Propositional Calculus - Basic and Derived Argument Forms

Basic and Derived Argument Forms
Name Sequent Description
Modus Ponens If then ; ; therefore
Modus Tollens If then ; not ; therefore not
Hypothetical Syllogism If then ; if then ; therefore, if then
Disjunctive Syllogism Either or, or both; not ; therefore,
Constructive Dilemma If then ; and if then ; but or ; therefore or
Destructive Dilemma If then ; and if then ; but not or not ; therefore not or not
Bidirectional Dilemma If then ; and if then ; but or not ; therefore or not
Simplification and are true; therefore is true
Conjunction and are true separately; therefore they are true conjointly
Addition is true; therefore the disjunction ( or ) is true
Composition If then ; and if then ; therefore if is true then and are true
De Morgan's Theorem (1) The negation of ( and ) is equiv. to (not or not )
De Morgan's Theorem (2) The negation of ( or ) is equiv. to (not and not )
Commutation (1) ( or ) is equiv. to ( or )
Commutation (2) ( and ) is equiv. to ( and )
Commutation (3) ( is equiv. to ) is equiv. to ( is equiv. to )
Association (1) or ( or ) is equiv. to ( or ) or
Association (2) and ( and ) is equiv. to ( and ) and
Distribution (1) and ( or ) is equiv. to ( and ) or ( and )
Distribution (2) or ( and ) is equiv. to ( or ) and ( or )
Double Negation is equivalent to the negation of not
Transposition If then is equiv. to if not then not
Material Implication If then is equiv. to not or
Material Equivalence (1) ( is equiv. to ) means (if is true then is true) and (if is true then is true)
Material Equivalence (2) ( is equiv. to ) means either ( and are true) or (both and are false)
Material Equivalence (3) ( is equiv. to ) means, both ( or not is true) and (not or is true)
Exportation from (if and are true then is true) we can prove (if is true then is true, if is true)
Importation If then (if then ) is equivalent to if and then
Tautology (1) is true is equiv. to is true or is true
Tautology (2) is true is equiv. to is true and is true
Tertium non datur (Law of Excluded Middle) or not is true
Law of Non-Contradiction and not is false, is a true statement

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