Natural Deduction

Origins of the natural deduction

The history of the theories of the deduction followed a curious advance. Many logicians whose Bertrand Russell and Alfred North Whitehead with their Principia Mathematica developed logic in an axiomatic form. The logical laws are deduced starting from logical axioms, in a way which resembles the Euclidean method, at least to the eyes of those which made these deductions. This method could seem relatively natural as long as one chose as axioms of the laws whose logical truth was obvious. But the concern of formulating systems of axioms as reduced as possible led sometimes to formalisms where one deduces from the obvious logical truths from others which are not it. The methods of the logicians made it possible to justify all the reasoning whose validity was recognized but they were very far away from the ideal obviously natural until one waits of logic.

The concept of naturalness is employed by the logicians mainly in two directions. If they are like above obviously natural, or related concepts like the natural deduction, one wants to insist on their universal and necessary character for all to be rational. One can live without never knowing that 2+2 = 4 but one cannot be rational without recognizing that 2+2 = 4 is necessary, as soon as one learned how to us to say it. The second direction of the concept of naturalness intervenes in the distinction between the natural languages and the formal, or artificial languages. The first are the languages which are familiar for us, they were worked by centuries of culture. The seconds are invented by the logicians to develop theories and to give evidence.

Gerhard Gentzen is the first to have developed formalisms which give again with logic the character of a natural advance. The principal starting idea of Gentzen was simple: no logical axioms, that rules of deduction and as much as it is necessary some to reproduce all the elementary and natural forms of reasoning. To carry out this idea, Gentzen developed a formalism where the deductions are not continuations of sentences but trees, facts of columns of sentences which meet until the conclusion. This method is very suggestive for the intuition and it led Gentzen to make beautiful discoveries but it harms the original idea which was to reproduce the natural forms of reasoning. Fitch modified the method of Gentzen while renonçant with the arborescent deductions. The deductions of Fitch are however not simple continuations of sentences. To give to a deduction a natural character it is necessary to reveal relations of dependence between the sentences. Instead of an arborescent form the method of Fitch makes use only of vertical bars. An assertion depends on the sentences which are with the top of it except those which are shifted on the line by these bars. This simple rule makes it possible to make at the same time formal and natural deductions. In this encyclopedia, we will not note the bars, but only the shifts. A sentence will depend on all the assumptions which are with the top of it except those which are shifted towards the line. The assumptions are provisional axioms or assumptions. They will be indicated with the mention (hyp) or (axiom).

Dag Prawitz showed a theorem of elimination of the cuts in natural deduction.

Rules of deduction for the fundamental Boolean operators

The rules exposed in this paragraph are those of the natural deduction for the Calcul of the proposals. They make it possible to connect the sentences logically, i.e. to introduce new sentences as logical consequences of what was known as before. With each fundamental logical operator two rules with deduction are associated. One of the rules is a rule of introduction : she explains how to prove a proposal having the operator. The other rule is a rule of elimination : she explains how to use a proposal having the operator to continue the reasoning.

Introduction and elimination are necessary to be able to dismount and go up formulas. The search for a logical deduction precisely consists in analyzing the premises, i.e. to dismount them, and to reassemble the pieces to make formulas which one can logically connect until the conclusion. It is believed sometimes that it is very difficult to make mathematical evidence, but in its principle, it is not very different from a building set with cubes.

Rules relating to the implication

The rule of introduction of the implication or regulates abandonment of a provisional assumption states that, to prove an implication “if P then Q”, it is enough to proceed in the following way: one poses P like provisional assumption, one then makes deductions starting from all the former sentences plus P in order to reach Q Once. Q reached, one can then deduce from it “if P then Q”. Let us insist on a point: Q is shown under the assumption P but “if P then Q”, it, depends only on the former premises. If one uses the method of Fitch, one can introduce any time into a deduction a provisional assumption. It is enough to shift it towards the line compared to the other premises. All that is deduced under a provisional assumption must be under it or on its right-hand side but never on its left. The rule of introduction makes it possible to conclude that it was proven “if P then Q” without the assumption P. One can shift “if P then Q” on the left compared to P, but not compared to the other premises used in the demonstration of Q. One will note:

1 \ quad \ quad P

(provisional assumption)

2 \ quad \ quad Q

(property deduced from 1)

3 \ quad P \ to Q

(introduction of the implication enters two lines 1 and 2)

