logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

, Nicod's axiom (named after the

French French (french: français(e), link=no) may refer to: * Something of, from, or related to France ** French language, which originated in France, and its various dialects and accents ** French people, a nation and ethnic group identified with Franc ...

logician Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

and

philosopher A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...

Jean Nicod Jean George Pierre Nicod (1 June 1893, in France – 16 February 1924, in Geneva, Switzerland) was a French philosopher and logician, best known for his work on propositional logic and inductive reasoning, induction. Biography Nicod's main contr ...

) is a formula that can be used as the sole

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

of a semantically complete system of

propositional calculus Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...

. The only connective used in the formulation of Nicod's axiom is the

Sheffer's stroke In Boolean functions and propositional calculus, the Sheffer stroke denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called nand ("not and") or ...

. The

has the following form: ((φ , (χ , ψ)) , ((τ , (τ , τ)) , ((θ , χ) , ((φ , θ) , (φ , θ)))))

Nicod Jean George Pierre Nicod (1 June 1893, in France – 16 February 1924, in Geneva, Switzerland) was a French philosopher and logician, best known for his work on propositional logic and induction. Biography Nicod's main contribution to formal log ...

showed that the whole

propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...

of ''

Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...

'' could be derived from this

alone by using one inference rule, called "Nicod's modus ponens": 1. φ 2. (φ , (χ , ψ)) ∴ ψ In 1931, the

Polish Polish may refer to: * Anything from or related to Poland, a country in Europe * Polish language * Poles, people from Poland or of Polish descent * Polish chicken *Polish brothers (Mark Polish and Michael Polish, born 1970), American twin screenwr ...

Mordechaj Wajsberg discovered an equally powerful and easier-to-work-with alternative: ((φ , (ψ , χ)) , (((τ , χ) , ((φ , τ) , (φ , τ))) , (φ , (φ , ψ))))

References

External links

*{{wikisource-inline, A Reduction in the number of the Primitive Propositions of Logic Propositional calculus Theorems in propositional logic pl:aksjomat Nicoda-Łukasiewicza