Natural deduction for propositional logic

● Natural deduction is a method of deducing the logical consequences of a set of
premises using a set of inference rules. In propositional logic, natural deduction is
used to determine the validity of logical statements.


● The basic rules of natural deduction in propositional logic include:
○ Modus ponens: If P and P -> Q are both true, then Q is also true.
○ Modus tollens: If P -> Q is true and Q is false, then P is false.
○ Hypothetical syllogism: If P -> Q is true and Q -> R is true, then P -> R is true.
○ Disjunctive syllogism: If P V Q is true and P is false, then Q is true.
○ Conjuctive syllogism: If P ^ Q is true and P is true, then Q is true.
○ Addition: If P is true, then P V Q is true.
○ Simplification: If P ^ Q is true, then P is true.
○ Conjunction: If P is true and Q is true, then P ^ Q is true.
○ Resolution: If (P V Q) ^ (~Q V R) is true, then P V R is true.
○ Reductio ad absurdum: If ~P is true, then P -> Q is true for any statement Q.


● These rules can be used to construct natural deduction proofs for propositional
logic, which are valid arguments that demonstrate the logical consistency of a set of
premises. By using the rules in a systematic way, one can deduce new statements
that are logically entailed by the premises, and ultimately determine whether a given
statement is logically consistent with a set of premises.


● It is important to note that natural deduction is a formal system, and the truth or
falsity of statements can only be determined within the context of the system.
Therefore, natural deduction can only be used to determine the logical consistency