Predicate logic
Consider the following argument:
This argument is clearly valid. But notice that using the propositional symbols we have introduced so far, we cannot demonstrate this validity. The two premises and the conclusion of (26) each express different propositions. We have no way, in our existing symbolism, of showing that these propositions involve the recurrent elements Koko, primate and hairy. As things stand, we can only assign a different letter variable to each of the propositions, giving us the following symbolism for the argument:
The logical form ‘p, q, therefore r’ is thus the only way we have in propositional logic to symbolize the structure of the argument. But, in itself, this logical form is invalid. To see this, recall that p, q, and r can refer to any proposition; thus (28) is equally an instance of an argument with the form p, q, therefore r:
Clearly, wherever the validity of (27) comes from, it does not derive from its conformity to the logical form p, q therefore r; as demonstrated by (28), not all arguments of this form are valid. Instead, the validity of (27) springs principally from the meaning of the term all. In order to symbolize (27) in a way that makes its validity clear, we will need to go beyond a purely propositional notation so that the idea of ‘all’ can be captured in a logically rigorous way.
Now consider the argument in (29):
Propositionally, this argument has the form p, therefore (q ) r): again, a clearly invalid argument form. Yet (29) is obviously valid, and its validity derives from the meaning of the term some. In order to symbolize the validity of arguments like (29), we therefore also need some way of capturing the idea of ‘some’.
‘Some’ and ‘all’ are the basic notions in the other branch of logic with which we will be concerned in this chapter. This branch is predicate logic, also known as quantificational or first-order logic. What exactly are predicates? Let’s examine (27) again. From a logical point of view, (27) contains three basic types of term: terms referring to individuals, such as Koko, terms referring to quantities, like all, and general terms like primate and hairy. Terms referring to individuals are called singular terms or individual constants. We will symbolize them with lower case letters. Koko, for instance, can be symbolized simply by k. Terms referring to quantities like ‘all’ or ‘some’ are called quantifiers: we will introduce the symbols for them presently.
‘Primate’ and ‘hairy’ in (27) are predicates. ‘Predicate’ has rather a different meaning in logic from the meaning it typically has in syntax. In syntax, ‘predicate’ is often roughly synonymous with ‘verb’. In logic, however, predicates are terms which represent properties or relations: here, the properties of ‘primateness’ and ‘hairiness’. A logical predicate could thus be a general noun like primate, an adjective like hairy or a verb like adore in Koko adores the news. Whereas singular terms refer to specific individuals, predicates refer to general terms, terms which are potentially true of numerous individuals. Being a primate and being hairy are proper ties which any number of individuals can hold. By contrast, the term Koko picks out just a single individual. The properties and relations expressed by predicates can be quite complex and lengthy. For instance, as well as ‘is hairy’ and ‘is a primate’, the expressions ‘is a good student’, ‘is taller than the Eiffel tower’, ‘loves skiing’ and ‘bought a book on the giant sloth from Amazon’ are all predicates. We will discuss these different types of predicate below.
Predicates are typically symbolized by single capital letters. The predicate ‘is a primate’, for example, could be symbolized P, and the predicate ‘is hairy’ by H. When expressions containing predicates and singular terms are translated into logical notation, the capitalized predicate symbol is written first, followed by the symbol for the singular term to which the predicate applies. Thus, we can translate the expressions ‘Koko is a primate’ and ‘Koko is hairy’ as follows:
The individual a predicate applies to is called its argument: P and H in (30) each have a single argument. But this notation will only get us a certain way. Eventually, we want to be able to translate propositions like ‘All primates are hairy’. To do this, we need to examine quantifiers. Quantifiers are the logical expressions ‘some’ and ‘all’, symbolized by the operators and respectively. Inferences which, like (27) and (29), involve the notions of ‘some’ and ‘all’ are very common. Examine the following formula:
(31) reads as ‘For every x, x is a primate’. What this says is that every individual in the domain in question is a primate. (31) is thus the translation of ‘Everything is a primate’ (an obviously false statement). Compare this to (32):
This reads as ‘there exists at least one x, such that x is a primate’. This says that something (or someone) is a primate – an obviously true statement.
∀ is known as the universal quantifier. Universal quantification is the logical operation which says that a predicate is true of every entity in the domain under discussion. Including in a formula thus applies the predicate to every entity (argument) in the domain in question. In English, uni versal quantification can be expressed by the words all and every, and the phrases each and every and everything.