The rule of elimination of the implication , or regulates detachment or modus ponens states that, of the two sentences “P” and “if P then Q” one can deduce “Q”. It allows to pass from conditions already known, P, with conditions new, Q, provided that there is a law which authorizes it, “if P then Q”, which will be noted:

1 \ quad P

2 \ quad P \ to Q

3 \ quad Q

(elimination of the implication enters lines 1 and 2)

Let us show for example that with these two rules one can deduce “if P then R” starting from the two sentences “if P then Q” and “if Q then R”:

$1 \ quad P \ to Q$ (assumption)

2 \ quad Q \ to R

(assumption)

3 \ quad \ quad P

(provisional assumption)

4 \ quad \ quad Q

(modus ponens on 3 and 1)

5 \ quad \ quad R

(modus ponens on 4 and 2)

6 \ quad P \ to R

(rule of introduction of the implication between 3 and 5, abandonment of the provisional assumption P).

Rules relating to the conjunction

For the conjunction, the rules are very simple.

The rule of introduction of the conjunction states that, starting from the two sentences has and B one can deduce the sentence (has and B).

1 \ quad A

2 \ quad B

3 \ quad has \ Land B

(introduction of the conjunction between 1 and 2)

The rule of elimination of the conjunction states that, starting from the sentence (has and B), one can deduce the two sentences has and B taken separately.

1 \ quad has \ Land B

2 \ quad A

(elimination of conjunction 1)

3 \ quad B

(elimination of conjunction 1)

These rules make it possible to assemble or contrary to separating from the assertions which all are regarded as true. It is the logical form of the capacity of the reason to analyze the world, i.e. to study its parts separately, and to synthesize it, i.e. to gather the elements of a study in a whole. This is why these rules are also called the rules of the analysis and the synthesis.

Rules relating to disjunction

The two rules of deduction for disjunction are the following ones.

The rule of introduction of disjunction or regulates weakening of a thesis states that, starting from the sentence P one can deduce the sentence (P or Q) as well as the sentence (Q or P), whatever the sentence Q. This rule can seem not very interesting but it is in very important truth. Sometimes it is advantageous to deduce (P or Q) after having proven P, because one can know in addition that (P or Q) has other consequences.

1 \ quad P

2 \ quad P \ lor Q

(introduction of disjunction from 1)

and of the same

1 \ quad Q

2 \ quad P \ lor Q

(introduction of disjunction from 1)

Rule of elimination of disjunction or regulates disjunction of assumptions or rule of distinction of case, stipulates that, if one proved (P or Q) and that it was also proven (if P then R) like (if Q then R), then one proved R. This rule is useful when one examines several possible cases which lead to the same conclusion.

1 \ quad P \ lor Q

2 \ quad \ quad P

(provisional assumption)

3 \ quad \ quad R

(property deduced from 2)

4 \ quad P \ to R

(introduction of the implication between 2 and 3)

5 \ quad \ quad Q

(provisional assumption)

6 \ quad \ quad R

(property deduced from 5)

7 \ quad Q \ to R

(introduction of the implication between 5 and 6)

8 \ quad R

(elimination of disjunction between 1,4,7)

Rules relating to the negation

The rule of introduction of the negation states that, if one shows a contradiction starting from an assumption P then this one is necessarily false and thus its negation is true. Thus, so in a deduction under the provisional assumption P, one found a contradiction (Q and not Q), also noted $\ bot$ , then one can deduce nonP starting from all the former premises except P. With the method of Fitch, one thus shifts (not P) on the left compared to P, which is represented as follows:

1 \ quad \ quad P

2 \ quad \ quad \ bot

3 \ quad \ lnot P

(introduction of the negation between 1 and 2)

The rule of elimination of the negation or regulates suppression of the double-negation or reasoning by absurdity states that, from (not nonP) one can deduce P. It acts well of the reasoning by the absurdity because in, such a reasoning, to prove P, one supposes (not P) and one seeks to obtain a contradiction. If such is the case, one proved (not P) according to the rule of introduction of the negation, and it is well the rule of suppression of the double-negation which makes it possible to conclude to P:

1 \ quad \ lnot \ lnot P

2 \ quad P

(elimination of double negation 1)

1 \ quad \ quad \ lnot P

(provisional assumption by the absurdity)

