# Truth table, implication and equivalence

1. Jul 1, 2013

### 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?

2. Jul 1, 2013

### 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.

3. Jul 1, 2013

### bobby2k

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"

4. Jul 2, 2013

5. Jul 3, 2013

### MLP

Implication and Equivalence

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.

Last edited: Jul 3, 2013