Implication

[triclino]

In defining implication we say that:

 

p => q is false, if p is true and q false, and is true for all other cases.

 

The question is if the definition could be other wised.

 

For example instead of F =>T defined as T ,why not defined as F

[Mr Skeptic]

There's two forms of implication. One is (if p then q) <==> (q or not p). For example "if the moon is made of green cheese, then I'm superman" which would be true. Here, p and q may be anything.

 

Another is the sort used frequently in common speech, where to the above one would ask, "So how does the moon being made of green cheese make you superman?" rather than "Of course that will always be true, since in fact the moon is not made of green cheese". This one is harder to define. I think that what it comes down to is that you can logically deduce that if p then q. This would, however, be true only of very specific p and q. Example: "If a number is even, then it is a multiple of 2".

[triclino]

There's two forms of implication. One is (if p then q) <==> (q or not p). For example "if the moon is made of green cheese, then I'm superman" which would be true. Here, p and q may be anything.

 

Another is the sort used frequently in common speech, where to the above one would ask, "So how does the moon being made of green cheese make you superman?" rather than "Of course that will always be true, since in fact the moon is not made of green cheese". This one is harder to define. I think that what it comes down to is that you can logically deduce that if p then q. This would, however, be true only of very specific p and q. Example: "If a number is even, then it is a multiple of 2".

 

 

So the argument : if the moon is made of cheese then i am a superman.But since the moon is not made of cheese ,hence i am not a superman .

 

Is valid??

[Cap'n Refsmmat]

"If the moon is made of cheese, then I am a superman" is valid. But the rest of the argument you construct is not, because denying the antecedent is a logical fallacy. There could be other things that cause you to be Superman besides the Moon's cheesiness.

[Mr Skeptic]

So the argument : if the moon is made of cheese then i am a superman.But since the moon is not made of cheese ,hence i am not a superman .

 

Is valid??

 

No. You seem to keep having this same problem. I suggest reading up a little on logical fallacies.

 

The argument you can make, however is this: if the moon is made of green cheese, then I'm superman. I am not superman. Therefore, the moon is not made of green cheese. The other way, the argument is invalid and unsound.

 

For example, try this:

If something is a horse, then it has four legs.

My table is not a horse

Therefore, my table does not have four legs.

 

Is that valid? It is the same as the argument you keep getting confused by, with a specific example where the answer turns out not to be true.

Edited April 30, 2010 by Mr Skeptic
messed up my example

[triclino]

"If the moon is made of cheese, then I am a superman" is valid.

.

 

First of all this is not an argument and thus cannot be valid or invalid .It is simply an implication and can only be true or false.

 

It is true because we have false implying false

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Merged post follows:

Consecutive posts merged

No. You seem to keep having this same problem. I suggest reading up a little on logical fallacies.

 

I simply asked

 

Remember in this forum we have not come up with a theorem or axiom or definition deciding for the validity or invalidity of an argument

 

The argument you can make, however is this: if the moon is made of green cheese, then I'm superman. I am not superman. Therefore, the moon is not made of green cheese. The other way, the argument is invalid and unsound.

 

For example, try this:

If something is a horse, then it has four legs.

My table has four legs.

Therefore, my table is a horse.

 

Is that valid? It is the same as the argument you keep getting confused by, with a specific example where the answer turns out not to be true.

 

As i said examples do not make up a theory

[Mr Skeptic]

First of all, sorry I put the two of my sentences backwards.

 

As i said examples do not make up a theory

 

And you can say that all you like, but it will be just as false. A single example is all it takes to prove some theories (which is equivalent to disproving the negation of that theory).

 

For example, given the theory "There are no usernames that begin with 't'" and the negation, There is at least one usernames that begin with 't'" Only one example is needed to disprove the theory.

 

In science, to be useful theories have to say a lot. This makes them hard to prove, so that many many examples are needed. Philosophy is not similarly constrained.

[triclino]

This is not the subject of our discussion now. We discussed that in another thread ,let it rest.

[D H]

For example instead of F =>T defined as T ,why not defined as F

Short answer: Because that is the only value that makes sense.

 

 

Long answer: Let's start with the truth table for implication, if P then Q, or [math]P\rightarrow Q[/math], or [math]P \Rightarrow Q[/math]. Here I am using 0 to indicate false, 1 to indicate true.

 

[math]

\begin{array}{l|ll}

\multicolumn{1}{c}{}&\multicolumn{2}{c}{Q} \\

P & 0 & 1 \\

\hline

0& 1 & 1 \\

1& 0 & 1

\end{array}

[/math]

 

How to construct that?

 

There are two valid production rules associated with implication. These are modus ponens and modus tollens. The first is what lets us derive Q from knowledge that both P and if P then Q are true statements. The latter is what lets us derive ~P from knowledge that Q is a false statement but if P then Q is a true statement. Modus ponens tells us that the bottom row (P=1) of the truth table must be 0 for Q=0 and 1 for Q=1. Modus tollens tells us that the left column (Q=0) of the truth table must be 1 for P=0 and 0 for P=1. Thus the truth table for implication must look like

 

[math]

\begin{array}{l|ll}

\multicolumn{1}{c}{}&\multicolumn{2}{c}{Q} \\

P & 0 & 1 \\

\hline

0& 1 & ? \\

1& 0 & 1

\end{array}

[/math]

 

There are also two invalid production rules that people try to associate with implication. These are affirming the consequent and denying the antecedent. The first says that trying to draw a conclusion from if P then Q given that Q is a true statement is invalid, while the second says that trying to draw a conclusion from if P then Q given that P is a false statement is invalid. These mean respectively that the two entries in the top row (P=0) must be equal to one another and that the two entries in the right column (Q=1) must be equal to one another.

 

There is one unknown element in the as-constructed truth table, and the above says this must be 1 (true). The completed truth table is thus

 

[math]

\begin{array}{l|ll}

\multicolumn{1}{c}{}&\multicolumn{2}{c}{Q} \\

P & 0 & 1 \\

\hline

0& 1 & 1 \\

1& 0 & 1

\end{array}

[/math]

[triclino]

Short answer: Because that is the only value that makes sense.

 

 

So the sqrt(4) = 2 ,because this the only value that makes sense?

[uncool]

So the sqrt(4) = 2 ,because this the only value that makes sense?

 

Not quite. sqrt(4) = 2 because that is what the definition of sqrt implies. But what DH is saying is that implication is defined the way it is because that is the only way that makes sense. That is, you could choose to define it differently, but we choose not to because another definition would be nonsensical.

=Uncool-