible derivation from sentience, which is not associated with signs that have any original or primary intent of communication...
s to its symbols and truth-conditions to its sentences.
The truth conditions of various sentences we may encounter in argument
Argument
In philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.Argument may also refer to:-Mathematics and computer science:...
s 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
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
, an idealised sentence suitable for logical manipulation.
Until the advent of modern logic, Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
's Organon
Organon
The Organon is the name given by Aristotle's followers, the Peripatetics, to the standard collection of his six works on logic:* Categories* On Interpretation* Prior Analytics* Posterior Analytics...
, especially De Interpretatione, provided the basis for understanding the significance of logic. The introduction of quantification
Quantification
Quantification has several distinct senses. In mathematics and empirical science, it is the act of counting and measuring that maps human sense observations and experiences into members of some set of numbers. Quantification in this sense is fundamental to the scientific method.In logic,...
, needed to solve the problem of multiple generality
Problem of multiple generality
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:then it follows logically that:The syntax of traditional logic permits exactly four sentence types: "All As are Bs", "No As are...
, rendered impossible the kind of subject-predicate analysis that governed Aristotle's account, although there is a renewed interest in term logic
Term logic
In philosophy, term logic, also known as traditional logic or aristotelian logic, is a loose name for the way of doing logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late nineteenth century...
, 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 TarskiAlfred TarskiAlfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
's semantic theory of truthSemantic theory of truthA semantic theory of truth is a theory of truth in the philosophy of language which holds that truth is a property of sentences.-Origin:The semantic conception of truth, which is related in different ways to both the correspondence and deflationary conceptions, is due to work published by Polish...
, based on his T-schemaT-schemaThe T-schema or truth schema is used to give an inductive definition of truth which lies at the heart of any realisation of Alfred Tarski's semantic theory of truth...
, and is one of the founding concepts of model theoryModel theoryIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
. 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 interpretationInterpretation (logic)An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...
of first-order predicate logic is given by a mapping from terms to a universe of individualIndividualAn individual is a person or any specific object or thing in a collection. Individuality is the state or quality of being an individual; a person separate from other persons and possessing his or her own needs, goals, and desires. Being self expressive...
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 semanticsTruth-conditional semanticsTruth-conditional semantics is an approach to semantics of natural language that sees the meaning of assertions as being the same as, or reducible to, their truth conditions...
, which was pioneered by Donald DavidsonDonald Davidson (philosopher)Donald Herbert Davidson was an American philosopher born in Springfield, Massachusetts, who served as Slusser Professor of Philosophy at the University of California, Berkeley from 1981 to 2003 after having also held teaching appointments at Stanford University, Rockefeller University, Princeton...
. Kripke semanticsKripke semanticsKripke semantics is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke. It was first made for modal logics, and later adapted to intuitionistic logic and other non-classical systems...
introduces innovations, but is broadly in the Tarskian mold.
- Proof-theoretic semanticsProof-theoretic semanticsProof-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 within the system...
associates the meaning of propositions with the roles that they can play in inferences. Gerhard GentzenGerhard GentzenGerhard Karl Erich Gentzen was a German mathematician and logician. He had his major contributions in the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus...
, Dag PrawitzDag PrawitzDag Prawitz is a Swedish philosopher and logician. He is best known for his work on proof theory and the foundations of natural deduction....
and Michael DummettMichael DummettSir Michael Anthony Eardley Dummett FBA D.Litt is a British philosopher. He was, until 1992, Wykeham Professor of Logic at the University of Oxford...
are generally seen as the founders of this approach; it is heavily influenced by Ludwig WittgensteinLudwig WittgensteinLudwig Josef Johann Wittgenstein was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He was professor in philosophy at the University of Cambridge from 1939 until 1947...
's later philosophy, especially his aphorism "meaning is use".
- Truth-value semanticsTruth-value semanticsIn formal semantics, 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...
(also commonly referred to as substitutional quantification) was advocated by Ruth Barcan MarcusRuth Barcan MarcusRuth Barcan Marcus is the American 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...
for modal logics in the early 1960s and later championed by Dunn, Belnap, and Leblanc for standard first-order logic. James GarsonJames GarsonJames Garson is an American philosopher and logician. He has made significant contributions in the study of modal logic and formal semantics, and is author of Modal Logic for Philosophers by Cambridge University Press...
has given some results in the areas of adequacy for intensional logicIntensional logicIntensional logic is an approach to predicate logic that extends first-order logic, which has quantifiers that range over the individuals of a universe , by additional quantifiers that range over terms that may have such individuals as their value...
s 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 semanticsGame semanticsGame semantics is an approach to formal semantics that grounds the concepts of truth or validity on game-theoretic concepts, such as the existence of a winning strategy for a player, somewhat resembling Socratic dialogues or medieval theory of Obligationes. In the late 1950s Paul Lorenzen was the...
has made a resurgence lately mainly due to Jaakko HintikkaJaakko HintikkaKaarlo Jaakko Juhani Hintikka 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 at Boston University...
for logics of (finite) partially ordered quantification which were originally investigated by Leon HenkinLeon HenkinLeon Albert Henkin was a logician at the University of California, Berkeley. He was principally known for the "Henkin's completeness proof": his version of the proof of the semantic completeness of standard systems of first-order logic.-The completeness proof:Henkin's result was not novel; it had...
, 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.