Unit 2: Testing Validity

Study guide

Validity seems like hocus-pocus until you understand the notion of logical form and learn how to test "whether a logical form has any substitution instances in which all the premises are true but the conclusion false". By the end of this unit you will know how to do these things.

1. Logical form. Learn how to abbreviate distinct independent clauses of a sentence with distinct sentence-letters. Learn why one needs to do this. See section 2.1. You will be asked whether a given symbolization could or could not be used to represent the logical form of a given sentence. Try exercises 2.1, 2.2, 2.3, 2.19.

3. Constructing a truth table. Learn the standard truth table for each connective. Pay particular attention to the conditional, since it is the source of most of the trouble. (See 2.2, 2.4).

Learn how to construct a truth table for a complicated sentence with nested clauses (2.3). To do well in the test you will need to be able to evaluate a truth function such as
É ~ (q v ~s))
without having to think about it too long. You will be asked to fill out a truth table for at least one such sentence. See exercises 2.4-2.10.

2. Fundamentals of symbolizing conjunctions, disjunctions, negations, conditionals, and biconditionals. Sections 2.2, 2.4. Learn how to symbolize simple English sentences of all these types. You will be given roughly 20 sentences in English, and asked to do nothing but symbolize them, using a key that is provided. Try exercises 2.11-2.16. See also the "Test items" section, which follows all the regular exercises.

4. English variants of connectives. Learn how to recognize the folk variants for the five kinds of sentential connective. See section 2.5. Some of sentences mixed into the problems will use these variants.

It can be tricky to see the equivalences among some of these variant forms. Likewise, the "difficult combinations" (such as "not both p and not q" vs. "both not p and not q") can be hard to parse. One thing that will make both tasks easier is to understand some basic logical equivalences. See section 2.6. I strongly suggest you learn them, and test those you dont believe with a truth table, before you get too bogged down memorizing the English variants or the "difficult combinations". (Knowing the equivalences will cut down on the memory load!)

5. The goal of the test: given an argument, (a) symbolize the statements, (b) construct a truth table for them, and (c) use the truth table to test the argument for validity. This wraps together all the skills above. There will be at least two problems of this sort. Try exercises 2.11, 2.17.

New in this unit: actual test items. For symbolization and testing for validity, try pages from the actual test used in 1998. These follow the answers to the regular exercises. The "test answers" section is the coding frame used by TAs to grade that test, so you can see how partial credit gets assigned. What a deal!