Logic

Logic is initially one of the great disciplines of , it became with XXE century a part of . Today, it is moreover integral part of: , , cognitive and, .

Synopsis

General

logic is the study of nature, , of the truth, the judgements and, of the validity of the reasoning. It is spread thus today according to four large axes' which are:

This classification in four large axes, generally allowed, is that proposed by J. Barwise in its work Handbook of Mathematical Logic. Since, a fifth large axis seems to take shape with work on theory of the types.

Disciplines of logic

Philosophy

Antiquity

logic in the beginning a reflexion is on the agreement of the speech (logos) with itself. One can say that it is an effort of the thought to make its own expression noncontradictory. Consequently, it is a tool (organon) ensuring the coherence of the reflexion. is thus useful of logic to organize its speech and to ensure a relevance concerning its assumptions to him on the world.
The coherence of a speech has two aspects which correspond to the various directions of of truth :

  • Coherence interns speech itself: it is logic in its purely formal aspect.
  • External coherence: it is the definition material truth: « adequatio rei and intellectus ", the agreement of the contents with reality.

The first type of coherence can be done for the second, but is also detached some to constitute an autonomous conceptual field.
In philosophy, logic pose the problem of the relations between and thought : logic seems to be indeed at the same time the effect and the cause of the speech. It rises from logos in philosophy (the direction speech); but, in mathematics (the form), formal coherence seems to be generated of itself.

logic was very early used against itself, i.e. against the same conditions of the speech: the sophist Gorgias uses in sound Treaty of the non-being in order to prove that there is notontology possible: « it is not to be it which is the object of our thoughts ». truth material logic is thus ruined. The language thus acquires its own law, which is that of logic, independent of reality. But them sophists were isolated history of philosophy (sophist took a pejorative direction), so that logic, in comprehension that one had of it for example with , remained subjected to the thought ofto be.

XIXE century

Kant, as for him, logic defines like a science which explains in detail and proves in a strict way, only them formal rules of very thought. The?uvre of Aristote called itOrganon, where figure in particular the study of syllogism, was regarded a long time as the handbook of reference on this subject. But birth of a nonpredicative formal logic, from XIXE century, somewhat changed this established fact. Thus Frege replace it the predicative analysis by a distinction between function and concept.

Logic originates in the fight of truth and the forgery, to be it and of the non-being. It was necessary to await the beginning of XXE century so that the obviousness of this bivalence is called in question: trivalent logics, adding an indefinite value, are then invented (Kleene, Lukasiewicz, Bochvar). But those, spreading in general-purpose logics, did not call nevertheless in question the strict membership of one proposal to the one (and only one) of these values. It is from that Zadeh work out one fuzzy logic (fuzzy logic) in which a proposal is true according to a certain degree of probability (degree to which one assigns itself a degree of probability). Far from the distinct world of the traditional certainty, a fuzzy world appears in all its complexity.

Mathematics

In this last case, its position is a little particular from an epistemological point of view, since it is at the same time a tool of definition of , and a branch of this same mathematics, therefore an object.

Elementary concepts of formal logic

One logical is defined by one , i.e. a system of symbols and of rules to combine them in the forms of formulas. Moreover, one is associated the language. It makes it possible to interpret it, i.e. to attach to these formulas like to symbols a significance. A system of proof also allows us to calculate the significance of the formulas while building demonstrations.

Logic includes/understands classically:

Let us consider a logical language. This last is: that is to say a language propositional, one speaks then about logic of the proposals; that is to say one first order language, one speaks then about logic of the predicates. Obviously, these logical languages differ from their syntax.

Let us consider their respective syntaxes.

The syntax of the logic of the proposals is founded on variables of proposals called also atoms which we note with small letters (p, Q, R, S, etc.). These symbols represent properties which are, either true, or false. These variables are combined by means of logical connectors which are:

  1. the disjunctive binary connector (or),
  2. the conjunctive binary connector (and),
  3. the binary connector of the implication (- >),
  4. the monadic connector of the negation (not).

