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Claudio Conforti

martes, 27 de julio de 2010

Formal Semantics

Formal semantics is the study of the semantics, or interpretations, of formal and also natural languages. 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 is an assignment of meanings to these symbols and truth-values to its sentences.

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.

Linguists rarely employed formal semantics until Richard Montague showed how English (or any natural language) could be treated like a formal language. His contribution to linguistic semantics, which is now known as Montague grammar, forms the basis for what linguists now refer to as formal semantics.


The Cambridge Dictionary of Philosophy, Formal semantics

4 comentarios:

  1. GAME-THEORETIC SEMANTICS Game-theoretic semantics is a method of assigning semantic values to formulas in terms of idealized games played between a “verifier” and a “falsifier.” Atomic sentences are assigned truth values as usual, and then a compound formula receives a truth value based on playing games
    with respect to the formula in question. We define game play as follows. At each “move” in the game, a formula Φ is in play:
    If Φ is atomic, then the verifier wins if Φ is true, and the falsifier wins if Φ is false.
    If Φ is a conjunctionΨ∧Σ, then the falsifier picks one of Ψ or E and play continues on the selected formula.
    If Φ is a disjunction Ψ ∨ Σ, then the verifier picks one of Ψ or Σ and play continues on the selected formula.
    If Φ is a negation ~ Ψ, then the verifier and falsifier switch roles and play continues on Ψ.
    If Φ is a universally quantified formula (∀x)Ψx, then the falsifier picks a member of the domain b and play continues on Ψb.
    If Φ is a existentially quantified formula (∃x)Ψx, then the verifier picks a member of the domain b and play continues on Ψb.
    A compound formula is true if and only if there is a winning strategy for the verifier on that formula – that is, there is a set of moves such
    that the verifier can guarantee that he will win the game on that formula no matter what the falsifier does. Similarly, a formula is false
    if and only if there is a winning strategy for the falsifier.

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  2. BRANCHING QUANTIFIER A branching quantifier (or Henkin quantifier) is a matrix of first-order quantifiers that allows for variable dependencies that cannot be expressed in standard first order logic, which is written with a linear notation. For example, in
    the branching quantifier:
    ⎛(∀x)(∃y) ⎞
    ⎢ ⎢Φ
    ⎝(∀z)(∃w)⎠
    the existential variable y depends on the universal variable x,
    but not on the universal variable z, and the existential variable w
    depends on the universal variable z, but not on the universal variable x.
    A logic which allows branching quantifiers is expressively stronger than standard first-order logic, but expressively weaker than
    second-order logic. Certain statements in natural language, such as the Geach-Kaplan sentence, cannot be adequately
    formalized using first-order logic, but can be formalized using branching quantifiers. Independence-friendly logic provides a
    linear notation for branching quantifiers.

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  3. Claudio te puedo enviar un viejo artículo de filosofía de la lógica que escribí hace un tiempo.......... ¡Y no lo pude comentar con nadie! :-( ¿Puede ser este un espacio para debatirlo?

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  4. Gabriel obvio que sí !! Va a ser un placer leerlo y disfrutarlo
    Desde ya, gracias por compartirlo.

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