Discussion Forum

Unit 10: Reference card: Eight Valid Forms of Inference (and a definition :)

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Modus ponens 

Way that affirms

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1.  p ⊃ q
2.  p
3.  ∴ q

Affirming p therefore q

Modus tollens

Way that denies

1.  p ⊃ q
2.  ~q
3.  ∴ ~p

Denying q with ~q we can conclude ~p

Syllogism: 

An instance of a form of reasoning in which a conclusion is drawn (whether validly or not) from two given or assumed propositions (premises), each of which shares a term with the conclusion, and shares a common or middle term not present in the conclusion (e.g., all dogs are animals; all animals have four legs; therefore all dogs have four legs ).
(Oxford English Dictionary, 2022:online)

Disjunctive syllogism

(historically known as modus tollendo ponens (MTP),^[3] Latin for "mode that affirms by denying")^[4] is a valid argument form which is a syllogism having a disjunctive statement for one of its premises.^[5][6]

1.    p v q
2.    ~p
3.    ∴ q

if we have asserted a disjunction and we have asserted the negation of one of the disjuncts, then we are entitled to assert the other disjunct.

Hypothetical syllogism

chain argument

1.  p ⊃ q
2.  q ⊃ r
3.  ∴ p ⊃ r

 a hypothetical syllogism is a valid argument form, a syllogism with a conditional statement for one or both of its premises.

Simplification

Imply truth of either conjuct in a conjunction

1.    p ⋅ q
2.    ∴ p

If we have asserted a conjunction then we are entitled to infer either one of the conjuncts.

Conjunction

"Reverse Simplification"

1.  p
2.  q
3.  p ⋅ q

If you have asserted two different propositions, then you are entitled to assert the conjunction of those two propositions. Don't confuse the rule called conjunction with the type of complex proposition called a conjunction

Addition

Takes an known assertion and uses it in a disjunction with another unknown assertion

1.    p
3.    ∴  p v q

if we have asserted some proposition, p, then we are entitled to assert the disjunction of that proposition p and any other proposition q we wish. 

Constructive dilemma

Given p or q is true,

And they are used as the antecedents in conditionals the we can conclude the consequents in a disjunction.

1.    p v q
2.    p ⊃ r
3.    q ⊃ s
4.    ∴ r v s

The first premise is a disjunction.  The second premise is a conditional statement whose antecedent is the left disjunct of the disjunction in the first premise.  And the third premise is a conditional statement whose antecedent is the right disjunct of the disjunction in the first premise.  The conclusion is the disjunction of the consequents of the conditionals in premises 2 and 3.