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(i) Truth-functor

 j y j or y or both T  T T T  F T F  T T F  F F

(ii)
Truth-functor.

 j It is true that j T T F F

(iii)
Not a truth-functor.

 j It is a necessary truth that j T ? F F

To show that the "?" in the top row (where the constituent sentence is true) is right, we need a pair of examples. We need;

(a) an example of a true sentence which, when it replaces j in the sentence-functor, yields a sentence which is true and; (b) an example of a true sentence which, when it replaces j in the sentence-functor, yields a sentence which is false.

For (a)  we could take "2+2=4" and for (b) "Socrates is a philosopher" (which is true, but not a necessary truth).

(iv)
Not a truth-functor.

 j It could have been the case that j T T F ?

To show that the "?" in the bottom row is right we need a pair of false sentences, one of which (a) yields a sentence which is true, and one of which (b) yields a sentence which is false. We could take for  (a)  "Socrates is a Roman", and for (b) "2+2=5".

(v)
Not a truth-functor.

 j  y j, but not because y T  T ? T  F T F  T F F  F F

To show that the "?" in the first row is right we need a pair of examples, each of which must itself be a pair of true sentences. We want;

(a) one pair to yield a sentence which is true and

(b) another pair to yield a sentence which is false.

For (a) we could take as our pair "Bill Clinton was president of the United States" and "The battle of Hastings took place in 1066", and for (b) we could take "Bill Clinton was president of the United States" and "Bill Clinton secured most votes in the electoral college".

(vi)
Truth-functor.

 j It is true that j or it isn't T T F T

(vii)
Not a truth-functor.

 j j, but grass is not green T ? F F

Here we cannot, of course, produce a pair of examples of true sentences, one of which, when it replaces j, will yield a sentence which is true, and one of which will yield a sentence which is false. This is because the resulting sentence will always as a matter of fact be false. However, it is still right to put a "?" in the top row because the falsity of, for example, "2+2=4, but grass is not green" is not determined solely by the fact that 2+2=4, given the meaning of "j, but grass is not green". Grass might not have been green.

(viii)
Truth-functor.

 j j, whether or not grass is green T T F F

(Though one might dispute the truth of "Grass is green, whether or not grass is green".)

(ix) Not a truth-functor.

 j Everyone knows that it is true that j or it isn't T ? F ?

We certainly shouldn't have a "T" the top line, because it is not true that everyone knows that it is true that Bill Clinton is no longer President of the United States or that it isn't. Some people have never thought about it; indeed some lack the necessary understanding. (Think about a newborn baby.) But, given the case of the newborn baby, shouldn't there be an "F"?

No. Even if as a matter of fact there is no true sentence which could replace j and yield a true sentence, things could have been otherwise – it is just a matter of fact, not a consequence of what the words mean. The same applies to the "?" in the second row.

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