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Logic Exercises - Truth-Functional Logic

Hey

Currently struggling through an exercise on definitions in truth-functional logic. Anyone any good with the below question? I'd appreciate it so much!

Say whether the following statements are true or false, giving reasons for your answers:

(i) If the sentence X is a tautology, and the inference from X to Y is truth-functionally valid, then Y is also a tautology.

(ii) If the sentence Y is a tautology and the inference from X to Y is truth-functionally valid, then X is also a tautology.

(iii) If the sentence X is a contradiction, and the inference from X to Y is truth-functionally valid, then Y is also a contradiction.

xxx
kissmekate
Hey

Currently struggling through an exercise on definitions in truth-functional logic. Anyone any good with the below question? I'd appreciate it so much!

Say whether the following statements are true or false, giving reasons for your answers:

(i) If the sentence X is a tautology, and the inference from X to Y is truth-functionally valid, then Y is also a tautology.


If the premise of an argument is a tautology then it is true in all structures. An argument is truth functionally valid if there exists no structure in which all premises can be true and the conclusion false. If the premises of an argument are true in all structures, and the argument is logically valid, then the conclusion must be true in all structures, which makes the conclusion a tautology. Therefore the above sentence is true.

(ii) If the sentence Y is a tautology and the inference from X to Y is truth-functionally valid, then X is also a tautology.

The definition of logical validity above is: An argument is logically valid if and only if there exists no structure in which all premises can be true and the conclusion false. From this, when the premises of a valid argument are true the conclusion must be true, but if one of the premises (X) is false it is possible for the conclusion (Y) to be either true or false. Therefore, when the conclusion is false we know that one of the premises must be false, but when the conclusion is true it tells us nothing about the truth value of the premises. Therefore it does not follow that a tautological conclusion must have a tautological premise.

(iii) If the sentence X is a contradiction, and the inference from X to Y is truth-functionally valid, then Y is also a contradiction.

xxx

If the premise of an argument is a contradiction then it is false in all structures. As a result, there exists no structure in which all premises can be true and the conclusion false, because the premises can never be true, which means that all arguments with contradictions as premises are logically valid. Y does not have to be a contradiction; in fact, it can be anything, true or false, and the argument is still logically valid. Anything follows from a contradiction according to the principle of explosion.

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