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.

see also:

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