# 3 no namers – logical implications

(P=>Q) => ((Q => R) =>(P => R))

This is actually the hypothetical syllogism in another form. For by considering (P Q) as a proposition S, (Q R) as a proposition T, and (P R) as a proposition U in the hypothetical syllogism above, and then by applying the “exportation” from the identities, this is obtained

((P => Q) ^ (R => S)) => ((P ^R) => (Q ^ S))

For example, if the statements “If the wind blows hard, the beach erodes.” and “If it rains heavily, the streets get flooded.” are true, then the statement “If the wind blows hard and it rains heavily, then the beach erodes and the streets get flooded.” is also true.

((P <=> Q) ^( Q <=> R)) => (P <=> R) This is saying that you can infer that P is equal to R when ((P <=> Q) ^( Q <=> R)).

This just says that the logical equivalence is transitive, that is, if P and Q are equivalent, and if Q and R are also equivalent, then P and R are equivalent.

# Hypothetical Syllogism – a logical implication

((P=>Q) ^ (Q => R)) => (P => R) named Hypothetical Syllogism

P => Q (if P imples Q and…)
Q => R (if Q implies R…)
Then, P must imply R.

P = I do not wake up
Q = I cannot go to work.
R = I will not get paid.

If I do not wake up, then I cannot go to work. (P => Q)
If I cannot go to work, then I will not get paid. (Q => R)
Therefore, if I do not wake up, then I will not get paid. (P => R)

# 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.

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.