Formal Logic, Part II

Part I is here, along with the text we're using. We'll begin today with Section 1.10, because I want to talk about logical equivalence.

Last time we talked about validity, and the difference between Aristotle's ideas of validity and modern ones. But there's another way of talking about validity in logic, which is this: an argument can be said to be valid if it is truth-preserving. That doesn't mean that it guarantees truth (see the section on "soundness"), but that a valid form will preserve whatever truth is there. If your propositions are true, a valid form will ensure that your conclusion is also true.

Truth-preservation is also why we can say that two apparently very different arguments are logically equivalent. What it means for two arguments to be logically equivalent is that the two arguments are true or false together, 100% of the time. If you go through the exercises of building truth tables, you'll see that the truth tables for the two arguments will be exactly, precisely the same in every case.

Consider one of his examples:

1) Either it is raining, or it is snowing but not raining.

2) Either it is raining or it is snowing.

Since this is the inclusive "or," if it is both raining and snowing both of these sentences are considered true (because the "either it is raining" part is true). Symbolically, (1) would be rendered (R v (S & ~R)), and (2) would be (R v S). If the "R" is satisfied, the sentence is true; if the S is satisfied, both sentences are true if it is not raining. Both sentences are false if it is neither raining nor snowing.

There's a shorthand way of saying that two logical sentences are equivalent, which is called the biconditional. It is rendered in natural language "if and only if," or in philosophical shortcut, "iff." It has several logical symbols, but where I come from we use the triple bar: ≡.

Note, though, that this kind of equivalence goes both ways. 1≡2 means 2≡1. That is not the case for every sort of sentence we would render in natural language with "only if." "He is a bachelor if and only if he is an unmarried male human" is a biconditional (as well, in this case, as a tautology) because wherever one set of things will be true, the other will, and vice versa.

The other kind of 'only if' is a material conditional. You could say "If John gets hired, then Mary will get hired." But that does not mean that the truth of John's hiring is equivalent to the truth of Mary's hiring. It means that "John will get hired only if Mary gets hired."

That's properly:

3) J -> M


4) J ≡ M

We can see they are not equivalent by building the truth table.

J M | (3) | (4)
T T | T | T
T F | F | F
F T | T | F
F F | T | T

Because the truth values of the claims do not hold together, the material conditional form of the statement is not logically equivalent to the biconditional. And while (J ≡ M) is the same as (M ≡ J), (J -> M) is not the same as (M -> J). The table for (M -> J) I will leave you to work out on your own as an exercise, if you choose, but you will see it comes apart from (J -> M).

Preserving the truth is what this is all about.


Eric Blair said...

That last bit about John and Mary I see in a different way.

That is, the IF [condition 1 (is true)] THEN [condition 2 (will occur)] construction, which is something I use all the time, does not mean that John gets hired only if Mary gets hired.

It is the opposite, I think.

Mary only gets hired if John gets hired first.

Perhaps this is only a matter of how you wrote it, but in any sort of conditional comparison, there is precedence of conditions that then allow other conditions.

For example,
if considering a set of employees that belong to a union, and have a date of service, and a uniform allowance is based on that date of service, something like this will be said:


Grim said...

It's a counter-intuitive finding, but that's how it's usually taught. Symbolic logic is different from programming in that it isn't necessarily interested in order of operations. You're talking about time and modality, in other words: it is currently possible that John will be hired, and if he is hired, that makes necessary Mary's hiring.

If we were talking about programming instructions, that would make sense. But one of the things modern logic wants to do is escape from time -- along with analytic philosophy more broadly, it wants to talk about atemporal rules. It's an odd concept, for ordinary human beings who are mired in time (and something like order of operations).

From this perspective, then, the claim is that John's hiring actually depends on Mary's -- not the other way around, as the natural-language reading of the logic would suggest.

That's another point that is quite different from Aristotle's approach. It's nice that you brought it out.

Yu-Ain Gonnano said...

The other way to think about it, Eric, is that if the ALLOWANCE-AMT = 250, the *only* way that happens is *if* the Union was "IBEW" and the DATE-OF-SERVICE is before '01JUL2010'.

In Grim's example, it is not that Mary's hiring *causes* John's hiring. Rather that John's hiring cannot happen without Mary's hiring. If Mary is not hired, you know for certain that John was not hired. Mary could be hired without John, but John cannot be hired without Mary.

Formally, "If A then B" is logically equivalent to If NOT B then NOT A"