2 \ quad \ quad \ bot

3 \ quad \ lnot \ lnot P

(introduction of the negation between 1 and 2)

4 \ quad P

(elimination of double negation 3)

When it is considered that any sentence is necessarily or true or false, the validity of this last rule is obvious. It is characteristic of the traditional Logique, which is presented here and is used by the large majority of the scientists. She is sometimes disputed because of an important problem, that of the evidence of existence by the absurdity. It happens that one can prove that a problem has a solution without being able to find it. For that it is enough to prove that the absence of solution leads to a contradiction. The reasoning by the absurdity then makes it possible to conclude that it is not true that the problem does not have a solution: not (the problem has a solution). In traditional logic, one removes the double negation to conclude from it that the problem has a solution. However, the step thus followed does not provide any constructive process of the sought solution. This objection was raised by certain mathematicians and logicians, of which Brouwer, which disputed this method and were opposed to Hilbert which defended it. The mathematicians constructivists or intuitionalists estimate that a proof of existence does not have a direction if it also does not provide a constructive process of the object in question. Also the latter reject the rule of suppression of the double negation to substitute the following rule to him: from P and (not P), one can deduce a contradiction, and of this contradiction any proposal Q.

1 \ quad P

2 \ quad \ lnot P

3 \ quad \ bot

4 \ quad Q

(elimination of negation 2 in logic intuitionalist)

In the examples, we will use this second rule of elimination when it is possible to do without the rule of elimination of the double negation

One can introduce other rules for the other Boolean operators, in particular the operator of equivalence, but it is not necessary, because these operators can be defined starting from the precedents and that their rules of deduction can be then deduced starting from the preceding rules. (P is equivalent to Q) is defined by ((if P then Q) and (if Q then P)).

Examples

We give below some examples of use of the natural deduction. In the first part, we will not use the rule of elimination of the double negation. In the second part, we will use this rule. The formulas deduced in this second part are not recognized valid by the mathematicians intuitionalists. We will use the following symbols:

\ to

for if… then… ,

\ lor

for or ,

\ land

for and ,

\ lnot

for not . The symbol

\ bot

indicates a contradiction, i.e. a false proposal.

Examples not using the elimination of the double negation

Example 1 : $(\ lnot has \ Land \ lnot B) \ to \ lnot (has \ lor B)$

$01 \ quad \ quad \ lnot has \ Land \ lnot B$ (provisional assumption)

02 \ quad \ quad \ lnot A

(elimination of conjunction 01)

03 \ quad \ quad \ lnot B

(elimination of conjunction 01)

04 \ quad \ quad \ quad has \ lor B

(provisional assumption)

05 \ quad \ quad \ quad \ quad has

(provisional assumption)

06 \ quad \ quad \ quad \ quad \ bot

contradiction between 02 and 05

07 \ quad \ quad \ quad has \ to \ bot

(introduction of the implication between 05 and 06)

08 \ quad \ quad \ quad \ quad B

(provisional assumption)

09 \ quad \ quad \ quad \ quad \ bot

contradiction between 03 and 08

10 \ quad \ quad \ quad B \ to \ bot

(introduction of the implication between 08 and 09)

11 \ quad \ quad \ quad \ bot

(elimination of disjunction between 04,07,10)

12 \ quad \ quad \ lnot (has \ lor B)

(introduction of the negation between 04 and 11)

13 \ quad (\ lnot has \ Land \ lnot B) \ to \ lnot (has \ lor B)

(introduction ofimplication between 01 and 12 and end of the deduction)

It is shown that reciprocal the

\ lnot (has \ lor B) \ to (\ lnot has \ Land \ lnot B)

is also valid. It is also shown that

(\ lnot has \ lor \ lnot B) \ to \ lnot (has \ B)

Land, but for the reciprocal one of this last property, one uses the elimination of the double negation.

Example 2 : $A \ to \ lnot \ lnot A$

01 \ quad \ quad A

(provisional assumption)

02 \ quad \ quad \ quad \ lnot A

(provisional assumption)

03 \ quad \ quad \ quad \ bot

(elimination of the negation between 01 and 02)

04 \ quad \ quad \ lnot \ lnot A

(introduction of the negation between 02 and 03)

05 \ quad has \ to \ lnot \ lnot A

(introduction of the implication between 01 and 04, end of the deduction)

Reciprocal the $\ lnot \ lnot has \ to A$ rises directly from the elimination of the double negation and is rejected by the intuitionalists. But curiously, it is not the same of $\ lnot \ lnot \ lnot has \ to \ lnot A$ which is proven without this assumption, and which, it, is accepted by the intuitionalists.

