Mathematical logic

The logical mathematics was born at the end of the 19th century from the Logique to the philosophical direction from the term. Its beginnings were marked by the meeting between two ideas nouvelles :

• will at Frege, Russell or at Hilbert to later give a axiomatic foundation to the mathématiques  ;
• the discovery by George Boole of the existence of algebraic structures allowing to define a “calculation of truth”.

Mathematical logic thus took again the objective of the Logique, to study the reasoning, but while being restricted with the language of mathematics which has the advantage of being extremely standardized. It is what returned possible the definition of various “logical systems” formalizing the reasoning in mathematics and the very fast development of mathematical logic during the XXe century.

Before finding its name current, allotted to Giuseppe Peano, mathematical logic was called “symbolic logic” (in opposition to philosophical logic) and “metamathematic” (terminology of Hilbert). Peano used its notations of mathematical logic for its Formulaire of mathematics, an attempt to formalize those.

In XXIe century, mathematical logic ramified in many under-fields, whose majority have to only see very little with the initial objectives of the mathematicians of the XIXe century, but are mathematical disciplines with whole share. Notamment  is counted;:

• the Set theory   ;
• the Theory of the demonstration   ;
• the Theory of the models   ;
• the Theory of the calculability.

This classification in four main roads, generally allowed, is that proposed by Jon Barwise in its work Handbook off Mathematical Logic . Since, fifth main roads seem to take shape with work on the Théorie of the types.

Some data of history

The first attempts at formal treatment of mathematics are due to Leibniz and Lambert; Leibniz introduced most of the modern mathematical notation in particular (use of the quantifiers, symbol of integration, etc). However one can speak about mathematical logic only starting from the medium of the XIXe century with work of George Boole (and to a lesser extent of Auguste De Morgan) which introduces a calculation of truth where logical combinations like the conjunction, disjunction and the implication, are operations similar to the addition or multiplication of the entireties, but bearing on the values of truth false and true (or 0 and 1); these Boolean operations are defined by means of truth tables .

The calculation of Boole seemed a profitable track in order to solve the problems of foundation of mathematics due to their complexification and the appearance of the Paradoxe S, but it did not make it possible to take into account the concept of variable. Consequently many mathematicians, sought to extend it to the general framework of the mathematical reasoning and one saw appearing the formalized logical systems   ; one of the first is due to Frege with the turning of the 20th century.

In 1900 during a very famous conference with the international Congress of mathematics in Paris, David Hilbert proposed a list of the 23 unsolved problems most important of mathematics of then. The second was that of the coherence of the Arithmétique, i.e. to show by means finitists the non-contradiction of the axioms of the arithmetic one.

The program of Hilbert caused many work in logic in the first quarter of the century, in particular the system development of axioms for the mathématiques : the Axioms of Peano for the Arithmetic , those of Zermelo supplemented by Skolem and Fraenkel for the Set theory and the development by Whitehead and Russell of a programme of formalization of mathematics, the Principia Mathematica. It is also the period when appear the principles founders of the Théorie of the models  : concept of model of a theory, Theorem of Löwenheim-Skolem.

In 1929 Kurt Gödel watch in his thesis of doctorate its Theorem of complétude which states the success of the company of formalization of the mathématiques : any mathematical reasoning can be formalized in theory in the Calcul of the predicates. This theorem was accommodated like a notable projection towards the resolution of the program of Hilbert, but one year later Gödel showed the Théorème of incomplétude (published in 1931) which irrefutably showed impossibility of carrying out this program.

This negative result however did not stop the rise of mathematical logic. The Thirties saw arriving a new generation of English and American logician, in particular Alonzo Church, Alan Turing, Stephen Kleene, Haskell Curry and Emil Post, which largely contributed to the definition of the concept of algorithm and to the development of the theory of the algorithmic complexity . The Théorie of the demonstration of Hilbert also continued to develop with work of Gerhard Gentzen which produced the first demonstration of relative coherence and thus initiated a programme of classification of the axiomatic theories.

