Symbolic Logic I
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
- 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
- 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
- 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
is NOT equivalent to
Large(x)). Know the other equivalences and non-equivalences in chapter 10
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!
- 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.
- 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).
Last update: October 20, 2014.
| Symbolic Logic | Philosophy
Department | Trinity