Example 3 : $\ lnot \ lnot \ lnot has \ to \ lnot A$

01 \ quad \ quad \ lnot \ lnot \ lnot A

(provisional assumption)

02 \ quad \ quad \ quad A

(provisional assumption)

03 \ quad \ quad \ quad has \ to \ lnot \ lnot A

(theorem of example 2)

04 \ quad \ quad \ quad \ lnot \ lnot A

(elimination of the implication or modus ponens between 02 and 03)

05 \ quad \ quad \ quad \ bot

(elimination of the negation between 01 and 04)

06 \ quad \ quad \ lnot A

(introduction of the negation between 02 and 05)

07 \ quad \ lnot \ lnot \ lnot has \ to \ lnot A

(introduction of the implication between 01 and 06, end of the deduction)

Examples using the elimination of the double negation

The examples which follow use the elimination of the double negation. One can show that this use is necessary. They are thus not accepted in Logique intuitionalist.

Example 4 : $\ lnot (has \ Land B) \ to (\ lnot has \ lor \ lnot B)$

01 \ quad \ quad \ lnot (has \ Land B)

(provisional assumption by the absurdity)

02 \ quad \ quad \ quad \ lnot (\ lnot has \ lor \ lnot B)

(provisional assumption)

03 \ quad \ quad \ quad \ lnot (\ lnot has \ lor \ lnot B) \ to (\ lnot \ lnot has \ Land \ lnot \ lnot B)

(theorem, reciprocal of example 1)

04 \ quad \ quad \ quad \ lnot \ lnot has \Land \ lnot \ lnot B

(elimination of the implication or modus ponens between 02 and 03)

05 \ quad \ quad \ quad has \ Land B

(elimination of the double negation in 04)

06 \ quad \ quad \ quad \ bot

(elimination of the negation between 01 and 05)

07 \ quad \ quad \ lnot \ lnot (\ lnot has \ lor \ lnot B)

(introduction of the negation between 02 and 06)

08 \ quad \ quad \ lnot has \ lor \ lnot B

(elimination of the double negation in 07)

09 \ quad \ lnot (has \ Land B) \ to (\ lnot has \ lor \ lnot B)

(introduction of the implication between 01 and 08)

Example 5 : if $(has \ to B) \ to (\ lnot B \ to \ lnot A)$ does not require the elimination of the double negation, this one is necessary to prove reciprocal the $(\ lnot B \ to \ lnot A) \ to (has \ to B)$ .

Example 6 : in the same way the demonstration of the validity of the third excluded $A \ lor \ lnot A$ uses the elimination of the double negation. By supposing that $\ lnot (has \ lor \ lnot A)$ , one of deduced $\ lnot has \ Land \ lnot \ lnot A$ (reciprocal of example 1), from where a contradiction and the validity of $\ lnot \ lnot (has \ lor \ lnot A)$ .

If the intuitionalists reject the excluded third, they on the other hand accept the principle of noncontradiction: $\ lnot (has \ Land \ lnot A)$ . Indeed, the assumption $A \ Land \ lnot A$ is a contradiction. There is thus $\ lnot (has \ Land \ lnot A)$ by introduction of the negation.

known Example 7 under the name of Law of Peirce : $((has \ to B) \ to A) \ to A$ . Curiously, although not containing negation, the deduction of this law requires the elimination of the double negation, for example via the use of the excluded third. It is supposed that one has $((has \ to B) \ to A)$ . According to the excluded third, or there is $(has \ to B)$ and the modus ponens involves that one has A. Or one has $\ lnot (has \ to B)$ , i.e. $A \ Land \ lnot B$ and thus one has A. well.

Rules of deduction for the quantifiers

The use of the names of variable in the first order theories

The rules of deduction for the universal and existential operators control the use of the names of variable. This use gives to a theory the power of the general information, i.e. the possibility of knowing not each invividu taken separately but all the individuals of a same kind, in only one sentence.