The most spectacular result of the post-war period is due to Paul Cohen which shows by using the method of the sustained pressure the independence of the assumption of continuous in set theory, thus solving the 1st problem of Hilbert. But mathematical logic also undergoes a revolution due to the appearance of the informatique  ; the discovery of the Correspondance of Curry-Howard which connects the formal evidences to the Lambda-calculation of Church and gives calculative contents to the demonstrations will start a research vast program.

Some fundamental results

Some important results were established during the years 1930:

• the Theorem of complétude of the Calculation of the predicates first order that Gödel showed in its thesis of doctorate, one year before its famous Théorème of incomplétude of Gödel. This theorem states that any mathematical demonstration can be represented in the formalism of the calculation of the predicates (which is thus complete ).

• the whole of the theorems of the calculation of the predicates is not not calculable, i.e. no algorithm makes it possible to check if a given statement is provable or not. There exists, however, an semi-algorithm: this one takes in entry a first order formula; it stops by answering “yes” if the formula is universally valid, and continues indefinitely if not. Even if this semi-algorithm is carried out during billion years, it can not conclude that a proposal is not universally valid. In other words, the whole of the first order formulas universally valid is “Récursivement énumérable”; it is said that it is “semi-décidable”.

• the coherence (not contradiction) of a system of axioms (for example: the Axiomes of Peano for the Arithmétique) is not a consequence of these only axioms safe in commonplace cases (in particular when the axioms are contradictory between them). It is the famous second Théorème of incomplétude of Gödel.

• the whole of all the universally valid formulas of the second order is not énumérable, even recursively. This is a consequence of the theorem of incomplétude.

• Any theorem purely logical can be shown by using constantly only proposals which are subformulas of its statement. Known under the name of theorem of the subformula , it is a consequence of the theorem of elimination of the cuts in Calcul of the séquents of Gerhard Gentzen.

• Incidentally, this theorem of the subformula provides a theorem of coherence for logic, because it prohibits the derivation of the empty formula (identified with the nonsense).

Other important results were established after the years 1930:

• the independence of the Hypothèse of continuous the compared to the other axioms of the set theory (ZF) is completed in 1963 by Paul Cohen what was worth the Médaille Fields to him.

• the développemnt of the theory of the Calculabilité.
• In the year 1980 the theory of the types was revivified by the Correspondance of Curry-Howard between evidence and programs which establishes the bond between what is done in mathematics and what is done in data processing.
• a major discovery in this field (1990) is that the law of Peirce, therefore traditional logic in addition to only the logical intuitionalist, is prone to this correspondence and thus that traditional logic is also constructive.

Logical system

Definition

A logical system or system of deduction consists of three components. The two first define its Syntaxe, the third its Sémantique  :

• a whole of formulas, or made   ; in the systems of Logical traditional or intuitionalist, the formulas represents mathematical statements expressed formally. The formulas are defined by combinatoires  means;: Continuation S of symbols, Tree S labelled, Graph S…

• a whole of Deduction s  ; the deductions are also defined by combinative means. A deduction makes it possible to derive from the formulas (the provable formulas or theorems ) starting from starting formulas (the axioms ) by means of rules (the rules of inference ).

• a interpretation of the formules  ; it is about a function associating with any Formula One an object in an abstract structure called model , which makes it possible to define the validity formulas.

Syntax and semantics

The principal characteristic of the formulas and the deductions is that they are finished objects   ; more still, each one of the whole of formulas and deductions is recursive, i.e. one has an algorithm which determines if a given object is a correct formula or a correct deduction of the system.

The Semantic , on the contrary, escapes the Combinatoire by calling (in general) upon infinite objects. It was initially used “to define” the truth. For example, in Calculation of the proposals, the formulas are interpreted by elements of a Boolean algebra; the valid formulas are those which are interpreted by the largest element.

A warning statement is of rigor here, because Kurt Gödel (followed by Alfred Tarski) showed with its Théorème of incomplétude impossibility of defining the mathematical truth mathematically. This is why it is more adapted to see semantics like a métaphore : the formulas - syntactic objects - are projected in another world, more abstract, for example the Boolean algebra. This technique - to plunge the objects in a vaster world for better reasoning above - is usually used in mathematics and amply showed its effectiveness.

