Polysemy and monosemy
6.2 Propositional logic
6. Koko is hairy.
All primates are hairy.
therefore
Koko is a primate.
As we have observed, the properties of sentences which make them true are linguistic properties. This suggests that logic and semantics are closely related. Some scholars, indeed, such as McCawley (1981: 2), have assumed that logic and semantics share an identical subject matter: the meanings of natural language sentences. As we will see, not everyone would agree with this: the degree of correspondence between logic and natural lan- guage has often been questioned, and with good reason. Nevertheless, as McCawley (1981: 2) notes, logic requires semantic analysis: the meanings of sentences must be identified before their logical properties can be dis- cussed. If we do not know the meanings of are and all in (4) we are not in a position to determine the validity of the arguments involving them.
The link between logic and semantics is further revealed by the fact that it is meanings, not sentences, that function as the premises and con- clusions of arguments. Thus, assuming (perhaps wrongly) that unhappy and discontented are synonyms, we can substitute any of the synonymous expressions in (7) for the premise of (6), and the synonymous expression in (8) for the conclusion of (6):
(7) All humans born on a Tuesday are unhappy.
All people born on a Tuesday are discontented.
All people who were, are or will be born on a Tuesday are unhappy.
(8) Bogomil is discontented.
These variations do not affect the underlying logical form of the argument.
considered a proposition as long as the referent of the noun ‘Koko’ has been fixed. Only if we know who ‘Koko’ refers to can we know whether a proposition in which she is mentioned is true or not.
Strictly, the notion of a proposition belongs to logic. We can, however, see it in mental terms. A series of experiments by psychologists has shown that people are very bad at remembering the actual words of utterances.
About twenty seconds after hearing or reading an utterance, all people remember is its content or gist: the actual words used usually can’t be remembered accurately. Given this, the propositions discussed here would be one possible representation of this remembered content or gist (see Barsalou et al. 1993 for discussion).
Natural language is not a collection of brute propositional statements without any mutual interrelations: a single statement like (10a) or (10b) can serve as the basis for a whole series of additional statements, depend- ing on the additional linguistic elements added to it. Some examples of these additional statements are given in (10c–h):
(10) a. Daryl Tarte grew up to publish a raunchy family saga in 1988.
b. Patsy Page is telling the truth.
c. Someone suspects that Daryl Tarte grew up to publish a raunchy family saga in 1988.
d. It is probable that Daryl Tarte grew up to publish a raunchy family saga in 1988.
e. Daryl Tarte did not grow up to publish a raunchy family saga in 1988.
f. Daryl Tarte grew up to publish a raunchy family saga in 1988, and Patsy Page is telling the truth.
g. Either Daryl Tarte grew up to publish a raunchy family saga in 1988, or Patsy Page is telling the truth.
h. If Daryl Tarte grew up to publish a raunchy family saga in 1988, then Patsy Page is telling the truth.
It is the italicized elements in (10c–h) which chiefly serve to insert the original propositions (10a–b) into a new, longer one. Among these ele- ments, propositional logic attaches special importance to the four found in (10 e–h). In English, these four elements are expressed by the words and, or, not and if . . . then. We will refer to these as the propositional connec- tives or logical operators (already mentioned in 4.3.1). These four differ from others, such as those in (10c–d), in that they are truth-functional.
This means that whether the larger propositions they are part of are true or not depends solely on the truth of the original basic propositions to which they have been added: the logical operators do not add anything true or false to the basic propositions themselves; all they do is generate additional propositions from the basic ones.
Let’s demonstrate truth-functionality by considering the operator not.
Let’s grant that (10a) ‘Daryl Tarte grew up to publish a raunchy family saga in 1988’ is true. Then, (10e) ‘Daryl Tarte did not grow up to publish a raunchy family saga in 1988’ cannot be true: the two propositions are contradictory,
and we cannot imagine a world in which they could be simultaneously possible. Conversely, if (10e) is true, then (10a) must be false. We can deduce the truth or falsity of one proposition from the other: if one is true, the other can only be false. Similarly, if (10a–b) are true, then (10f) must also be true. But if one or both of (10a–b) are false, then (10f) as a whole must likewise be false.
