Predicate logic in the context of "Finitary relation"

Jacques Marie Émile Lacan (UK: /læˈkɒ̃/, US: /ləˈkɑːn/ lə-KAHN; French: [ʒak maʁi emil lakɑ̃]; 13 April 1901 – 9 September 1981) was a French psychoanalyst and psychiatrist. Described as "the most controversial psycho-analyst since Freud", Lacan gave yearly seminars in Paris, from 1953 to 1981, and published papers that were later collected in the book Écrits. Transcriptions of his seminars, given between 1954 and 1976, were also published. His work made a significant impact on continental philosophy and cultural theory in areas such as post-structuralism, critical theory, feminist theory and film theory, as well as on the practice of psychoanalysis itself.

Lacan took up and discussed the whole range of Freudian concepts, emphasizing the philosophical dimension of Freud's thought and applying concepts derived from structuralism in linguistics and anthropology to its development in his own work, which he would further augment by employing formulae from predicate logic and topology. Taking this new direction, and introducing controversial innovations in clinical practice, led to expulsion for Lacan and his followers from the International Psychoanalytic Association. In consequence, Lacan went on to establish new psychoanalytic institutions to promote and develop his work, which he declared to be a "return to Freud", in opposition to prevalent trends in psychology and institutional psychoanalysis collusive of adaptation to social norms.

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. The general study of interpretations of formal languages is called formal semantics.

The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a function that provides the extension of symbols and strings of an object language. For example, an interpretation function could take the predicate symbol ${\displaystyle T}$ and assign it the extension ${\displaystyle \{(\mathrm {a} )\}}$. All our interpretation does is assign the extension ${\displaystyle \{(\mathrm {a} )\}}$ to the non-logical symbol ${\displaystyle T}$, and does not make a claim about whether ${\displaystyle T}$ is to stand for tall and ${\displaystyle \mathrm {a} }$ for Abraham Lincoln. On the other hand, an interpretation does not have anything to say about logical symbols, e.g. logical connectives "${\displaystyle \mathrm {and} }$", "${\displaystyle \mathrm {or} }$" and "${\displaystyle \mathrm {not} }$". Though we may take these symbols to stand for certain things or concepts, this is not determined by the interpretation function.

Analytic philosophy is a broad movement and methodology within contemporary Western philosophy, especially anglophone philosophy, focused on: analysis as a philosophical method; clarity of prose; rigor in arguments; and making use of formal logic, mathematics, and to a lesser degree the natural sciences. It is further characterized by the linguistic turn, or a concern with language and meaning. Analytic philosophy has developed several new branches of philosophy and logic, notably philosophy of language, philosophy of mathematics, philosophy of science, modern predicate logic and mathematical logic.

The proliferation of analysis in philosophy began around the turn of the 20th century and has been dominant since the latter half of the 20th century. Central figures in its historical development are Gottlob Frege, Bertrand Russell, G. E. Moore, and Ludwig Wittgenstein. Other important figures in its history include Franz Brentano, the logical positivists (especially Rudolf Carnap), the ordinary language philosophers, W. V. O. Quine, and Karl Popper. After the decline of logical positivism, Saul Kripke, David Lewis, and others led a revival in metaphysics.

In predicate logic, an existential quantification is a type of quantifier which asserts the existence of an object with a given property. It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier ("∃x" or "∃(x)" or "(∃x)"), read as "there exists", "there is at least one", or "for some". Existential quantification is distinct from universal quantification ("for all"), which asserts that the property or relation holds for all members of the domain. Some sources use the term existentialization to refer to existential quantification.

Quantification in general is covered in the article on quantification (logic). The existential quantifier is encoded as U+2203 ∃ THERE EXISTS in Unicode, and as \exists in LaTeX and related formula editors.

Formal semantics is the scientific study of linguistic meaning through formal tools from logic and mathematics. It is an interdisciplinary field, sometimes regarded as a subfield of both linguistics and philosophy of language. Formal semanticists rely on diverse methods to analyze natural language. Many examine the meaning of a sentence by studying the circumstances in which it would be true. They describe these circumstances using abstract mathematical models to represent entities and their features. The principle of compositionality helps them link the meaning of expressions to abstract objects in these models. This principle asserts that the meaning of a compound expression is determined by the meanings of its parts.

Propositional and predicate logic are formal systems used to analyze the semantic structure of sentences. They introduce concepts like singular terms, predicates, quantifiers, and logical connectives to represent the logical form of natural language expressions. Type theory is another approach utilized to describe sentences as nested functions with precisely defined input and output types. Various theoretical frameworks build on these systems. Possible world semantics and situation semantics evaluate truth across different hypothetical scenarios. Dynamic semantics analyzes the meaning of a sentence as the information contribution it makes.

In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language.

The abbreviation wff is pronounced "woof", or sometimes "wiff", "weff", or "whiff".

In predicate logic, existential generalization (also known as existential introduction, ∃I) is a valid rule of inference that allows one to move from a specific statement, or one instance, to a quantified generalized statement, or existential proposition. In first-order logic, it is often used as a rule for the existential quantifier (${\displaystyle \exists }$) in formal proofs.

Example: "Rover loves to wag his tail. Therefore, something loves to wag its tail."

Non-classical logics (and sometimes alternative logics or non-Aristotelian 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.

Philosophical logic is understood to encompass and focus on non-classical logics, although the term has other meanings as well. 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.