Addition – a logical implication

P => (P v Q) meaning, P implies (P or Q)

If the compound statement P=>(P v Q) is given and is true (it is always true) then we can conclude P by just taking P.

If P = the traffic light is green
and Q = the traffic light is red.

Then in English, P=>(P v Q ) will be “If the traffic light is green then the traffic light is green or (otherwise) the traffic light is red”. As mentioned, this compound statement is always true.

So, If the compound expression (P => (P v Q)) is given, we know it is true, so we can always conclude P by just taking P.

Suppose that, it is false that (P) the traffic light is green,
Suppose that, it is true that (Q) the traffic light is red,

Then we can conclude that it is false that the traffic light is green, because P is false.

Suppose that, it is true that (P) the traffic light is green,
Suppose that, it is false that (Q) the traffic light is red,

Then we can conclude that it is true that the traffic light is green, because P is true.

Suppose that it is true that (P) the traffic light is green,
Suppose that it is true that (Q) the traffic light is red,

Then normally we would conclude that the traffic light is broken, but in this case, logically we would conclude that the traffic light is green because P is true.

Suppose that it is true that (P) the traffic false is green,
Suppose that it is true that (Q) the traffic false is red,

Again, normally we would conclude that the traffic light is broken, but in this case, logically we would conclude that it is false that the traffic light is green because P is false.

Picture 17

In the truth table we can see the tautology in P=>(P v Q) – the compound statement is always true – which proves what we have said in the examples.

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