This is an addendum to our earlier discussion which I hope will advance it a step or two. We heard Alan Rhoda claim that the following sentence is false: 'If nothing exists, then it is true that nothing exists.' Let's think further about this. We first note that 'If nothing exists, then it is true that nothing exists' can be parsed in two ways:
1. If nothing exists, then it is true that (nothing exists).
2. If nothing exists, then it is true (that nothing exists).
Call (1) the operator construal. 'It is true that ( )' is a sentential operator the operand of which is a sentence. The result of the operation is itself a sentence. If the operand is true, then the resulting sentence is true. If the operand is false, then the resulting sentence is false. Please note that prefixing 'It is true that' to a sentence cannot change the truth-value of the sentence. In this respect, the truth operator 'It is true that ( )' is unlike the negation operator 'It is not the case that ( ).' Assuming Bivalence -- as I have been doing throughout -- if you negate a true sentence you get a false one, and vice versa.
Call (2) the predicate construal. The consequent of (2) is of course a sentence, but it is not the result or product of a sentential operator operating upon a sentence. For what is within the parentheses is not a sentence. 'That nothing exists' is not a sentence. It does not have a truth-value. If I assertively utter it I do not convey a complete thought to my audience. 'That nothing exists' is the name of a proposition. It follows that 'it is true' in the consequent of (2) functions as a predicate as one can more clearly see from the equivalent
3. If nothing exists, then that nothing exists is true.
In (2) and (3) a predicate is attached to a name, whereas in (1) this is not the case: a sentential operator is attached to a sentence.
Not only are the parsings different, the ontological commitments are as well. (2) commits us to propositions while (1) doesn't. And (1) seems to commit us to operators while (2) doesn't.
Here is the place to comment on my asterisks convention. Putting asterisks around a declarative sentence forms a name of the proposition expressed by the sentence. 'The Moon is uninhabited' is a declarative sentence. '*The Moon is uninhabited*' is not a sentence but a name. It names an entity that has a truth-value, but it itself does not have a truth-value. (2) and (3) can also be rendered as
4. If nothing exists, then *Nothing exists* is true.
With the operator/predicate distinction under our belts we may be in a position to see how one philosopher (Alan) could reasonably reject 'If nothing exists, then it is true that nothing exists' while another accepts it. The one philosopher gives the original sentence the predicate construal which is committed to propositions. This philosopher then reasons that, if nothing exists, then no propositions exist either, and are therefore not available to instantiate the property of being true. The other philosopher gives the original sentence the operator construal and finds it impossible to understand how anyone could reject the original sentence so construed. This philosopher insists that if nothing exists, then it is true that nothing exists; that this truth is not nothing, and that therefore it is something, which implies that it cannot be the case that nothing exists.