Propositional variables and connectives

Propositional variables are the letters we use

to stand for the most basic sentences we might make in expressing our thoughts. Sometimes, borrowing from physics, these are called atomic sentences while all other wff's that involve connecting them are said to be molecular sentences. By convention in our translations we restrict such sentences to being positive statements (even if the idea is a negative one, as when we might say that somebody fails). (I'll leave it up to you to decide whether we should then call ~J an atomic sentence or a molecular one.)

Let's work with just these few thoughts: logic is easy (E), logic is fun (F), and logic is interesting (I)

When we assert E we do mean to say that it is true that logic is easy. If logic in fact is hard, then the truth value of the proposition E is false. Supposing I use the sentence ~F to express the thought that logic is not fun (and we will not build the "not" into the atomic sentence represented by F), we would see that if I'm wrong we would call ~F false (so that F is true) but if I'm right then we would call ~F true (so that F is false).

What this means, if we think about it, is that what we can call the truth value of any proposition that is not just a single letter is a function of the truth value of its components. Our connectives or operators set up what we mean by these truth-functional relationships. Keep in mind, though, that this already restricts the kinds of things that could be expressed. Whenever we say "might" (Jack might pass the test) or "ought" (Jill ought to study) we are not really thinking of something black or white, since one involves probability and the other could be seen as possibly true from one point of view but false from another. Symbolic logic, then, will actually be very limited as a technique for representing our thoughts.

Clearly nothing gets interesting here

until we begin representing things that are more than just simple combinations of a noun and a verb. What we will be concerned with are how five basic relationships are expressed.

negation (what we hear with "not"): ~
conjunction (what we hear with "and" or "but" or "although"): &
disjunction (what we hear with "unless" and with "or" when at least one of the items has to be true): v
implication (what we hear with phrases such as "if...then" or "...only if..."): ->
equivalence or mutual implication (what we hear with phrases such as "if and only if"): <->

Obviously we are stripping down all the additional levels of meaning in our sentences. Time, for instance, is not usually a factor, so F could be a reference to past, present, or future. F could equally well express "logic was fun," "logic is fun," and "logic will be fun." If there is some reason to make a distinction based on time then we would use different variables, so that the thought that logic was easy yesterday but not today might be expressed as Y&~T

In the same way all the ideas of contrast or even of causality can disappear to focus simply on the thought that two things are true at the same time. "Logic is easy because it is fun" would be E&F and so would not point out the difference in meaning from "Logic is fun because it is easy." (Strictly speaking, causal relationships are not truth-functional.)

We can explain these truth-functional relationships by setting up what we call truth tables.

P	~P
true	false
false	true

P	Q	P&Q	PvQ	P->Q	P<->Q
true	true	true	true	true	true
true	false	false	true	false	false
false	true	false	true	true	false
false	false	false	false	true	true

If we think about what we are shown we can see that there are some key differences between what we might have in mind in conversation and the way our ideas would be expressed in a symbolic notation. When we say, for instance, I could order either fish or chicken at the restaurant, I do not ordinarily mean to say I have only two choices, but logical disjunction does mean there are just these choices. In the same way, it seems rather odd to have a logical implication be true when there might not be any further connection between the ideas, but when I say to a girl "If you are a boy then you will grow up to be President" I have a proposition that has to be classified as true just because the condition I'm setting up is false.

All of this may seem rather strange, but the point is that when we treat symbolic language as an artificial language we find that it does not completely map our natural languages (but then our written language does not map the spoken one, and our spoken language only very imperfectly maps our thoughts).

Let's do some exercises. First, we'll see how you might "translate" some simple English sentences, then we'll ask about whether the propositions would be true or false when we know something else about the atomic sentences making them up. On a piece of paper write out your own answers, then go on to another page where you will see mine.

Logic is both easy and fun.
Logic is not easy but it is fun.
Either logic is easy or it will not be fun.
If logic is easy then it will be interesting.
If logic is fun then it will be easy and interesting.
Logic will not be fun unless it is both interesting and easy.
If it is false that logic is fun only if it is easy, then it must be interesting.
Logic is both easy and fun if and only if it is interesting.

Now decide which sentences are true and which are false if I start you off with the ideas that it's true logic is fun and it's true that logic is interesting, but it is false that logic is easy.

click on for the answers