4.1 Why Another Deductive Logic?

Before diving into the the syntax and semantics of SL, a brief look at the evolution of the two deductive logics, Aristotelian and Sentential, will help highlight the usefulness of Sentential Logic.

4.1.1 Some Historical Context

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 that Aristotle did his work in the 4th century BCE), the great German philosopher Immanuel Kant remarked in his Critique of Pure Reason  (1781 and 1787) that

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 or category; and since terms in Aristotle’s logic refer to classes, his logic is often referred to as a term logic. For the Stoics, the fundamental logical unit was the proposition; and since propositions refer to sentences, we call this a sentential logic. These two approaches to logic were developed independently. Because of the twists and turns of intellectual history, 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.

4.1.2 The Value of the Propositional Approach

Why is this alternative approach advantageous? One of the concerns mentioned when we were introducing Aristotelian Logic in Unit 3 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 hopeful promise that, as a matter of fact, lots of sentences that were not in standard categorical form could be translated into that form. Furthermore, we assured ourselves that the restriction to categorical syllogisms was similarly unproblematic because lots of arguments that are not standard form syllogisms could be rendered as such arguments, possibly as a series of arguments.

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

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 Unit 3, Section 3.1.5) that there’s a simple trick for transforming sentences with singular terms (names like ‘Kennedy’ and ‘Nixon’) into categoricals: let those names be class terms referring to the unit class containing the character they refer to, then render the sentences as universals. So the conclusion, ‘John F. Kennedy won the election’ can be rewritten in standard form as ‘All John F. Kennedys are election winners..’, where ‘John F. Kennedys’ refers to the unit class containing only John F. Kennedy. Similarly, ‘Richard M. Nixon’ did not win the election’ could be rewritten as a universal negative: ‘No Richard M. Nixons 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 Kennedy and Nixon? Here’s an idea: stick them together into the same class, 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:

Well at least, all the propositions are now categoricals. The problem is, that this is not a categorical syllogism. Those involve exactly three classes; this argument has four—Kennedys, Nixons, election winners, and candidates. True, ‘candidates’ is just a composite class made by combining Kennedys and Nixons, so you can make a case that there are really only three classes here. But, in a categorical syllogism, each of the class terms is supposed to occur exactly twice. ‘Election winners’ occurs in all three, and there is no apparent way to eliminate one of those occurrences.

Ugh! This is headache material. 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 units of this course) to recognize the argument is valid. It’s easy for regular people to make such a judgment about either/or reasoning. 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 that determines its validity (!) —is something like this:

In this schema, ‘p’ stands for the proposition stating that Kennedy won the election and ‘q’ for the proposition stating that Nixon won the election.  It’s easy to see that this is a valid form.  (In fact, this is one of the common forms listed in the supplemental page on Common Deductive Argument Forms at the end of Module 1.  It’s called a Disjunctive Syllogism.)  A clear advantage of switching to a sentential, rather than a term logic is that it becomes easy to analyze this and many other argument forms.

With some appreciation of the benefit of the sentential language, we move on now to its bits and pieces — the syntax of our artificial language, SL.