In this chapter we mentioned some of the motivations for using formal logic as a semantic metalanguage. We discussed the notion of valid inference, and showed that valid patterns of reasoning guarantee a true conclusion only when the premises are true. We then showed how propositional logic accounts for certain kinds of inferences, namely those which are determined by the meanings of the logical operators ‘and’, ‘or’, ‘not’, and ‘if’. In this way propositional logic helps to explain certain kinds of tautology and contradiction, as well as certain types of meaning relations between sentences (entailment, paraphrase, etc.), namely those which arise due to the logical structure of the sentences involved. Finally we gave a brief introduction to predicate logic, which allows us to represent the meanings of the propositions, and an even brief introduction to the use of quantifiers, which will be the topic of Chapter 14.
Our emphasis in this chapter was on translating sentences of English (or some other object language) into logical notation. In Unit IV we will discuss how we can give an interpretation for these propositions in terms of set theory, and how this helps us understand the compositional nature of sentence meanings.
Further reading
Good, brief introductions to propositional and predicate logic are provided in Allwood et al. (1977: chapters 4–5) and Kearns (2000: chapter 2). More detailed introductions are provided in J. N. Martin (1987) andGamut (1991a).a
aL. T. F. Gamut is a collective pen-name for the Dutch logicians Johan van Benthem, Jeroen Groenendijk, Dick de Jongh, Martin Stokhof and Henk Verkuyl.
Discussion exercises
A. Create a truth table to prove each of the following tautologies:
a. Law of Double Negation: ¬(¬p) ↔ p
b. Law of Contradiction: ¬(p ∧ ¬p)
c. Modus Tollens: [(p → q) ∧ ¬q] → ¬p
B. Construct syllogisms, using English sentences, to illustrate each of the following patterns of inference:
a. Modus Ponens: [(p → q) ∧ p] → q
b. Modus Tollens: [(p → q) ∧ ¬q] → ¬p
c. Hypothetical Syllogism: [(p → q) ∧ (q → r)] → (p → r)
d. Disjunctive Syllogism: [(p ∨ q) ∧ ¬p] → q
C. Translate the following sentences into logical notation:
a. All unicorns are herbivores.
b. No philosophers admire Nietzsche.
c. Some green apples are edible.
d. Bill feeds all stray cats.
Homework exercises
A. Using truth tables. Arthur has been selected to be a juror in a case which has generated a lot of local publicity. He is asked to promise not to read the newspaper or watch television until the trial is finished. There are two different ways in which he can make this commitment:
(1) a. I will not read the newspaper or watch television until the trial is finished.
b. I will not read the newspaper and I will not watch television until the trial is finished.
Construct truth tables for these two sentences to show why they are logically equivalent. You may omit the adverbial clause (until the trial is finished) from your table. (Hint: Let p stand for I will read the newspaper and q stand for I will watch television. Assume the following translation for sentence (a): ¬(p ∨ q). Construct a truth table for this proposition, and a second truth table for sentence (b). If the right-most column of the two tables is identical, that means that the two propositions must have the same truth value under any circumstances.)
(1′ ) a. 
b. 
B. Translate the following sentences into logical notation:
a. All famous linguists quote Chomsky.
b. David tutors some struggling students.
c. No president was Buddhist or Hindu.
d. Alice and Betty married Charlie and David, respectively.