The rules of use of the variables specify in which conditions one can introduce names of variable and what one can then say on their subject. These rules are natural but there are some technical difficulties in connection with the concepts of term, of free or dependant occurrence of a variable, of substitution of a term to the occurrences of a variable and of substitution of a variable to the occurrences of a term.

So that a theory can use first order logic it is necessary to have defined a field of objects and it is necessary that the predicates allotted by the theory to its objects are not themselves of the objects of the theory.

The significance of a name of variable of object, it is to represent an unspecified object of the theory: that is to say X a number. Often one introduces a name of variable into premises which determine general terms on this object. X is an unspecified object of the theory only provided which it satisfies these conditions: that is to say X a prime number… A theory in general contains names for its objects. The theory of the integers contains for example names for all the numbers: 0,1,2,…, -1,-2,…

The terms can be simple or made up. They are the names of object, the names of variable of object, and all the made up expressions which one can form from them by applying the operators of objects of the theory. For example, 1, X, x+1et (x+x) +1 is terms of the theory of the numbers.

Let us point out initially the very important distinction between the variables bound by an operator and the others, the free variables. The occurrences of a name of variable in a sentence are all the places where this name appears. An occurrence can be free or dependant. When an existential or universal operator in X is applied to a complex sentence, all the occurrences of X become dependant by this operator. All the occurrences which are not thus bound are free.

If a sentence contains several existential and universal operators, it is desirable that these operators carry all on different names of variable. This rule is not essential. It is not respected when one puts the calculation of the predicates in the form of a cylindrical algebra (a cylindrical algebra is a Boolean algebra supplemented by laws particular to the universal and existential operators). But it is always respected in practice because it makes it possible to avoid confusions.

A sentence is obtained from an other by substitution of a term T to the occurrences of a variable X when all the free occurrences of X were replaced per T. For example, (the father of X is human and the father of X is honest) is obtained by substitution of the term the father of X to the variable there in the formula (is human there and is honest there).

These preliminaries make it possible to formulate the rules of deduction for the universal and existential operators.

Rules relating to the universal quantifier

The rule of introduction of the universal quantifier or regulates generalization states that, from P (X) one can deduce (for any X, P (X)) provided that the name of variable X never appears on the assumptions whose P (X) depends.

1 \ quad P (X)

2 \ quad \ forall X \; P (X)

(introduction of whatever the , where X is not free in the premises of P)

Very often one introduces variables into provisional assumptions. One then reasons on them as if they were objects. One can then deduce some from the general laws, because what one deduced is true for all the objects which check the same assumptions. They are the rules of abandonment of a provisional assumption and generalization which make it possible to conclude.

The rule of elimination of the universal quantifier or regulates singularisation states that, starting from a sentence of the form (for any X) P (X), one can deduce P (T), for any term T whose variables are not dependant in P (X). P (X) indicates an unspecified sentence which contains X like free variable, P (T) indicates the sentence obtained starting from P (X) in there substituent T with all the free occurrences of X. The rule of singularisation simply consists in applying a universal law to a singular case.

1 \ quad \ forall X \; P (X)

2 \ quad P (T)

(elimination of whatever the )

Rules relating to the existential quantifier

The rule of introduction of the existential quantifier , or regulates direct evidence of the existence, states that, starting from the sentence P (T), one can deduce it exists X such as P (X) for any variable X which do not appear in P (T) and for any term T whose names of variables are not dependant in P (X).

P (T) indicates an unspecified sentence which contains at least once the term T.P (X) is obtained starting from P (T) in substituent X with T with one or more of its occurrences. In this rule it is not necessary to substitute X for all the occurrences of T.

1 \ quad P (T)

2 \ quad \ exists X \; P (X)

(introduction of there exists )

The rule of elimination of the existential quantifier or regulates consequences of the existence allows, starting from a proposal (there exists X, P (X)), to introduce a new provisional assumption P (there) where there is a name of variable which used forever before and where P (there) is obtained starting from P (X) in substituent there with all the occurrences of X. One can then reason under this assumption. The rule of the consequences of the existence then makes it possible to conclude in the following way: sentence (there exists X such as P (X)) one can deduce R when R was deduced under the single provisional assumption additional P (there) and that there appears neither in the premises former to P (there) nor in R. is a kind to be there hypothetical. It does nothing but be used as intermediary in a deduction but it appears neither in its premises, nor in its conclusion.

