Exam 2 Review

Here are some things to be familiar with for the second exam.

**1. Basic concepts**

- be familiar with basic logical equivalences for the conditional and biconditional (especially those in exercises 7.1-7.4 and 7.6).
- know whether the argument forms in exercises 8.18-8.25 are valid or not; be able to prove them if they are valid and give counterexamples if they aren't.
- Suppose the only propositional logical connectives our language included were those for negation and conjunction. Would our language lose any expressive power? Explain.
- Be able to explain the difference between a wff (well-formed formula) and a sentence and, given an example, be able to tell whether it is a wff and whether it is a sentence. (Remember that every sentence is a wff, but not vice versa.)
- Be able to explain the difference between bound and unbound variables, and, given a formula, be able to tell whether an occurrence of a given variable is free or bound.
- Know the four Aristotelian forms for quantifier sentences, both how to recognize them in English and also what the standard translation is for each form. Also know equivalences involving those forms and negation (for example, that the negation of a universal affirmative is equivalent to a particular negative).
- what are FO validity, FO equivalence, and FO consequence? (So far we have only explained these ideas informally, in terms of logical relations that still hold regardless of what substitutions are made for predicates, including substitution of predicates with unknown interpretations like 'Mimsy(x)' and 'Outgrabe(x,y)'.)
- logical equivalences: know what the DeMorgan's laws for the quantifiers are, and how they are related to the DeMorgan's laws for propositional logic. Also know the Aristotelian form equivalences: Ax (A(x) -> B(x)) <=> ~Ex (A(x) & ~B(x)), Ex (A(x) & B(x)) <=> ~Ax (A(x) -> ~B(x)), Ax (A(x) -> ~B(x)) <=> ~Ex (A(x) & B(x)), Ex (A(x) & ~B(x)) <=> ~Ax (A(x) -> B(x)). what other equivalences do (and do not!) hold for first-order logic. For example, you should know that ∀x (Tet(x) ∧ Large(x)) ⇔ (∀x Tet(x) ∧ ∀x Large(x)), but ∃x (Tet(x) ∧ Large(x)) is NOT equivalent to (∃x Tet(x) ∧ ∃x Large(x)). Know the other equivalences and non-equivalences in chapter 10 also.

**2. Truth Tables**

- be able to identify the
**truth-functional form**of quantifier sentences, and use a truth table to determine whether a quantifier sentence is a tautology, is tautologically equivalent to other another sentence, or is a tautological consequence of a set of other sentences. - know how to use truth tables to determine whether sentences involving conditionals and biconditionals are tautologies, whether pairs of sentences are tautologically equivalent, and whether a conclusion is a tautological consequence of a collection of premises.
- keep in mind that your answer will be counted wrong if you don't follow our text's convention for filling in the base columns of a truth table!

**3. Translations**

- know how to translate sentences from English into symbolic notation, including conditionals, biconditionals, and single quantifiers. (The English sentences will either use English equivalents of Tarski's World predicates, which you should already know how to translate, or I will give you a list of predicates you can use in the translation.)
- Remember how to translate English sentences of the forms if P then Q, P if and only if Q, P only if Q, Q if P, P is necessary for Q, P is sufficient for Q, P unless Q, etc.
- Translations will include sentences with one quantifier of a single type, but no sentences with multiple quantifiers.

**4. Proofs**

- know how to do proofs using all the rules we have covered so far: in addition to the rules used on the first exam, this will include the Intro and Elim rules for the conditional and biconditional.
- If you've gotten rusty on propositional proofs, or if there were homework exercises you never did get, it would be a very good idea to go back over these proofs! You should be able to do all the homework exercises. If you would like additional practice on proofs, there is a handout with additional propositional proofs on the TLEARN site (plus a second handout with solutions to the odd-numbered proofs).

Curtis Brown | Symbolic Logic | Philosophy Department | Trinity University

cbrown@trinity.edu