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.”

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Let,

p = Weather is too hot

q = Weather is too cold

r = game will held

s = VC will give a speech

As per the statements,

"If the weather is not too hot or not too cold" = ~p v ~q

"then the game will held and a prize giving ceremony will occur" = (~p v ~q) -> r

"If the game is held then the VC will give a speech" = r -> s

Finally,

"The VC didn't give a speech" = ~s

Conclusion is "The weather was too hot" = ~s => p

Writing the above statements altogether as,

~s => p

s is being tend from r, so back substituting s to r as,

~r => p

r is being tend from ~p v ~q, so back substituting r to ~p v ~q as,

~(~p v ~q) => p

Therefore, p ^ q => p

Let us construct Truth Table for the above statement p ^ q => p

(p ^ q) => p is one of the rules under Rules of Inference, the rule name is Simplification.

p q p ^ q (p ^ q) -> p

T T T T

T F F T

F T F T

F F F T

The values of the last column says all the conditions are True, therefore the statement (p ^ q) => p holds True.