1 \ quad \ exists X \; P (X)

(assumption)

2 \ quad \ quad P (there)

( there variable news)

3 \ quad \ quad R

(deduction from 2, not utilizing there )

4 \ quad R

(elimination of there exists in 1 and 3)

Examples

Example 1 $(\ exists X \; \ lnot has (X)) \ to (\ lnot \ forall X \; With (X))$

1 \ quad \ quad \ exists X \; \ lnot has (X)

(provisional assumption)

2 \ quad \ quad \ quad \ forall X \; With (X)

(provisional assumption)

3 \ quad \ quad \ quad \ quad \ lnot has (there)

( there variable news from 1)

4 \ quad \ quad \ quad \ quad has (there)

(elimination of whatever the in 2)

5 \ quad \ quad \ quad \ quad \ bot

(elimination of negation 3,4)

6 \ quad \ quad \ quad \ bot

(elimination of there exists between 1 and 5)

7 \ quad \ quad \ lnot (\ forall X \; With (X))

(introduction of the negation between 2 and 6)

8 \ quad (\ exists X \; \ lnot has (X)) \ to (\ lnot \ forall X \; With (X))

(introduction of implication 1,7)

Example 2 : there is also $\ lnot (\ forall X \; With (X)) \ to \ exists X (\ lnot has (X))$ , but the proof requires an elimination of the double negation, and this theorem of traditional logic is not accepted in logic intuitionalist.

How to check the correction of a reasoning?

For these twelve rules, or fourteen, if it is counted that the rule of eliminination of the conjunction is in fact a double rule, and the same for the rule of introduction of disjunction, it is necessary to add one of them fifteenth, very simple, the rule of repetition: one can always repeat a former thesis, except if it depends on an assumption which was abandoned. One can thus repeat all the theses as long as one does not shift them on the left. This rule is necessary when one wants to repeat a premise in a conclusion or when one needs a former thesis in the body of a deduction under provisional assumption to observe a rule.

The list of the fifteen preceding rules is complete with the direction where it is enough to make all the correct deductions. Kurt Gödel proved, (Théorème of complétude) for a different logical system, but equivalent with that which has just been presented, that it is enough to formalize all the correct deductions of the Calcul of the first order predicates.

The correct deductions are initially all those which comply with these fundamental rules rigorously. All the sentences must be or many axioms, or provisional assumptions, or consequences of the sentences which precede them under the terms of one by the fifteen rules. By preoccupation with a speed, one can also use derived rules (for example to make follow $\ lnot (\ lor B)$ by $\ lnot has \ Land \ lnot B$ ). A derived rule is correct when one can show that all that can be deduced with it can also be deduced without it with only the fundamental rules.

As long as the deductions are formalized, it is always possible to recognize, including by a program, if a deduction is correct because the rules of deduction are of finished number and that it is always possible to check in a finished time if a formula is the consequence of one or more others under the terms of one of these rules. This step carries out part of the program called sometimes finitaire proposed by David Hilbert to solve the problems of the bases of mathematics. Hilbert sought an effective method which makes it possible to decide for very stated mathematical if it is a theorem, i.e. a mathematical truth. The calculation of the predicates made it possible to bring back the problem to the question of knowing if a formula is a logical law or not.

But if it is possible to check the validity of a deduction leading to a formula, contrary, the question of knowing if a given formula is a logical law or not cannot be solved in general. Indeed, there does not exist universal method making it possible to say if a formula is a logical law or not. This inexistence results from the Théorème of incomplétude of Gödel. It is said that the question of knowing if a formula is a logical law is indécidable.

The checking of the correction of the deductions in the current language raises more difficulties. It is initially necessary to translate the sentences familiar in a formalized language of the calculation of the predicates. This stage poses sometimes problem if one has doubts about the fidelity of the translation. But the largest difficulty comes from what the current deductions do in general a broad place with the implicit one. Even the relations of dependence compared to an assumption are not always clarified. The formal deductions on the contrary do not leave anything in the shade. Those which are proposed here are very similar to the current deductions except that them clarify all. In order to certify correct the usual reasoning and demonstrations it is necessary to clarify of all that is implicit. This is why often it is necessary much to study and know all the implicit one before knowing if a reasoning is correct.

See too

Calculation of the predicates
traditional Calculation of the proposals
Logical
Calculation of the séquents
Logical intuitionalist
Theorem of complétude

A prouvor of formulas is with the address

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