Modus ponens – a logical implication

(P ^ (P => Q)) => Q named Modus ponens meaning affirms by affirming – affirming the antecedent/premise P

If the compound statement (P ^ (P => Q)) => Q is given and is true (it is always true) and P => Q is true then P must be true – therefore Q must be true (see truth table to follow the reasoning).

P => Q (If P => Q is true then...)
P (must be true)
therefore Q must be true.

In English:
P = today is Tuesday
Q = I will go to work

If today is Tuesday, then I will go to work.
Today is Tuesday.
Therefore, I will go to work.

Picture 15

In instances of modus ponens we assume as premises that p => q is true and p is true. Only one line of the truth table – the last – satisfies these two conditions. On this line, q is also true. Therefore, whenever p => q is true and p is true, q must also be true

Picture 6

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

see Wikipedia

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