Formal semantics (logic)

In logic, formal semantics is the study of the semantics, or interpretations, of formal and (idealizations of) natural languages 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 idioms of no logical interest.^[1])

A formal language can be defined apart from any interpretation of it. This is done by designating a set of symbols (also called an alphabet) and a set of formation rules (also called a formal grammar) which determine which strings of symbols are well-formed formulas. When transformation rules (also called rules of inference) are added, and certain sentences are accepted as axioms (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 meanings to its symbols and truth-conditions to its sentences.^[2]

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.

Notes

^ Mieszko Talasiewicz (2009). Philosophy of Syntax - Foundational Topics. Springer. p. 12. ISBN 9789048132874.
^ The Cambridge Dictionary of Philosophy, Formal semantics

Logic

Overview

Academic areas	Argumentation theory Axiology Critical thinking Computability theory Formal semantics History of logic Informal logic Logic in computer science Mathematical logic Mathematics Metalogic Metamathematics Model theory Philosophical logic Philosophy Philosophy of logic Philosophy of mathematics Proof theory Set theory

Foundational concepts	Abduction Analytic truth Antinomy A priori Deduction Definition Description Entailment Induction Inference Logical consequence Logical form Logical implication Logical truth Name Necessity Meaning Paradox Possible world Presupposition Probability Reason Reasoning Reference Semantics Statement Strict implication Substitution Syntax Truth Truth value Validity

Philosophical logic

Critical thinking and Informal logic	Analysis Ambiguity Argument Belief Bias Credibility Evidence Explanation Explanatory power Fact Fallacy Inquiry Opinion Parsimony Premise Propaganda Prudence Reasoning Relevance Rhetoric Rigor Vagueness

Theories of deduction	Constructivism Dialetheism Fictionalism Finitism Formalism Intuitionism Logical atomism Logicism Nominalism Platonic realism Pragmatism Realism

Metalogic and metamathematics

Cantor's theorem Church's theorem Church's thesis Consistency Effective method Foundations of mathematics Gödel's completeness theorem Gödel's incompleteness theorems Soundness Completeness Decidability Interpretation Löwenheim–Skolem theorem Metatheorem Satisfiability Independence Type–token distinction Use–mention distinction

Mathematical logic

General	Formal language Formation rule Formal system Deductive system Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Natural deduction Rule of inference Relation Theorem Logical consequence Axiomatic system Type theory Symbol Syntax Theory

Traditional logic	Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram

Propositional calculus and Boolean logic	Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables

Predicate	First-order Quantifiers Predicate Second-order Monadic predicate calculus

Set theory	Set Empty set Enumeration Extensionality Finite set Function Subset Power set Countable set Recursive set Domain Range Ordered pair Uncountable set

Model theory	Model Interpretation Non-standard model Finite model theory Truth value Validity

Proof theory	Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax

Computability theory	Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function

Non-classical logic

Modal logic	Alethic Axiologic Deontic Doxastic Epistemic Temporal

Intuitionism	Intuitionistic logic Constructive analysis Heyting arithmetic Intuitionistic type theory Constructive set theory

Fuzzy logic	Degree of truth Fuzzy rule Fuzzy set Fuzzy finite element Fuzzy set operations

Substructural logic	Structural rule Relevance logic Linear logic

Paraconsistent logic	Dialetheism

Description logic	Ontology Ontology language

Logicians

Anderson Aristotle Averroes Avicenna Bain Barwise Bernays Boole Boolos Cantor Carnap Church Chrysippus Curry De Morgan Frege Geach Gentzen Gödel Hilbert Kleene Kripke Leibniz Löwenheim Peano Peirce Putnam Quine Russell Schröder Scotus Skolem Smullyan Tarski Turing Whitehead William of Ockham Wittgenstein Zermelo

Lists

Topics	Outline of logic Index of logic articles Mathematical logic Boolean algebra Set theory

Other	Logicians Rules of inference Paradoxes Fallacies Logic symbols

Portal
Category
Outline
WikiProject
Talk
changes

Philosophy of language

Philosophers	Plato (Cratylus) • Confucius • Xun Zi • Aristotle • Stoics • Pyrrhonists • Scholasticism • Ibn Rushd • Ibn Khaldun • Thomas Hobbes • Gottfried Leibniz • Johann Herder • Wilhelm von Humboldt • Fritz Mauthner • Paul Ricœur • Ferdinand de Saussure • Gottlob Frege • Franz Boas • Paul Tillich • Edward Sapir • Leonard Bloomfield • Zhuangzi • Henri Bergson • Lev Vygotsky • Ludwig Wittgenstein (Philosophical Investigations • Tractatus Logico-Philosophicus) • Bertrand Russell • Rudolf Carnap • Jacques Derrida (Of Grammatology • Limited Inc) • Benjamin Lee Whorf • Gustav Bergmann • J. L. Austin • Noam Chomsky • Hans-Georg Gadamer • Saul Kripke • Alfred Jules Ayer • Donald Davidson • Paul Grice • Gilbert Ryle • P. F. Strawson

Theories	Causal theory of reference • Contrast theory of meaning • Contrastivism • Conventionalism • Cratylism • Deconstruction • Descriptivist theory of names • Direct reference theory • Dramatism • Expressivism • Linguistic determinism • Logical atomism • Logical positivism • Mediated reference theory • Nominalism • Non-cognitivism • Phallogocentrism • Quietism • Relevance theory • Semantic externalism • Semantic holism • Structuralism • Supposition theory • Symbiosism • Theological noncognitivism • Theory of descriptions • Verification theory

Concepts	Ambiguity • Linguistic relativity • Meaning • Language • Truthbearer • Proposition • Use–mention distinction • Concept • Categories • Set • Class • Intension • Logical form • Metalanguage • Mental representation • Principle of compositionality • Property • Sign • Sense and reference • Speech act • Symbol • Entity • Sentence • Statement • more...

Related articles	Analytic philosophy • Language • Philosophy of information • Philosophical logic • Linguistics • Pragmatics • Rhetoric • Semantics • Formal semantics • Semiotics

Category · Task Force · Discussion