Answer to Question #300190 in Discrete Mathematics for Sphafy

Question

Consider the following premises:

1.A-->(B-->A) is a theorem of proportional calculus for all statements A and B.

2.suppose then that the following are the temporary axioms.

a)w

b)y

c)y-->z

Using the logical rules of inference, modus ponens and hypothetical syllogism, show that x-->z is diducible from the given premises.

Answer 1

Solution:

Y is the given axiom, and Y → Z is the other given axiom. Then applying modus ponens (MP), Z can be obtained. Now, we have two axioms, X, and Z with us. Premise 1 says that for all the statements form A and B, A → (B → A) id valid. Hypothetical syllogism can be understood like: if p, the q. If q, then r. So, if p, then r.

So, assuming X as A, and Z as B, putting in hypothetical syllogism to obtain the result:

X → (Z → X)

(Z → X) → Z, then

X → Z is a tautology or valid