Polysemy and monosemy
6.1 Validity, soundness and logical form
Logic may be defined as the study of valid reasoning and inference. On this definition, logic investigates the properties of valid chains of reason- ing, and specifies the conditions which these chains must meet in order to be valid, in order to work as arguments. Consider the following exchange:
(1) A: Koko is a primate, so she likes daytime television.
B: What? I don’t get it.
A: All primates like daytime television
Initially, B is unable to follow A’s line of thought, and as a result A is forced to state the general principle on which her conclusion rests. This allows us to reconstruct A’s original train of thought as the following argument or syllogism:
(2) 1. All primates like daytime television.
2. Koko is a primate.
therefore
3. Koko likes daytime television.
Argument (2) thus reveals the explicit logical structure of A’s comment in (1). As Kneale and Kneale explain (1962: 12), the ‘first tentative steps towards logical thinking are taken when men try to generalize about valid arguments and to extract from some particular valid argument a form or principle which is common to a whole class of valid arguments’. Given the meanings of the words all, like and is, the conclusion Koko likes daytime tel- evision just has to be true as long as we accept the truth of the proposition All primates like daytime television. It seems likely that it was in domains like mathematics, especially geometry, that the need to make the principles of valid reasoning explicit first arose (Kneale and Kneale 1962: 2); in modern times, the study of logic has been particularly undertaken in the attempt to symbolize the types of reasoning that underlie mathematical argu- ments.
Logic is important to linguistics for at least three reasons. First, the study of logic is one of the oldest comprehensive treatments of questions of meaning. When people first began to think systematically about the meanings of language and the relations between these meanings, it was logical concepts to which they often appealed for explanations. As a result, the tradition of logical analysis, which we can trace as far back as Aristotle, provides a rich body of reflection on meaning, and most schol- ars who have studied meaning in the Western tradition have had at least some knowledge of logical principles. The relevance of logic to linguistics is far from simply historical, however. Logical concepts inform a wide range of modern formal theories of semantics, and are also crucial in research in computational theories of language and meaning. We will not be exploring formal theories in themselves here, but our exposition of
some fundamental logical ideas will provide some background for those wanting to do so. Lastly, logical concepts provide an enlightening point of contrast with natural language. The basic logical concepts are accessible to practically anyone; indeed, many philosophers have seen in logical principles the universal ‘laws of thought’ which constitute the basic grounds of human rationality: for Immanuel Kant, for example, ‘logic is the science that exhaustively presents and strictly proves nothing but the formal rules of all thinking’ (1998 [1787]: 106). Yet, as we will see, logical meanings often differ strikingly from the types of meaning found in natu- ral language. Studying logic therefore provides a window onto a body of apparently universal concepts with strikingly different behaviour from natu- ral language, which provide a rigorous and enlightening way of disambigu- ating certain types of natural language expression.
Formal theories
A formal theory is one which offers an analysis of meaning in a tech- nical, usually symbolic, metalanguage, according to principles which can be expressed in mathematical terms. A formal representation of meaning avoids the ambiguities contained in natural language by enforcing a strict correspondence between symbols and meanings:
whereas natural languages always contain ambiguous or polysemous terms, in which a single form stands for several meanings (think of English step, match or get), a formal language has a strictly one-to-one relation with its meanings, so that each symbol of the formalism has one and only one interpretation.
As the above quotation from Kant suggests, the principles of valid argu- ment have typically been taken, in the logical tradition, as the very prin- ciples governing rational human thought. Logic can be seen, from this perspective, as the science of the laws of rational thought. On this view, logic is the science which tries to specify all the conclusions that can val- idly be reached from a given set of propositions. It is logical principles which thus describe the process of valid reasoning.
The first two propositions in (2) are called the premises. An argument’s premise may be defined as its starting-point, one of the propositions from which the conclusion follows. In (2), the last proposition is the conclu- sion. Note that the validity of arguments or of chains of reasoning has a special relationship to the words in which the premises and conclusion may be stated: substitute different words, and the argument may not be valid. None of the following arguments, for instance, is valid:
(3) a. Most primates like daytime television.
Koko is a primate.
therefore
Koko likes daytime television.
b. All primates may like daytime television.
Koko is a primate.
therefore
Koko likes daytime television.
c. All primates used to like daytime television.
Koko is a primate.
therefore
Koko likes daytime television.
d. Some primates like daytime television.
Koko is a primate.
therefore
Koko likes daytime television.
e. All primates like daytime television.
Koko wants to be a primate.
therefore
Koko likes daytime television.
QUESTION For each of these arguments, explain why it is not valid.
We may therefore say that the validity of this argument type is a function of the meaning of the terms in the syllogism, in particular the meanings of the determiner all, and of the verbs like and is. In (4), if we retain the ‘logical’ words all and are, but substitute words whose meaning is unknown, the resulting argument is still valid: if the premises are true, the conclusion must be true:
(4) All sclaps are hillories Steven is a sclap, therefore
Steven is a hillory.
Any argument which conforms to the pattern of (4) will be valid. All the arguments below contain the same pattern of reasoning as (4):
(5) a. All primates like daytime television.
Humans are primates.
therefore
Humans like daytime television.
b. All computers will break down.
Laptops are computers.
therefore
Laptops will break down.
c. All gladiators feared defeat.
The Thracian was a gladiator.
therefore
The Thracian feared defeat.
This allows us to see that the pattern of valid inference exemplified in (4) and (5) is systematic and does not depend on the details of the material
inserted into the logical formula. The validity of these arguments is a function purely of the meaning of the predicates and of the term all and of the argument’s logical form, its underlying logical structure.
The fact, however, that an argument conforms to the pattern of (4) may make it valid, but this does not make its conclusion true. Logical validity and truth are quite different properties: validity is a property of argu- ments, truth a property of sentences. The following argument is valid in logic, but its first premise is not true.
(6) All people born on a Tuesday are unhappy.
Bogomil was born on a Tuesday.
therefore
Bogomil is unhappy.
This means that (6) is valid in logic, but it is not true. Valid arguments whose premises are true are referred to as sound. Arguments like (6) which are valid, but which do not have true premises, are thus unsound.
We can tell that (6) is valid, since if the first premise were true, then the conclusion would also necessarily be true. (Note that whether Bogomil is, as a matter of fact, unhappy, has nothing to do with the soundness of (6).
Bogomil may well actually be unhappy, but this is not proven by the argu- ment in (6).)
QUESTION Assess the following arguments, stating whether they are valid and sound, valid but unsound, or invalid.
1. All people need oxygen.
Harry is a person.
therefore
Harry needs oxygen.
(Assume that Harry is a parrot.) 2. Brazil is a country.
All countries belong to the UN.
therefore
Brazil belongs to the UN.
3. Henry has fallen in the pool.
If someone has fallen in the pool, then they are wet.
therefore Henry is wet.
(Assume that the first premise is true.) 4. Gin and tonic is a popular drink.
Popular drinks are cheap.
therefore Water is cheap.
5. Mirrors reflect things.
This glass reflects things.
therefore
This glass is a mirror.
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.