As we will see, the other two connectives are also truth-functional.
Before showing this, however, we need to abstract away from the English words which express them. Observe that (10e) is not the only way in which negation is expressed in English. The following sentences all involve nega- tions or denials, but unlike (10e), they do not use the grammatical means of do/did not to express this:
(11) a. Neither the newspaper nor the radio gave more details.
b. She has not been an opera enthusiast all her life.
c. The Post Office had taken no notice of her death.
d. He was unable to tell the difference between Schumann and Schubert.
e. He failed the driving test for the third time.
Intuitively, however, it seems obvious that all these sentences contain a denial or a negation, but under different grammatical guises. Examples like these show that language makes differing means available to express what is, intuitively, a single logical operation, negation. Let’s further assume that propositions like those in (11a–e) can be expressed in every language. Let’s assume, in other words, that there is no reason that the propositions have to be stated in English: speakers of any language can negate propositions in a way that is semantically identical to the English negations in (11a–e).
QUESTION What are some alternative ways in which the other operators could be expressed in English?
Considerations like these mean that we need to find some other way of symbolizing the operators which abstracts away from their translations into any single natural language. To do this, we will adopt a set of sym- bols for negation and the other operators. Negation, for example, will be symbolized with the symbol ¬. We will introduce the other symbols in the rest of this section. Note that the symbols apply uniquely to entire propositions. If the small letters p, q, r. . . stand for given proposi- tions, ¬p, ¬q and ¬r stand for their negations. We cannot use ¬ to sym- bolize negations of non-propositional elements like not tomorrow, not again, etc.
The values or meanings of the operators can be specified in the form of diagrams called truth tables. Truth tables display the way in which logical operators affect the truth of the propositions to which they are added.
(The use of truth tables is a fairly recent innovation in logic: they are implicitly present in Frege, but first overtly used by Wittgenstein.) The truth table for ¬ is very simple, and is given in Table 6.1:
All this says, reading left to right and top to bottom, is that if p is true,
¬p (‘not p’) is false, and that if p is false, ¬p is true. Let’s say that p is the proposition ‘Marie Bashir is governor of New South Wales’. If this is true, then ¬p, ‘Marie Bashir is not governor of NSW’ must be false; conversely, if it is false, then ¬p must be true. The truth table can be read in either direc- tion. It is equally true, then, that if ¬p is false, then p is true, and if ¬p is true, then p is false.
The next logical operator is conjunction. As its name implies, this denotes the conjunction or union of two propositions. The conjoined propositions are called conjuncts. The symbol for conjunction is the ampersand, &. The lexical realizations of conjunction are quite various. In particular, the logi- cal operator translates English and and but, as well as other contrastive conjunctions like in spite of and although. If p stands for the proposition ‘The Emperor has no money’ and q for ‘he has 400 000 soldiers’, then p & q can stand for any one of the following complex propositions:
(12) The Emperor has no money, and he has 400 000 soldiers.
The Emperor has no money, he has 400 000 soldiers.
The Emperor has no money, but he has 400 000 soldiers.
The Emperor has no money, although he has 400 000 soldiers.
The Emperor has no money even though he has 400 000 soldiers.
The Emperor has no money, in spite of which he has 400 000 soldiers.
The truth-table for & is given in Table 6.2:
Table 6.2. Truth table for &.
p q p & q a. T T T b. T F F c. F T F d. F F F Table 6.1. Truth table for ¬.
p ¬ p T F F T
If two propositions are both true, then their conjunction is also true (case (a) in Table 6.2). If the proposition apricots are fruit and the proposition beans are vegetables are both true (as, indeed, they are), then the compound proposition apricots are fruit and beans are vegetables must also be true. But if one of the conjoined propositions (conjuncts) is false, then the entire con- junction is also false (cases (b) and (c)). For example, let’s take the two propositions apricots are fruit (which is true) and beans are fruit (which is
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