If - then: or the truth about truth tables

General Academics Science, Technology and Medicine
1

Friday the thirteenth surprised by bringing something brand new to be skeptical at. 8) And from a most unlikely source too! First period, and we were in class learning all about Logical connectives. From what I could gather, Logic is used by mathematicians to prove stuff, and also in more mundane applications such as electronics and programming (or is it coding these days?) The lecturer kicked off by explaining that a statement has to be either true or false. One can then join two or more statements in a way such that the combination of the statements must also be true or false. The truth values of these combinations can be written concisely in a truth table. In the case of a conjuction (two statements joined by “and”) the truth table will look like this:

A B A and B
T T T
T F F
F T F
F F F

This means, for example, that if statement A is true, and statement B is also true, and you join the two statements by “and”, you will find that the result is also true. But whenever a false statement is included the conjuction will be false. For instance, if I were to say that “6 is a prime number and Colorado is in the USA”, one of the statements is false, and therefore the conjuction is also false. This makes perfect sense.

The truth table for the disjunction (two statements joined by “or”) seems similarly intuitive:

A B A or B
T T T
T F T
F T T
F F F

So you need only a minimum of one of the statements to be true in order for the disjunction to be true. “6 is a prime number or Colorado is in the USA” is therefore perfectly acceptable as true. No problems so far.

But the part that made me take issue was the table describing the truth values for the implication (two statements joined up by “implies” or “If … then …”:

A B If A then B
T T T
T F F
F T T
F F T

Here is why I’m not convinced by this particular table. If two true statements always result in the implication being true, I could link up any two true statements, each of which has nothing to do with the other, and suddenly “If Six is an even number then Colorado is in the USA” is true! Yet, I somehow doubt that the early frontiersmen had any concerns about the divisibility of integers when they named the state. :stuck_out_tongue:

In short, the table says that: ANY TRUE STATEMENT IMPLIES ANY OTHER TRUE STATEMENT.

I queried this, and was told that Logic focuses on the structure of the arguments, and not the content of the statements.

Hmmm. :confused:

I’m not sure if I’m letting too much philosophy seep into my rudimentary understanding of mathematics, but still : if no consideration is paid to the content, then an implication is inherently meaningless. And of what use will a meaningless implication be in proofs?

Rigil

2

I’m afraid that you have completely missed the essence of the logical implication. One must think in terms — almost — of meta-truth here, and you have it back-to-front by focussing on the terms (i.e. the antecedent and the consequent). The way to start looking at logical implications is as whole statements without reference to its particular components, i.e. the starting point is the truth or falsity of the whole implication, not the clauses that constitute it.

The statement “if P then Q” (or “S = P → Q” for short) is either true or false. That is to say, S is true if it is universally true that the truth of P invariably implies the truth of Q. The latter is called modus ponens: If S is true, when we observe P to be true, we must conclude that Q is also true. Your “6/Colorado” example is an S that is clearly not true because there is no logical or causal link between its terms.

Thus, you need to start with an S that appears to be true or at least provisionally true, for example P = “this is a duck” and Q = “it can quack”. In this case (and unless we’re dealing with a lame or sick or dead one), S = P → Q is clearly true. So, whenever I show you a duck, your conclusion that it can quack is logically assured if S is true.

However, this is not the only circumstance in which S is true. The implication still remains true if we observe a case where P is false but Q is true. So, if you see something that can quack, it need not be a duck. It could be a hunter with a duck whistle. The third possibility that retains the truth of S is the one where whenever Q is false, P is also false, which is known as modus tollens. Formally, S → S’ where S’ = ~Q → ~P. Using our example, if I show you something that can’t quack then your conclusion that it is not a duck is logically impeccable provided S is true.

The only case where S becomes false is when you see a case that directly contradicts it, namely a P that is true with a Q that is false. So if I show you a healthy duck that can’t quack then S, the implication itself, is false.

'Luthon64

3

Ah … ok. I wasn’t aware of that prerequisite! Must’ve jumped the gun a bit. I must admit that I am secretly relieved that I’m not to be the one responsible for bringing the entire body of mathematical proofs crashing down…

Mefiante, thanks so much for the explanation and examples, you are such a sweetheart! I’m starting to regret trying to bully you into changing your signature. :wink:

Rigil

4

Just a belated close-out to the saga … we’ve since written a test on this and it went very well. I actually plugged in that duck analogy several times in preparation for the test, and it was invaluable in making sense of the ponens/ tollens type rules of inference. Thanks again, M. :slight_smile:

Rigil

5

Cool, well done, I liked doing truth tables they’re fun.

6

Cheers Cr1t … yes, they are rather enjoyable. At least, once you’ve stopped being angry at them! ;D

Rigil