Logic Tutorial, Part 8.
Sustained Arguments

(c) 2017 by Barton Paul Levenson

To make the rules of inference more generally applicable, it would be nice to be able to treat compound or complex logical statements as well as simple ones. This is easily done. We just take capital letters to represent compound statements, as lower-case letters represented atomic or simple statements. (Remember that compound statements are ones put together using ¬, ∧, ∨, and optional parentheses). All the rules of inference we have learned apply to both type of statement. Thus modus ponens can be listed as:

[(p → q) ∧ p] → q

or as

[(A → B) ∧ A] → B

Now, analyzing sustained arguments (ones longer than syllogisms) can be done using the multi-line "argument form" we saw earlier. Take a group of three rather complicated premises:

1. If it is true that I am going either to Bath or to London, then I had better pack lightly, and not be late at the bus station.
2. Packing heavily would mean I'd be late.
3. I most definitely am going either to Bath or to London.

Just to show where I got these, I'll break the compound statements down into their constituent atomic statements:

Listing all three premises in argument form, we could apply modus ponens to the 1st and 3rd premises:

So far, we've done nothing with statement #2. But in theory the argument could go on from here and #2 might be used later on. By doing this kind of thing we can extend a logical argument to arbitrary size.

Whoop-te-do, you may be thinking. What's the point of all this?

The point is, once we learn about the predicate calculus as well, we will be able analyze anybody's argument and rewrite it in logical calculus, making it easier to analyze, and helping us see logical fallacies. To start doing this, the last chapter on logic will talk about how to translate English into logical calculus.

But before we do that, here are three more rules of inference, much more general than the highly technical ones we've listed already:

Premise. A premise can be written down as part of a proof. Among logicians, one school says all the premises should be written down at the beginning of the proof, and the other says you should just introduce them when and where they are needed. Either method is allowable; which you choose is purely a matter of style or habit. But stick to one or the other, or you'll confuse your readers.

Substitution. Any part of a premise or conclusion in a sustained argument can be replaced with a logically equivalent one, according to the rules of inference. Here's an example:

Conjunction. Any two lines in a proof can be written down as another line using the conjunction operator ( ∧ ).