Changes

::: <math>\Omega_1 = \{ \lnot \} \,,</math>

::: <math>\Omega_1 = \{ \lnot \} \,,</math>

::: <math>\Omega_2 = \{ \rightarrow \} \,.</math>

::: <math>\Omega_2 = \{ \Rightarrow \} \,.</math>

An axiom system discovered by [[Jan &#321;ukasiewicz|Jan Lukasiewicz]] formulates a propositional calculus in this language as follows:

An axiom system discovered by [[Jan &#321;ukasiewicz|Jan Lukasiewicz]] formulates a propositional calculus in this language as follows:

::* <math>p \to (q \to p)</math>

::* <math>p \Rightarrow (q \Rightarrow p)</math>

::* <math>(p \to (q \to r)) \to ((p \to q) \to (p \to r))</math>

::* <math>(p \Rightarrow (q \Rightarrow r)) \Rightarrow ((p \Rightarrow q) \Rightarrow (p \Rightarrow r))</math>

::* <math>(\neg p \to \neg q) \to (q \to p)</math>

::* <math>(\neg p \Rightarrow \neg q) \Rightarrow (q \Rightarrow p)</math>

The inference rule is [[modus ponens]], from p, (p &#8594; q), infer q. Then a &#8744; b is defined as ¬a &#8594; b, and a &#8743; b is defined as ¬(a &#8594; ¬b).

The inference rule is ''[[modus ponens]]'':

 

::* From ''p'', (''p'' &rArr; ''q''), infer ''q''.

 

Then we have the following definitions:

 

::* ''p'' &or; ''q'' is defined as &not;''p'' &rArr; ''q''.

 

::* ''p'' &and; ''q'' is defined as &not;(''p'' &rArr; &not;''q'').

==Example 2.  Natural deduction system==

==Example 2.  Natural deduction system==