We have now met most of the the pieces we use in symbolic logic, propositional and predicate variables and a small set of connectives. It might seem at first that the rules for the connectives, which we explain with truth tables, are "natural," especially when we see how they fit in with certain things we do in working with numbers. Conjunction, for instance, can be seen as logical multiplication while disjunction is logical addition. To explain that, let's preview something we will see more of later.
If I want to have a computer decide the truth table values for an expression, an easy way of doing it is to use a mod 2 number system (like a clock with a hand that just moves between 0 and 1) and treat any expression as having an algebraic equivalent. Negation, for instance, is done by adding 1 (and in mod 2 we know that 1+1=0). Conjunction multiplies values, and inclusive disjunction (which permits the possibility of both elements being true at the same time) adds the value of the conjunction and the values of the two elements. We start off by thinking of 1 as expressing "true" or "on."
| p | q | ~p | ~q | p&q | pvq |
| 1 | 1 | 1+1=0 | 1+1=0 | 1x1=1 | (1x1)+1+1=1 |
| 1 | 0 | 1+1=0 | 0+1=1 | 1x0=0 | (1x0)+1+0=1 |
| 0 | 1 | 0+1=1 | 1+1=0 | 0x1=0 | (0x1)+0+1=1 |
| 0 | 0 | 0+1=1 | 0+1=1 | 0x0=0 | (0x0)+0+0=0 |
Then we look at the connective that we use in expressing conditional relationships, the things we mean when we say one thing depends on another. For instance, having an electric car run depends on its batteries being charged and it also depends on turning it on. Having the batteries charged is what we mean by a necessary condition: have it be false that the batteries are charged and it will be false that the car runs. Turning it on (assuming the batteries are charged and nothing else is wrong) is a sufficient condition for getting the car to run: if it's true that we turn it on, then it's true that it will run.
Remember the truth table for a conditional looks like this (using 1 for "true" and "0" for false):
| p | q | p->q |
| 1 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 1 |
| 0 | 0 | 1 |
There might be a bit of a surprise in finding that having both elements false automatically allows the conditional relationship to be true. There's not much of a problem when we have a sentence such "If Jack studies then he will pass" since if Jack fails when he does not study we have not been shown to be wrong in our original statement. What does seem odd is that I can also come up with a sentence such as "If Abraham Lincoln is still President, then the moon is made of green cheese" and automatically I can see that it is "true." We talk about this as material implication and it should remind us that we are working with a game that may move us away from our more commonsensical ideas of what sentences we would count as true or false.
What we also have is an interesting link between a sufficient condition and a necessary condition. If we say that logic is fun if it is easy (and the fact that the idea of it being easy is expressed second here does not change the fact that we would still see being easy as a sufficient condition for its being fun), we could just as easily say that it is easy only if it is fun. Either way I have the sentence E->F. The shades of meaning may differ in the English sentences, but logically they are equivalent. A better way of putting this is that the wff's translating these sentences are logically equivalent.
| p | q | p->q | ~pvq | ~(p&~q) |
| 1 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 1 | 1 | 1 |
Think about the fact that with just 4 rows and two possibilities for each cell (1 or 0) we must have only 16 possibilities for how we could make up a column. This means that no matter how long a wff I have with just two variables, it will still have to have the same value as some other, possibly shorter wff.
Let's look at all our possibilities.
| p | q | ~p | ~q | p&q | ~(p&q) | pvq | ~(pvq) | p->q | q->p | ~(p->q) | ~(q->p) | p<->q | ~(p<->q) | pv~p | p&~p |
| 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 |
| 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
| 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 0 |
The last two columns should be particularly interesting. It is possible for a wff to be always "true" and we call it a tautology when that happens. It is also possible for a wff to be always "false" and then we call it a contradiction. Otherwise the truth value of any wff depends on its composition and it is considered contingent. When two wffs would never be true at the same time (such as ~p and p&q) they are inconsistent, but they would not be contradictory unless they could not be false at the same time (such as p and ~p).
A last thing to notice is that all tautologies by definition are
equivalent. The tautologies that most interest us are the wffs used
to translate valid arguments (ones in which the truth
of the premises is a sufficient condition for the truth of the conclusion).
Imagine we set up the hypothesis that logic is fun if it is easy and in
fact it is easy, and this implies that in fact it is fun. See
what happens in the truth table below, and compare this with the truth
table for the argument in which I change the word "if" to the phrase "only
if."
| E | F | ((E->F)&E)->F | ((F->E)&E)->F |
| 1 | 1 | 1 | 1 |
| 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 1 |
| 0 | 0 | 1 | 1 |