Minimal logic

The minimal logical is, like the Logique intuitionalist, an alternative of the traditional Logique. Three logics differ on the way of treating the negation and contradiction in the Calculation of the proposals or the Calculation of the predicates. To a certain extent, minimal logic does not approach this concept and represents a logic without true negation.

Comparison enters various logics

One will utisera like notation the following symbols:

\ lor

for disjunction,

\ land

for the conjunction,

\ to

for the implication,

\ lnot

for the negation,

\ leftrightarrow

for equivalence.

Common rules

In three logics, one has the two following rules, relating to the negation:

the rule of elimination of the negation : If there is at the same time a $A$ proposal and his negation $\ lnot A$ , then there is a contradiction, noted $\ bot$ .
the rule of introduction of the negation : If a $A$ proposal leads to a contradiction, then it is that $\ lnot A$ is valid. This rule can be taken as definition of the negation besides: $\ lnot A$ is a synonym of $A$

\to \bot.

Differences

Three logics differ on the consequences to draw from a contradiction.

traditional logic uses the reasoning by the absurdity and deduced from $\ lnot \ to \ bot$ the fact has that $A$ is valid. It is in fact a rule of elimination of the double negation, since $\ lnot has \ to \ bot$ is a synonym of $\ lnot \ lnot A$ .
logic intuitionalist deduces from a contradiction any proposal: $\ club-footed \ to B$ , which one summarizes by the formula ex falso sequitur quodlibet .
minimal logic does not envisage any treatment related to $\ bot$ .

It results from it that minimal logic does not establish a difference between the formula $\ bot$ and any other formula. Let us consider for example an unspecified formula $C$ . Let us define $\ sim A$ like synonym of $A \ to C$ . One has then:

If there is at the same time $A$ and $\ sim A$ , then one has $C$ . Indeed, from $A$ and $A \ to C$ , one can deduce $C$ . It is the rule of the modus ponens .
If a $A$ proposal leads to $C$ , then one has $A \ to C$ and thus $\ sim A$ .

One thus sees that, if one allots no role particular to contradiction, one can make play the part of this contradiction to any formula

C

, by defining the negation as being

A \ to C

, and that conversely, one can remove any reference to the negation in minimal logic.

By preoccupation with a comparison with other logics, we will continue nevertheless to use the symbols $\ lnot$ and $\ bot$

Valid examples of formulas in minimal logic

Example 1 :

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

Indeed, let us suppose that one has $\ lnot has \ Land \ lnot B$ (in other words, one has at the same time $\ lnot A$ and $\ lnot B$ ). Let us show that one has $\ lnot (has \ lor B)$ , in other words, show that the assumption $A \ lor B$ led to a contradiction. Let us distinguish the cases: either there is $A$ which is quite contradictory with the assumption $\ lnot A$ , or there is $B$ which is contradictory with $\ lnot B$ . In all the cases, there are a contradiction, CQFD.

Reciprocally, let us suppose that one has $\ lnot (has \ lor B)$ and show that one has $\ lnot A$ , in other words that $A$ leads to a contradiction. But $A$ involves $A \ lor B$ which is contradictory with the assumption. CQFD. One proceeds in the same way to show $\ lnot B$ .

On the other hand, there is only $(\ lnot has \ lor \ lnot B) \ to \ lnot (has \ Land B)$ , the reciprocal one being only true in traditional logic.

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

Let us suppose that $A$ is had. Then the additional assumption $\ lnot A$ led to a contradiction. There is thus $\ lnot \ lnot A$ . CQFD

It should be noted that the reciprocal one is invalid in minimal logic, as well as in logic intuitionalist. One has however $\ lnot \ lnot \ lnot has \ to \ lnot A$ . Indeed, let us suppose that $\ lnot \ lnot \ lnot A$ . The additional assumption $A$ involves $\ lnot \ lnot A$ which is contradictory with $\ lnot \ lnot \ lnot A$ , therefore one has $\ lnot A$ .

Example 3 : One can show the validity in minimal logic of $\ lnot \ lnot (has \ to B) \ to (\ lnot \ lnot has \ to \ lnot \ lnot B)$ . But the reciprocal one is only valid in logic intuitionalist or traditional logic.

Invalid examples of formulas

Example 1 : The formula $\ lnot \ lnot has \ to A$ is invalid in minimal logic. Indeed, if it were provable, then one could also prove, by replacing $\ bot$ by an unspecified proposal $C$ that $((has \ to C) \ to C) \ to A$ , but this last formula is not even provable in traditional logic, without additional assumption on $C$ .

Example 2 : The formula $(\ lnot has \ lor B) \ to (has \ to B)$ is valid in logic intuitionalist and traditional logic, but not in minimal logic. Indeed, a proof would require to suppose $\ lnot has \ lor B$ and to deduce $A \ to B$ from it, in thus to suppose $A$ and to deduce $B$ from it. The use of $\ lnot has \ lor B$ by disjunction of the cases and $A$ to prove $B$ would require to prove that $\ lnot A$ and $A$ prove $B$ , and that $B$ and $A$ prove $B$ . But the proof of $B$ starting from $\ lnot A$ and $A$ does not exist in minimal logic. It exists in logic intuitionalist, since, from contradiction $A \ Land \ lnot A$ , one can deduce $B$ .

Minimal logic and traditional logic

Translation of Gödel

The minimal logic, amputated by the treatment of the negation, seems quite poor in front of traditional logic or logic intuitionalist. It of it is however not so distant. It is indeed shown, that, for any formula has, it exists a A' formula, equivalent to has in traditional logic, such as has is provable in traditional logic if and only if A' is provable in minimal logic. A' is obtained by means of the translation of Gödel , defined repeatedly as follows:

\ bot' = \ bot

p' = \ lnot \ lnot p

for any atomic formula different of

\ bot

(\ lnot A) “= \ lnot A'

(has \ Land B)” = A' \ Land B'

(has \ to B) “= has” \ to B'

(has \ lor B) “= \ lnot \ lnot (has” \ lor B')

(\ forall X \; A) “= \ forall X \; A'

(\ exists X \; A)” = \ lnot \ lnot \ exists X \; A'

In other words, the translation of Gödel of a formula consists in adding double negations in front of the atomic formulas, the existential disjunctions and quantifiers. That means that in traditional logic, it is enough to call upon reasoning by the absurdity only in front of atomic formulas, existential disjunctions or quantifiers.

Examples

For example, the third excluded

A \ lor \ lnot A

is a theorem of traditional logic, but not of minimal logic. On the other hand, in minimal logic the formula

\ lnot \ lnot (\ lnot \ lnot has \ lor \ lnot \ lnot \ lnot A)

is valid. Indeed, this one is equivalent, in minimal logic, with

\ lnot \ lnot (\ lnot \ lnot has \ lor \ lnot A)

or with

\ lnot (\ lnot \ lnot \ lnot has \ Land \ lnot \ lnot A)

or with

\ lnot \ bot

, i.e. with

\ club-footed \ to \ bot

which is a valid formula.

See too

Calculation of the proposals
traditional Calculation of the predicates
Logical
Logical intuitionalist

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