Thus semantics is not only used “to define” the truth. For example, the dénotationnelle Sémantique is an interpretation, not formulas, but of the deductions aiming at capturing to them contained calculative .

Systems of deduction

In logics traditional and intuitionalist, one distinguishes two types of Axiome s : the logical axioms which express purely logical properties such as for example $A\; \backslash \; lor\; \backslash \; lnot\; A$ (Principe of the third excluded, valid in traditional logic but not in logic intuitionalist) and the axioms extra-logics which define mathematical objects, for example the Axiomes of Peano which define the Arithmétique or the axioms of Zermelo-Fraenkel which define the Set theory. When the system has axioms extra-logics, one speaks about axiomatic theory. The study of the various models of the same axiomatic theory is the object of the Théorie of the models.

Two systems of deduction can be equivalent to the direction where they have the same theorems exactly. One shows this equivalence by providing translations of the deductions of the one in the deductions of the other. For example, there exist (at least) three types of equivalent systems of deduction for the Calcul of the predicates  : the Systems in Hilbert, the natural Deduction and the Calculation of the séquents. One adds to it sometimes the style of Fitch which is an alternative of the natural deduction in which the demonstrations are presented in a linear way.

Whereas the theory of the numbers shows properties of the numbers, one will note the main feature of logic as a mathematical theory: it “shows” properties of systems of deduction in which the objects are “theorems”. There is thus a double level in which one should not be lost. To clarify the things, the “theorems” stating of the properties of the systems of deduction are sometimes called “métathéorèmes”. The system of deduction which one studies is called the “theory” and the system in which one states and shows the métathéorèmes is called the “métathéorie”.

The fundamental properties of the systems of deduction are the following ones.

• the correction : The correction states that the theorems are valid in all the models. That means that the rules of inference are well adapted to these models, in other words that the system of deduction is a manner of reasoning well in these models.

• the coherence : A system of deduction is coherent (it is also said that it is consistent by imitation with English) if it admits a noncommonplace model, i.e. a model which has at least two elements. That amounts affirming that in this system of deduction, there exist proposals which are not theorems.

• the complétude : It is defined compared to a family of models. A system of deduction is complete compared to a family of models, if any proposal which is valid in all the models of the family is a theorem. In short, a sytème is complete if all that is valid is demonstrable.

Another property of the systems of deduction connected with the complétude is the maximum coherence . A system of deduction is maximalement coherent, if the addition of a new axiom which is not itself a theorem makes the system incoherent.

To affirm “such system is complete for such family of models” is a typical example of métathéorème.

It will be noted that within this framework, the intuitive concept of truth gives place to two formal concepts: validity and demonstrability. The three properties of correction, coherence and complétude specify how these concepts can be connected, hoping that thus the truth, searches of the logician, can be encircled.

The calculation of the proposals

See also: Calculation of the proposals

The mathematical proposals and objects, are assemblies of symbols and letters formed while following certain rules of Syntaxe. The principal symbols of logic are called connectors , they are primarily used to create new proposals starting from proposals already created. In the calculation of the proposals, the basic proposals that one calls also the variable propositional do not have contents (do not have significance) a priori. One can replace a propositional variable by “it rains” , but it is not the weather contents which interest the logician, but the way in which the basic proposals are combined to build reasoning.

Dans the continuation of this section, one supposes that one places in the traditional Logique, because the reduction of the connectors to one or two is not possible in Logique intuitionalist and the limitation with only one disjunction and only one conjunction is not true in linear Logique.

P a proposal and X and are there of the unspecified objects.

• (∀ X ) (∀ there ) P ⇔ (∀ there ) (∀ X ) P
• (∃ X ) (∃ there ) P ⇔ (∃ there ) (∃ X ) P
• (∃ X ) (∀ there ) P ⇒ (∀ there ) (∃ X ) P
  (reciprocal Implication distorts in general)