Simplification – a logical implication

(P ^ Q) => P named Simplification meaning if (P ^ Q) then P
(P ^ Q) => Q named Simplification meaning if (P ^ Q) then P

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

The same reasoning applies to (P ^ Q) => Q.

P = I cannot get hold of any money
Q = The bank will not lend me any money

The the compound statement: “if I cannot get hold of any money and the bank will not lend me any money then I cannot get hold of any money” is always true.

Suppose that, it is true that (P) I cannot get hold of any money
Suppose that, it is true that (Q) The bank will not lend me any money

Then we can conclude that it is true that: “I cannot get hold of any money – if I cannot get hold of any money and the bank will not lend me any money”. You can see that just more truth (The bank will not lend me money) to some original truth (I cannot get hold of any money) – So we can take it that the original truth is still true if both the original truth (I cannot get hold of any money) and the added truth (the bank will not lend me any money) are both true.

Picture 19

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