Discrete Math – Relationship Between Argument Forms, Rules of Inference, and Tautologies : askmath

I am currently teaching myself discrete math in my spare time. Lately, I have had trouble understanding the relationship between valid argument forms, rules of inference and tautologies.

In the book that I am using it states, “We say this form of argument is valid because whenever all its premises(all statements in the argument other than the final one, the conclusion) are true, the conclusion must also be true.”

The book later talks about fallacies, “These fallacies resemble rules of inference, but are based on contingencies rather than tautologies.

The proposition ((P -> Q) ^ Q) -> P is not a tautology, because it is false when p is false and q is true. However, there are many incorrect arguments that treat this as a tautology.”

In a purely logical sense, I understand why this proposition would lead to an incorrect conclusion. Particularly, it is easy for me to see why

P -> Q


therefore Q is valid and why

P -> Q


therefore P is not.

What confuses me is when its in the form of

((P -> Q) ^ Q) -> P


((P -> Q) ^ P) -> P

Why must this statement be a tautology for it to be used as a rule of inference. Are valid forms and rules of inference the same thing? Also, since a contingency is true when all its premises are true, why can’t we use contingencies for deriving rules of inference?

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