Logical axiom

The axiomatic method makes it possible to define the whole of the logical laws first order starting from logical axioms and of rules of deduction in such way that all the logical laws are or an axiom or a formula derived from the axioms with a finished number of applications of the rules of deduction.

This presentation, purely syntactic, is equivalent to the semantic presentation of the Théorie of the models, which makes it possible to define a logical law as a true formula in all the possible worlds. This equivalence is the subject of a theorem of complétude.

Logical axioms of Principia Mathematica

The logical laws are obtained inside the system of Whitehead and Russell (1910) starting from six diagrams of axioms and two rules of deduction, the rule of detachment and the rule of generalization.

Diagrams of axioms

These diagrams of axioms are the following. p, Q, and R can be replaced by unspecified formulas (with or without free variables) of the Calcul of the predicates to the first order.

if (p or p) then p

if p then (p or Q)

if (p or Q) then (Q or p)

if (if p then Q) then (if (p or R) then (Q or R))

if (any X is such as p) then p'

where p' is obtained starting from p in substituent a variable there, nondependant in p, with all the free occurrences of X in p.

if (any X is such as (p or Q)) then (p or any X is such as Q)

where p is a formula which does not contain X like free variable

Two rules of deduction

The rule of detachment or modus ponens says that of the two premises p and (if p then Q) one can deduce Q.

The rule of generalization says that single premise p one can deduce (any X is such as p)

Equivalence with the natural deduction

One can prove that all the anhypothetic truths, within the meaning of the natural deduction, are or many axioms obtained starting from these diagrams, or many consequences which one can deduce in a finished number of stages starting from these axioms with the two rules of deduction.

All the evidence that one can formalize in the natural deduction can be formalized in the logical calculation (with the first order) of Whitehead and Russell and conversely.

Complétude of the system

Gödel proved a theorem of complétude which affirms that these six diagrams of axioms and these two rules of deduction are enough to obtain all the logical laws.

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