These variables then form formulas called also proposals. We note them by Greek letters small (phi, psi, theta, etc.).

The syntax of the logic of order one, contrary to that of order zero, considers on the one hand the terms which represent them studied objects, and in addition the formulas which are properties on these objects. In the continuation of this manuscript, we will note V the whole of variables (X, y, Z...), F the whole of the symbols of functions (f,g...) and P the whole of the symbols of predicates (P,Q...). One also has an application known as of arity m.

What happenhappen does significance of a formula? It is the object of semantics. There still, it differs according to the language considered.

In propositional logic, a formula is either true or false. More formally, the whole of the values of truth is a unit B of two Boolean : truth (1) and the forgery (0). Significance of Boolean is defined using functions Boolean worms of Boolean. These functions can be represented in the shape of truth table.

The significance of a formula thus depends on the value on truth of its variables. One speaks about interpretation or assignment.

As in the propositional case, the semantics of the logic of first order is described by an interpretation. However the language first order logic is richer. Consequently, of new definitions are necessary. Contrary to the language propositional, interpretations and the assignments are different objects. An assignment gives a value to each variable, whereas an interpretation describes the field of the values and semantics of the symbols of functions and predicates.

We equipped propositional logic as well as first order logic with a semantics. However, it is difficult, within the meaning of complexity algorithmic, to use it to decide if a formula is satisfiable (or not) even valid (or not). It would be necessary for that to enumerate all interpretations. Their number is exponential. An alternative consists in examining the well formed evidence, and considering their conclusions. For that we use a system of proof.

A system of proof is a couple (A,R), where A is a whole of formulas called axioms and R a whole of rules ofinference, i.e. of relations between sets of formulas (premises) and formulas (the conclusion).

One calls derivation starting from a whole ofassumptions a nonempty succession of formulas which are: maybe of axioms, that is to say formulas deduced from the preceding formulas of the continuation.

A proof of a formula phi starting from a whole of Gamma formulas is a derivation starting from Gamma whose last formula is phi.

Quantification

One introduces primarily two quantifiers in traditional logic:

  • <maths>\exists</maths> (there is at least one), called existential quantifier.
  • <maths>\forall</maths> (for all), called universal quantifier.

A third quantifier, which can be defined starting from the preceding quantifiers, is often introduced:

  • <maths>\exists</maths>! (there is only one).


Thanks to the negation, the existential and universal quantifiers play of the dual roles and thus, in traditional logic, one can found it calculation of the predicates on only one quatificator.

Automatism and Data processing

In these two fields logic is omnipresent and represents the base of these diciplines.

  • In automatism, in order to be able to order processes according to precise conditions, a logical operation is necessary. Usinglogical operators simple and combined, the combinatory logic makes it possible to determine conditions and automated decision-makings. Formerly, them automats multiples contained relay providing these functions. Today, it is in fact of microcomputers specialized having a part ofelectronics of power to interact with its environment and, one interface man/machine adapted.
  • In :
    • in the part numerical electronics, the same logical operators are used into large numbers;
    • in the part , operators of logic of of programming is very much used, as system of comparison and decision-making;
    • on the level of , there exist major relations between logic intuitionalist and it lambda-calculation (and thus functional languages). correspondence of Curry-Howard propose to see the proposals like types, and a proof of a proposal P like a term having the type P. One then obtains rules identical to those used for the typing of the terms of lambda-calculation. This approach is used in a certain number of software of assistance to the proof, like Cock or HOL. Lastly, the addition of continuations the language allows to find traditional logic, the type of these new terms which can be brought closer third-excluded.
    • In order to specify a system (protocol, software...), in particular in model-checking, one calls upon temporal logics.

See too


Theory of knowledge

· Conscience · · Dialectical · Empiricism · · Space · Imagination · Judgement · · Logic · · · Thought · Phenomenology · Philosophy of the language · Cognitive psychology · Reason · Rationalism · Reality · Science · · · Truth

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