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- Wisconsin
- University of Wisconsin - Milwaukee
- Philosophy
- Philosophy 211
- Tierney
- Eight basic Inference rules

Chris M.

Modus Ponens M.P.

Modus Ponendo Ponens

Modus Ponendo Ponens

p [horseshoe] q

p

____________

/therefore q

p

____________

/therefore q

Modus Tollens M.T.

Modus Tollendo tollens

Modus Tollendo tollens

p [horseshoe] q

~q

____________

/therefore ~p

~q

____________

/therefore ~p

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Hypothetical Syllogism H.S.

p [horseshoe] q

q [horseshoe]r

_____________

/therefore p [horseshoe] r

q [horseshoe]r

_____________

/therefore p [horseshoe] r

Simplification Simp.

p * q

____

/therefore p

p*q

___

/therefore q

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____

/therefore p

p*q

___

/therefore q

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Conjunction Conj.

p

q

_______

/ therefore p * q

q

_______

/ therefore p * q

Disjunctive Syllogism D.S.

p v q

~p

_____

/therefore q

p v q

~q

_____

/therefore p

~p

_____

/therefore q

p v q

~q

_____

/therefore p

Addition Add.

p

_____

/therefore p v q

q

_____

/therefore p v q

_____

/therefore p v q

q

_____

/therefore p v q

Dilemma Rule Dil.

p [horseshoe] q

r [horseshoe] s

p v r

_____________

/therefore q v s

r [horseshoe] s

p v r

_____________

/therefore q v s

"Given a conditional and the antecedent of that conditional, you are permitted to infer the consequent of the conditional"

Modus Ponens M.P.

p [horseshoe] q

p

____________

/therefore q

p [horseshoe] q

p

____________

/therefore q

"Given a conditional and the negation of the consequent of the conditional, you are permitted to infer the negation of the antecedent of that conditional"

Modus Tollens M.T.

p [horseshoe] q

~q

____________

/therefore ~p

p [horseshoe] q

~q

____________

/therefore ~p

"Given two conditionals in which the consequent of the first is identical to the antecedent of the second, you may infer the conditional whose antecedent is the antecedent of he first and whose consequent is the consequent of the second"

Hypothetical Syllogism H.S.

p [horseshoe] q

q [horseshoe]r

_____________

/therefore p [horseshoe] r

p [horseshoe] q

q [horseshoe]r

_____________

/therefore p [horseshoe] r

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"From a conjunction as premise, we may infer either of the conjuncts separately as a conclusion"

Simplification Simp.

p * q

____

/therefore p

p*q

___

/therefore q

p * q

____

/therefore p

p*q

___

/therefore q

"Given any two statements, we may infer their conjunction"

Conjuntion Conj.

p

q

_______

/ therefore p * q

p

q

_______

/ therefore p * q

"Given a disjunction and the negation of one of the disjuncts, you may infer the other disjunct"

Disjunctive Syllogism D.S.

p v q

~p

_____

/therefore q

p v q

~p

_____

/therefore q

"Given any statement, you may infer any disjunction that includes that statement as one of the disjuncts"

Addition Add.

p

_____

/therefore p v q

q

_____

/therefore p v q

p

_____

/therefore p v q

q

_____

/therefore p v q

"Given two conditionals and the disjunction of the antecedents of those conditionals, we may infer the disjunction of the consequences of the conditionals"

Dilemma Dil.

p [horseshoe] q

r [horseshoe] s

p v r

_____________

/therefore q v s

p [horseshoe] q

r [horseshoe] s

p v r

_____________

/therefore q v s

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