4.1: Why Another Deductive Logic?

Aristotle’s logic was great. It had a two-plus millennium run as the only game in town. As recently as the late 18th century (remember, Aristotle did his work in the 4th century BCE), the great German philosopher Immanuel Kant remarked that “since the time of Aristotle [logic] has not had to go a single step backwards... [and] it has also been unable to take a single step forward, and therefore seems to all appearance to be finished and complete.” (Kant, I. 1997. Critique of Pure Reason. Guyer, P. and Wood, A. (tr.). Cambridge: Cambridge University Press. p. 106)

That may have been the appearance in Kant’s time, but only because of an accident of history. In his own time, in ancient Greece, Aristotle’s system had a rival—the logic of the Stoic school, culminating in the work of Chrysippus. Recall, for Aristotle, the fundamental logical unit was the class; and since terms pick out classes, his logic is often referred to as a “term logic”. For the Stoics, the fundamental logical unit was the proposition; since sentences pick out propositions, we could call this a “sentential logic”. These two approaches to logic were developed independently. Because of the vicissitudes of intellectual history (more later commentators promoted Aristotelian Logic, original writings from Chrysippus didn’t survive, etc.), it turned out that Aristotle’s approach was the one passed on to future generations, while the Stoic approach lay dormant. However, in the 19th century, thanks to work by logicians like George Boole (and many others), the propositional approach was revived and developed into a formal system.

Why is this alternative approach valuable? One of the concerns we had when we were introducing Aristotelian Logic was that, because of the restriction to categorical propositions, we would be limited in the number and variety of actual arguments we could evaluate. We brushed aside these concerns with a (somewhat vague) promise that, as a matter of fact, lots of sentences that were not standard form categoricals could be translated into that form. Furthermore, the restriction to categorical syllogisms was similarly unproblematic (we assured ourselves), because lots of arguments that are not standard form syllogisms could be rendered as (possibly a series of) such arguments.

These assurances are true in a large number of cases. But there are some very simple arguments that resist translation into strict Aristotelian form, and for which we would like to have a simple method for judging them valid. Here is one example:

Either Clinton will win the election or Trump will win the election.
Trump will not win the election.
Therefore, Clinton will win the election.

None of the sentences in this argument is in standard form. And while the argument has two premises and a conclusion, it is not a categorical syllogism. Could we translate it into that form? Well, we can make some progress on the second premise and the conclusion, noting, as we did in Chapter 3, that there’s a simple trick for transforming sentences with singular terms (names like ‘Clinton’ and ‘Trump’) into categoricals: let those names be class terms referring to the unit class containing the person they refer to, then render the sentences as universals. So the conclusion, ‘Clinton will win the election’ can be rewritten in standard form as ‘All Clintons are election-winners’, where ‘Clintons’ refers to the unit class containing only Hillary Clinton. Similarly, ‘Trump will not win the election’ could be rewritten as a universal negative: ‘No Trumps are election-winners’. The first premise, however, presents some difficulty: how do I render an either/or claim as a categorical? What are my two classes? Well, election-winners is still in the mix, apparently. But what to do with Clinton and Trump? Here’s an idea: stick them together into the same class (they’re not gonna like this), a class containing just the two of them. Let’s call the class ‘candidates’. Then this universal affirmative plausibly captures the meaning of the original premise: ‘All election-winners are candidates’. So now we have this:

All election-winners are candidates.
No Trumps are election-winners.
Therefore, all Clintons are election-winners.

At least all the propositions are now categoricals. The problem is, this is not a categorical syllogism. Those are supposed to involve exactly three classes; this argument has four—Clintons, Trumps, election-winners, and candidates. True, candidates is just a composite class made by combining Clintons and Trumps, so you can make a case that there are really only three classes here. But, in a categorical syllogism, each of the class terms in supposed to occur exactly twice. ‘Election-winners’ occurs in all three, and I don’t see how I can eliminate one of those occurrences.

Ugh. This is giving me a headache. It shouldn’t be this hard to analyze this argument. You don’t have to be a logician (or a logic student who’s made it through three chapters of this book) to recognize that the Trump/Clinton argument is a valid one. Pick a random person off the street, show them that argument, and ask them if it’s any good. They’ll say it is. It’s easy for regular people to make such a judgment; shouldn’t it be easy for a logic to make that judgment, too? Aristotle’s logic doesn’t seem to be up to the task. We need an alternative approach.

This particular example is exactly the kind of argument that begs for a proposition-focused logic, as opposed to a class-focused logic like Aristotle’s. If we take whole propositions as our fundamental logical unit, we can see that the form of this argument—the thing, remember, that determines its validity—is something like this:

Either p or q
Not q
Therefore, p

In this schema, ‘p’ stands for the proposition that Clinton will win and ‘q’ for the proposition that Trump will win. It’s easy to see that this is a valid form. (This form is often called the “Disjunctive Syllogism”. Notice that the word ‘syllogism’ is used there. By the Middle Ages, Stoic Logic hadn’t disappeared entirely. Rather, bits of it were simply added on the Aristotelian system. So, it was traditional (and still is in many logic textbooks), when discussing Aristotelian Logic, to present this form, along with some others, as additional valid forms (supplementing Barbara, Datisi, and the rest). But this conflation of the two traditions obscures the fundamental difference between a class-centered approach to logic and one focused on propositions. These should be kept distinct.) This is the advantage of switching to a sentential, rather than a term, logic. It makes it easy to analyze this and many other argument forms.

In this chapter, we will discuss the basics of the proposition-centered approach to deductive logic—Sentential Logic. As was the case with Aristotle’s logic, Sentential Logic must accomplish three tasks:

1. Tame natural language.
2. Precisely define logical form.
3. Develop a way to test logical forms for validity.

The approach to the first task—taming natural language—will differ substantially from Aristotle’s. Whereas Aristotelian Logic worked within a well-behaved portion of natural language—the sentences expressing categorical propositions—Sentential Logic steps outside of natural language entirely, constructing an artificial language and only evaluating arguments expressed in its terms. This move, of course, raises the concern we had about the applicability to everyday arguments even more acutely: what good is a logic if it doesn’t evaluate English arguments at all? What we must show to alleviate this concern is that there is a systematic relationship between our artificial language and our natural one (English); we must show how to translate between the two—and how translating from English into the artificial language results in the removal of imprecision and unruliness, the taming of natural language.

We will call our artificial language “SL,” short for ‘Sentential Logic’. In constructing a language, we must specify its syntax and its semantics. The syntax of a language is the rules governing what counts as a well-formed construction within that language; that is, syntax is the language’s grammar. Syntax is what tells me that ‘What a handsome poodle you have there.’ is a well-formed English construction, while ‘Poodle a handsome there you what have.’ is not. The semantics of a language is an account of the meanings of its well-formed bits. If you know what a sentence means, then you know what it takes for it to express a truth or a falsehood. So semantics tells you under what conditions a given proposition is true or false. (That’s actually a controversial claim about the role of semantics. Your humble author, for example, is one of the weirdos who thinks it not true (of natural language, at least). But let’s leave those deviant linguists and philosophers (and their abstruse arguments) to one side and just say: semantics gives you truth-conditions. That’s certainly true of our artificial language SL.) Our discussion of the semantics of SL will reveal its relationship to English and tell us how to translate between the two languages.