Question

Use rules of inference to show that the hypotheses “If the weather is not too hot or not too cold, then the game will be held and a prize-giving
ceremony will occur,” “If the game is held then the VC will give a
speech,” “The VC did not give a speech,” imply the conclusion “The
weather was too hot.

Answer 1

: a mode of reasoning from a hypothetical proposition according to which if the antecedent be affirmed the consequent is affirmed (as, if A is true, B is true; but A is true; therefore, B is true)

Answer 2

So if all conditions of the given hypothesis are true then the conclusion also be.

Given,

The hypotheses, "If the weather is not too hot or not too cold, then the game will be held and a prize-giving ceremony will occur,", “If the game is held then the VC will give a speech”, “The VC did not give a speech”. And The conclusion, "The weather was too hot."

To Find,

The proof of the conclusion of that hypothesis.

Solution,

We can solve this reasoning problem using the following method.

At first, just assume A indicates Weather is too hot, B indicates Weather is too cold, C indicates the game will held, and D indicates VC will give a speech.

∴We can write "If the weather is not too hot or not too cold" is ~A $\vee$ ~B

"then the game will hold and a prize-giving ceremony will occur" is (~A v ~B)$\rightarrow$ C

"If the game is held then the VC will give a speech" is C $\rightarrow$ D

"The VC didn't give a speech" is ~D

And the conclusion "The weather was too hot" is ~D ⇒ A.

We need to proof (A  $\wedge$ B) ⇒ A.

So we can obtain ~D ⇒ A, by back substituting D to C as, ~C ⇒ A.

C is being tended from ~A  $\vee$ ~B, now again back substituting r to ~A  $\vee$  ~B as, ~(~A v ~B)⇒A

Which means A  $\wedge$ B ⇒ A.

Now let's build Truth Table for the  A  $\wedge$  B  ⇒ A, using Simplification Rules of Inference.

A   B   A $\wedge$ B   (A  $\wedge$ B) ⇒ A

T   T      T            T

T   F      F             T

F   T       F             T

F   F      F             T

In the last line, we can say that the statement (A  $\wedge$  B) ⇒ holds True.

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