Tautological Equivalence...

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Reply Tue 14 Jul, 2009 11:29 am
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?
 
Theages
 
Reply Sat 8 Aug, 2009 11:36 am
@Horace phil,
For #2, what do you mean by "downward arrow"? There are a lot of different notations floating around. I think that the arrow is normally used for "true iff ~a.~b" (NOR), whereas the stroke means "false iff ~(ab)" (NAND). If this is the case, then #2 would not be an equivalence, although ab would imply ~(a down b). Of course, I could be wrong. Just make sure that you're using the right operator.
 
Owen phil
 
Reply Thu 15 Apr, 2010 07:14 am
@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.
 
Jason Kaufman
 
Reply Thu 15 Apr, 2010 02:55 pm
@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?

It seems that you have a decent grasp on what a tautology is. A simpler way of viewing a tautology is that a tautology is simply circular reasoning. It validates itself.
 
Owen phil
 
Reply Thu 15 Apr, 2010 03:09 pm
@Jason Kaufman,
Jason Kaufman;152474 wrote:
It seems that you have a decent grasp on what a tautology is. A simpler way of viewing a tautology is that a tautology is simply circular reasoning. It validates itself.


What? There is no proposition that validates itself.

Tautology cannot be circular reasoning. Circular reasoning is a particular fallacy of reasoning.
 
kennethamy
 
Reply Thu 15 Apr, 2010 03:18 pm
@Jason Kaufman,
Jason Kaufman;152474 wrote:
It seems that you have a decent grasp on what a tautology is. A simpler way of viewing a tautology is that a tautology is simply circular reasoning. It validates itself.


A tautology is not reasoning at all. It is a statement. It is not an argument. All bachelors are unmarried is a tautology. It is a true statement, as all tautologies are. But statements are not a kind of reasoning. Arguments are a kind of reasoning.
 
 

 
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