@Horace phil,
Horace;77232 wrote:A sentence "a" is tautologically equivalent to a sentence "b" if and only if the sentence "a if and only if b" is a tautology...
So I have to say whether these are true or false...
1. A sentence -[A or B] is tautologically equivalent to the sentence [-A or -B]
This statement is true. I made a truth table.
2. A sentence - [A downward arrow B] is tautologically equivalent to the sentence [A and B].
This statement is true. I made a truth table to prove it.
3. A sentence [A if and only if B] is not tautologically equivalent to any sentence containing a negation sign (i.e. "-").
I haven't tested this but I believe this statement would also be true. It won't be tautologically equivalent.
4. Any consistent set of sentences which contains a sentence A could be extended to form a consistent set containing the sentence [A or B], where B is any sentence whatsoever.
Not entirely sure about this one.
5. The expression "it is necessary that A" is a unary sentence functor but not a truth functor.
Well I know it's not a truth-functor, but I dont know what a unary sentence functor is yet. So I'll have to check.
6. If the conditional correspondent to an argument is true, then the argument is valid.
I believe this is true to but I need to think through it a little more.
So out of six, how did I do?
Very badly.
1. A sentence -[A or B] is tautologically equivalent to the sentence [-A or -B]
False. -[A or B] <-> [-A and -B].
2. A sentence - [A downward arrow B] is tautologically equivalent to the sentence [A and B].
False, -[A nor B] <-> [A or B].
3. A sentence [A if and only if B] is not tautologically equivalent to any sentence containing a negation sign (i.e. "-").
False.
[A <-> B] <-> [-A <-> -B]
[A <-> B] <-> -[A <-> -B]
[A <-> B] <-> -[-A <-> B].
[A <-> B] <-> [-[A or B] or [A and B]].
4. Any consistent set of sentences which contains a sentence A could be extended to form a consistent set containing the sentence [A or B], where B is any sentence whatsoever.
True. A -> [A or B] is tautologous.
5. The expression "it is necessary that A" is a unary sentence functor but not a truth functor.
imo, (5) is also false, Necessity and possibility are both unary functors and truth functional. See:
http://www.philosophyforum.com/philosophy-forums/branches-philosophy/logic/7050-truth-table-method-decision-modal-logic.html
6. If the conditional correspondent to an argument is true, then the argument is valid.
If you mean:
The argument [A, B] therefore [C] can be stated in the form
(A and B) -> C, is a tautology ...then I agree.
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