8.2 Classifying and Comparing Statements

8.2  Classifying and Comparing Statements

 

Another application of truth tables allows us to classify every truth-functional statement as falling in to one of three categories. As you will have noticed, the truth-values of the simple statements in 8.1.2 did not change, but the truth-values of the main operators did. That’s because they are the kinds of statements in which the truth-values of the statements are said to matter, and to determine the truth-value of the compound, given the meanings of the operators. That kind of statement is called “contingent,” which means in this context that the value of the whole is dependent (contingent) upon the value of the parts. (There is another philosophical sense of “contingency,” of an existential nature, which has nothing to do with this logical notion of it.)

But there are also statements which have their truth-value as a consequence of their structure rather than as a consequence of their content. Some statements are true because their structure makes them be true, and it doesn’t matter whether they are about Wittgenstein or Leonardo or Humpty Dumpty. They are called “tautologies.” And then there’s a third group, the ones that are false as a result of their structure, and which, again, can’t be anything other than false regardless of what content you give them. These are called “self-contradictions.”

A simple example of a self-contradiction is this: “I think you’re right but I think you’re wrong.” The form of this is R ∙ ~R. It’s pretty clear intuitively that there’s something wrong with that claim, i.e., that it is false.

R ~R
T F F
F F T

As that table makes plain, there’s always going to be an F under the dot in a conjunction that joins a statement with its own negation.

Here’s a tautology:

Every class is either a member of itself or not.

We could write it as M v ~M .

M v ~M
T T F
F T T

Every statement disjoined from its negation will be true. “Either a statement is true or false” is a tautology too (since “false” and “not true” are synonyms).

I’ve presented it in very simple examples, but here are some more challenging cases you can work on, to practice calculating, and get accustomed to classifying statements into contingencies, tautologies and self-contradictions:

Before you can do these, you’ll have to refresh yourself on how many rows a truth-table requires. The formula is “Number of rows = 2 nth power” where “n” is the number of simple statements. This means that if there is just one simple statement, only two rows are needed (one row shows what happens when it is true, the other shows what happens when it’s false). So # 1 below requires just two rows. A statement with two simple statements requires 4, one with three requires 8, one with four requires 16, one with five requires 32, one with six requires 64.

1. M ⊃ (M ⊃ M)

2. (G ⊃ G) ⊃ G

3. (S ⊃ R) ∙ (S ∙ ~R)

4. [(Q ⊃ P) ∙ (~Q ⊃ R)] ∙ ~(P v R)

5. {[(G ∙ N) ⊃ H] ∙ [(G ⊃ H) ⊃ P]} ⊃ (N ⊃ P)

Besides doing the truth tables for these, make sure you can put them into words: 1 and 2 don’t say the same thing, for example, but what do they say? 1 says “M implies that M implies M.” (It can also be read as “If M is true then M implies M.”  Or as “If M, then if M then M.”  What does 2 say?

Translate this one and do a table.

The balance of payments will decrease if and only if interest rates remain steady; however it is not the case that either interest rates will not remain steady or that the balance of payments will decrease.

 

 

Now that you know how to calculate values and how to build truth tables, you can apply it to another task, which is to compare statements to other statements. When you have a set of compound and/ or simple statements, you can make a table up that shows all the possibilities of their truth-values, and judge from that whether any two or more of them are equivalent to each other (like the triple bar statement and a biconditional), or whether they contradict each other (which is not the same thing as a statement contradicting itself), or whether they are consistent or inconsistent with one another. These features can be read off a truth table mechanically. If two or more statements always have the same truth-value under their main operators, they are equivalent. If they have opposite values under their main operators, they are contradictory (like A and O in categorical logic). If they never show a True on the same line, they are inconsistent (meaning they cannot both be true), and if they show a True on at least one line under their main operator, they are consistent with each other (meaning that under certain contingent circumstances, they can both be true).

 

Here are a few examples you can try this out on. Make a truth table for each of the expressions and then compare them line by line under their main operators. See what you find.

1. ~(p ∙ q)               ~p ∙ ~q

2. F ∙ M               ~( F v M)

3. ~A ≡ X            (X ∙ ~A) v (A ∙ ~X)

4. Q ⊃ ~( K v F)         (K ∙ Q) v (F ∙ Q)

5. from Hurley’s Concise Introduction to Logic:

Christina and Thomas are having a discussion about their plans for the evening. Christina: “If you don’t love me, then I’m certainly not going to have sex with you.” Thomas: “Well, that means that if I do love you, then you will have sex with me, right?” Is Thomas correct? (Hint: Construct a truth table for each statement, and compare them.)

Another hint: next time there’s an awkward silence in a romantic conversation, propose building truth tables!

 

 

8-Truth-Tables-4

 

 

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