Disjunctive Syllogism – a logical implication

(not P ^ (P v Q) => Q named Disjunctive Syllogism formerly known as modus tollendo ponens meaning if not P and P or Q then we can conclude Q.

P v Q (if P or Q is true and….
not P (is true)
then Q must be true.

if P = The car is fast
and Q = The car comes in first

Then the compound statement: “If the car is not fast and (the car is fast or the car comes in first) then the car comes in first” is always true.

Suppose that (P) is false, the car is fast
Suppose that (Q) is true, the car comes in first

Then if the car is not fast and (the car is fast or the car comes in first) then we can conclude that the car comes in first. What happened was that the statement the car is not fast removes the statement the car is fast – all that is left is the “the car comes in first” – if that statement is true then the conclusion can only be that it is true that that the car comes in first.

So you might say as a human, wow the car was not first, but it came in first.

Picture 20

3 no namers – logical implications
Hypothetical Syllogism – a logical implication
Disjunctive Syllogism – a logical implication
Simplification – a logical implication
Addition – a logical implication
Modus tollens – a logical implication
Modus ponens – a logical implication
Logical Implications

Logical Implications

logical implications are used as rules of inference. Implications are (tautologies) of propositional logic. They are simple to prove by constructing truth tables for them that show the tautologies. more implication explanations and some simple exercises at the end of that page


Examples of usage:

Law of Excluded Middle: Proof in Tarski’s propositional calculus.

3 no namers – logical implications
Hypothetical Syllogism – a logical implication
Disjunctive Syllogism – a logical implication
Simplification – a logical implication
Addition – a logical implication
Modus tollens – a logical implication
Modus ponens – a logical implication
Logical Implications