1. Why are there two kinds of logic? Does each have its own advantage, or is one clearly better than the other?
2. What exactly does the conditional mean, if not the usual "if-then"?

1. Why are there two kinds of logic? Does each have its own advantage, or is one clearly better than the other?

Much as a disagreement between two experts leaves us doubting both of them, the existence of two kinds of logic can make us suspicious of logic in general. If the two work equally well, how can there be a logic? And if one is better than the other, why learn the worse one along with the better one, except as a historical curiosity?

Leave aside the historical questions of how each logic developed, and the technical questions of what theory justifies each one, matters beyond the scope of this course. What concerns us is the practical question of the work that each logic does and the advantage and disadvantage of each in evaluating arguments.

The principal difference comes down to the two logics' relationships to ordinary language. Categorical logic remains with the forms of natural sentences; truth-functional logic--like all modern logic, of which it is a part--departs from ordinary grammar.

As a result, categorical logic makes for easier translation out of ordinary language.

However, categorical logic suffers from its attachment to grammatical form. First, it needs to specify in advance which sentences will work in arguments, namely those that can be put into forms beginning with "all" or "some." As a result, the logic of categorical statements only works with a limited number of sentence structures.

Secondly, because we begin and end with recognizable grammatical forms instead of with the logical forms that lie beneath them, the rules for testing an argument are often unwieldy. Categorical logic uses general statements about kinds of sentences instead of penetrating to the principles that make them work.

Formal logic presents a greater challenge at the beginning. Translating a sentence into truth-functional logic poses problems that do not arise in Aristotle's logic. Take "or." ("If-then" makes another example--see the next question.) The English "or" flexibly adapts to mean different things. "You are safe if your door is bolted or padlocked shut" gives an example of the inclusive "or": If you both bolt and padlock your door, you will be safe. Either alternative, or both, will do. But sometimes we use "or" to mean one or the other alternative but not both. If you tell a child, "You can have pie or pudding," you imply that the child cannot have both.

When clarity is essential to English, we make up for this flexibility by adding words. "Either" often is a way of emphasizing the exclusive "or": "You can have either pie or pudding." The inclusive "or" has no traditional linguistic markers, so people sometimes use "and/or" to indicate that they mean one or the other or both. Not felicitous English, but it does the job.

In formal logic we have no place for clarifying words, because the "v" already means one or the other or both. Because its meaning is fixed, some English sentences that use "or" do not go easily into logic. Let A be "You can have pie" and B be "You can have pudding." We can't render the sentence to the child as "A v B," for that leaves open the possibility of a pie-and-pudding treat. We write "(A v B) & ~(A & B)." A disappointment for the sweet-toothed youngster and a clumsy animal for the logician, but clear.

This departure from ordinary language makes our first use of truth-functional logic harder. But it creates one clear practical advantage: The rules are easier to learn. Despite the number of rules in Chapter 9, they can all be derived from simpler rules, each one of them obvious once it's been stated. You do not have to remember what a distributed term is, or which one is the middle term. Each sign carries a fixed meaning and can always be treated in a single way.

2. What exactly does the conditional mean, if not the usual "if-then"?

The conditional, like disjunction, is a logical connective that only approximately matches a grammatical one. No one objects to the conditional where the antecedent is true: If the consequent is true, then so is the whole statement, and if the consequent is false it likewise makes the whole thing false. But what about those pesky sentences with false antecedents?

It helps to bear in mind that we really care only about if-then statements whose antecedent turns out true. "If we find the map, we'll find the treasure." If the map is not found, what does it matter how we categorize the sentence? The conditional becomes irrelevant.

It would make a better match with ordinary language if we assigned no truth value at all to conditionals with false antecedents. But logic will not permit such a move. Every connective must yield some constant truth value when the truth values of all its elements are known. We have to say something.

When the two component sentences of the if-then construction are both false, it makes more sense to call the conditional true than to call it false. Consider the sentence "If she's right, she's right." Surely that's true, even trivially true. But because she might be wrong in any given instance, the sentence could have a false antecedent and a false consequent. Nevertheless we have to call the compound true.

We are again asked to treat the conditional as true when it has a false antecedent and a true consequent. "If the reader of this study guide is an elephant, then the reader of this study guide is not an elephant." Logic calls the sentence true. Those who expect truth-functional logic to capture the nuances of natural language will rebel: Logic has turned raving nonsense--a sentence that sounds like a contradiction--into sturdy truth.

It is best to remember that we need the false-true combination to yield the same truth value in every case, whatever that value may be. We sacrifice flexibility to gain logical rigor and efficiency, somewhat as we sacrifice the pleasures of walking in order to get further, faster, in a car. The conditional should capture the core meaning of "if-then," even if subtleties fall by the wayside.

The crucial point is that calling such conditionals true makes more sense than calling them false. Logic gives one reason: If we were to call sentences of this form false, the truth table for "If A, then B" would turn out identical to the one for "If B, then A." (Work them out and see.) But we want the two to mean different things. "If that's a hawk, it's a bird" can't mean "If that's a bird, it's a hawk."

A less technical reason arises out of our customary use of conditionals. What really matters to us in if-then claims? Above all, what matters is that we not find ourselves with a true antecedent and a false consequent. Those are the unreliable conditionals. The claim says, "If the car has gas in it, it will run"; you put gas into the tank, and still it does not run. It is essential to any use of the words "if" and "then" to call that a false conditional, and no other truth value of the conditional matters as much as that falsity. So we leave the other cases aside, by letting them stand as true, in order to focus on this one thing that a conditional shall not do.

One final comment about what the conditional means. The arrow is read "if-then," as in, "If A, then B." You might want the arrow to correspond to a word you say between the two letters; then you may read the sentence, "A only if B." But the arrow does not mean "if" by itself ("A if B"); and it does not mean "implies," which describes a logical relationship between two sentences.