Definition Of Logical Connectives

[Bashyboy]

Hello,

I just began my Discrete Mathematics class. It is rather interesting, but I have a few questions regarding the definitions of logical connectives. For instance, my book states that the conditional statement,$p \rightarrow q$ serving as an example, is false when p is true and and q is false, and true otherwise.

Was there reasoning used to define this, or did the person arbitrarily define it?

 

[grmnsplx]

p→q is read "p implies q" or "if p then q"
That is, by definition, if p is true then so is q.

So if p were true, and q were not true then, p→q would not be a true statement.

Does that make sense?

We can go through a proof if you like.

 

[Stephen Tashi]

Was there reasoning used to define this, or did the person arbitrarily define it?

In mathematics (and perhaps in life) if someone claims "If A is true then B is true" and you wish to disprove it, then need an example when A is true and B is false. It doesn't help to cite an example when A is false. For example, if we claim if a figure is a triangle then the sum of it's interior angles is 180 deg. then we don't want to someone to disprove that by drawing a square.

If left to non-mathematicians ,when A is false, the statement "If A then B" might be declared to be "undecided" or something like that - something neither true nor false. But this doesn't work once you begin to consider logical functions with variables in them. To turn these into "statements" , you quantify the variables with modifiers like "for each" or "there exists". We regard the statement "For each x, if 0 < x < 3 then 0 < x^2 < 9" as true. We don't want to say it's "undecided" or false because of the case when x = 234. The "if..." part is rather like a filter. If a statement correctly filters out all cases that don't apply, then the statement is true.

 

[SteveL27]

Hello,

I just began my Discrete Mathematics class. It is rather interesting, but I have a few questions regarding the definitions of logical connectives. For instance, my book states that the conditional statement,$p \rightarrow q$ serving as an example, is false when p is true and and q is false, and true otherwise.

Was there reasoning used to define this, or did the person arbitrarily define it?

If 2+2 = 5 then I'm the Pope. That's a true statement.

How could you disprove it? You'd have to show that

1) 2 + 2 = 5; and

2) I'm not the Pope.

But you can't do that! You can't show that 2 + 2 = 5 because that's false.

So you see, if 2 + 2 = 5 then I'm the Pope. Any implication where the antecedent is false, is a true implication.

Hope this helps. This is certainly a common area of confusion. After a while you'll get used to it. False antecedent implies anything.

By the way if I happened to be the Pope -- which, on an anonymous forum, can't be completely ruled out -- then "if 2 + 2 = 5 then I'm the Pope" is also a true implication. If the consequent is true, then the implication is true.

Therefore to make my examples work, I have to actually assure you that I am not the Pope :-) But if 2 + 2 were 5, I certainly would be.