## 1

(i)

Correct

The only structure which makes ¬[P®Q] true (the second row) makes [¬P®Q] true too.

(ii)

Correct

The only structure which makes ¬[P®Q] true (the second row) makes [P®¬Q] true too.

(iii)

Incorrect.

Rows three (P is false and Q is true) and four (P is false and Q is false) both provide counterexamples.

(iv)

Correct.

It is easiest here to look at the rows in which the formula after the turnstile is false. There are three, and in each of them at least one of the formulae before the turnstile is false too. (In the third row both are false; in the fourth row [P®Q] is false; in the seventh row [R®Q] is false.) So there is no structure in which all the formulae before the turnstile are true and the formula after it is false.

(v)

Incorrect.

Rows three (P is true, Q is false, R is true) and seven (P is false, Q is false, R is true) both provide counterexamples.

(vi)

Incorrect.

The structure where P is false and Q is true is a counterexample. So is the structure where P is true and Q is false.

(vii)

Correct

(viii)

Correct. (Any sequent whose premises are inconsistent is correct. This corresponds to the fact that any argument with inconsistent premises is valid.)

(ix)

Correct. (Any sequent whose conclusion is a semantic theorem – i.e. is never false – is correct. This corresponds to the fact that any argument whose conclusion is a necessary truth is valid.)

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