Truth table, implication and equivalence

[bobby2k]

Hello, I have some questions about the truth tables for impliocation and equivalence.

for implication we have:

p | q | p=> q

T | T | T
T | F | F
F | T | T
F | F | T


Here I do not understand the last two lines, how can we say that p implies q when p is false, and q is either true or false, if we only know that p is false and q is true, shouldn't p=> be unknown instead of T?
The same for p is false and q is false?, shouldn't p=>q then be unknown.

I have the same problem for equivalence:

p | q | p<=> q

T | T | T
T | F | F
F | T | F
F | F | T

Here I only have the problem with the last line when both p and q are false. How can we then say that p implies q?

 

[MarneMath]

You have to move away from the English language idea of "implies" and into a more general mathematical view of it. In fact, it would behoove you to realize that "P implies Q" is just a convenient way of saying P -> Q, which to many people tends to imply that you can deduce Q from P, which as you can see isn't always true.

The idea that if given a false statement and a true or false conclusion, then the conclusion doesn't matter, because the statement is false. Think of it like a contract. If you run a mile, then I'll give you water. What if you ran half mile and I gave you water? What if you ran half a mile and I don't give you water? Well, I didn't lie, because my conclusion was only guaranteed when you fulfilled your obligation, so therefore no matter the case, I did a 'truthful' thing.

 

[bobby2k]

You have to move away from the English language idea of "implies" and into a more general mathematical view of it. In fact, it would behoove you to realize that "P implies Q" is just a convenient way of saying P -> Q, which to many people tends to imply that you can deduce Q from P, which as you can see isn't always true.

The idea that if given a false statement and a true or false conclusion, then the conclusion doesn't matter, because the statement is false. Think of it like a contract. If you run a mile, then I'll give you water. What if you ran half mile and I gave you water? What if you ran half a mile and I don't give you water? Well, I didn't lie, because my conclusion was only guaranteed when you fulfilled your obligation, so therefore no matter the case, I did a 'truthful' thing.

Thank you, I think I understand it now.

I have another question though. I think I used the wrong arrows, for what I was supposed to explain I should have use -> and <-> instead of => and <=>. Is there any easy way to explain the difference of these two kinds of implications?

Is it correct to use the implication a>0 -> "a is positive" or a >0 => "a is positive"

 

[Bacle2]

You may also be interested in:

http://en.wikipedia.org/wiki/Material_conditional#Philosophical_problems_with_material_conditional

 

[MLP]

Implication and Equivalence

Thank you, I think I understand it now.
I have another question though. I think I used the wrong arrows, for what I was supposed to explain I should have use -> and <-> instead of => and <=>. Is there any easy way to explain the difference of these two kinds of implications?

Quine has called this an unfortunate choice of terminology dating back at least to Russell of calling the statement connective '$\supset$' or '→' "implication". This invites confusion with the notion of "logical implication" which is the relationship between formulas A and B when it is not possible for A to be true and B false.

Similarly, by calling the sentence connective '$\leftrightarrow$' or '$\equiv$' "equivalence" we invite confusion with the notion of "logical equivalence" which is the relationship between formulas A and B when A logically implies B and B logically implies A.

Sometimes the symbol '$\Rightarrow$' is used for logical implication and the symbol '$\Leftrightarrow$' is used for logical equivalence. Notice, however, that in this case these symbols belong not to the object language (sentential calculus, predicate calculus, etc.) but to the meta-langauge. Logical implication and logical equivalence are relationships between formulas not sentence connectives.

Using '$\Rightarrow$' for logical implication and '$\Leftrightarrow$' for logical equivalence, we can capture their relationship with '→' and '$\leftrightarrow$' as follows:

A $\Rightarrow$ B if and only if 'A → B' is logically true.
A $\Leftrightarrow$ B if and only if 'A $\leftrightarrow$ B' is logically true.