ꓱ is known as the existential quantifier. Existential quantification is the logical operation which says that a predicate is true of at least one entity in the domain under discussion. Including in a formula applies a predicate to at least one entity (argument) in the domain in question. In English, existential quantification can be expressed by the words some, at least one, and something.
The quantifiers can be combined with the propositional operators. Some examples of this are given below. In (33), the abbreviation S stands for ‘is simple’, and F stands for ‘is fun’.
The most interesting combinations, however, result from the use of). Consider the following formula in conjunction with the explanations of the symbols:
This says that for all x’s, if x is a primate then it is hairy. This allows us to give the following translation of the argument in (27), with the justification for the steps shown at the right (k = Koko)
‘To be hairy’ and ‘to be a primate’ are one place predicates: this means that they can only be associated with a single individual constant at a time. (Recall that individual constants, or singular terms, are terms refer ring to a single individual. Individual constants are sometimes known as variables.) For example, the sentence ‘Koko and Wilma are primates’ can only be expressed logically as (36a), not as (36b).
The formula in (36b) is ill-formed. Since the property of being a primate only ever involves a single individual at a time, one of the constants in (36b) is left ‘floating’: it is not attached to any predicate, and nothing (even existence) is asserted of it.
Not all predicates are one-place predicates. The predicate ‘admire’, for example, is a two-place predicate: if admiring is going on, then two participants are necessarily involved, the admirer and the admiree. Using A for ‘admire’, we can express the sentence ‘Dietmar admires Horst’ as (37) and ‘Horst admires Dietmar’ as (38):
A two-place predicate can thus be interpreted as indicating a set of ordered pairs of individuals: here, the pair Dietmar and Horst. It is a set of ordered pairs precisely because the order in which the individuals occur is crucial: the first individual is the one who admires, the second the one who is admired.
There is no limit on the number of places a predicate may have. ‘Give’ is an example of a three-place predicate, as in G d, b, h ‘Dietmar gave the book to Horst’.
We have been defining ‘predicate’ as a general term expressing a property or a relation. But we may also think of predicates in terms of the individuals to which they apply. Thus, a one-place predicate may be interpreted as a set of individuals: those individuals to which the predicate applies (these are sometimes referred to as the individuals that ‘satisfy’ the predicate). A two-place predicate applies to an ordered pair of individuals, a three-place predicate to an ordered triple of individuals, and so on. Accordingly, a predicate can have as many places as the members of the ordered n-tuple of individuals that satisfy it.
We are now in a position to be able to produce translations into logical notation of some reasonably complex propositions. These examples involve one- and two-place predicates, and show how the propositional operators are used with them. We first give the logical formula, then a translation into ‘logiceeze’, then a translation into idiomatic English.
Note that the last example could also be translated as follows
The examples given so far involve only a single quantifier. But natural language frequently expresses propositions involving multiple quantification, i.e. expressions which refer to two or more quantities. A two-place predicate, for example, may be quantified in various different ways, some of which we will now illustrate with the two-place predicate R ‘remember’.
The simplest case of multiple quantification is where both variables have the same quantifier:
Note that this formula would be valid in the case where someone remembers themselves.
More complex are cases where one variable receives universal quantification and the other existential. Consider the following example:
Here we will say that y is in the scope of x. Let’s now consider what happens if we swap the order of the individual variables:
Here, ∀x is in the scope of ⱻy. The contrast between (42) and (43) is the difference between an active (42) and a passive (43) sentence. Importantly, the order of the variables in (43) is crucial: (43) is not logically equivalent to (44), which expresses a quite different proposition:
The difference between (43) and (44) is subtle but real. (43) says that there is at least one single individual whom everyone remembers. It is the same individual who is remembered by everyone: in a universe consisting of Nina, Andrew, Tom, Harry and Briony, Tom might be remembered by Nina, Andrew, Harry and Briony. (44), by contrast, says that every person remembers at least one person. This single person remembered by everybody may well differ from person to person: Briony may remember Harry, Nina may remember Andrew, Andrew may remember Tom. In (44), the existential quantifier is said to be in the scope of the universal quantifier.
To take another example of scope differences, consider the two-place predicate F ‘is the father of’ in the following two propositions (see Allwood et al. 1977: 67 for discussion):
The first proposition, (45), is true, the second, (46), is not. Yet the difference between them consists solely in the order of the existential and universal quantifier, and the consequent scope differences between the two.
Predicate logic notation can be used to precisely represent ambiguities in natural language. Sentence (47a), for example, has, among other readings, (47b) and (47c):
We can represent this difference concisely using the constant p for a per son and c for a pair of companies, and the predicate W ‘work for’:
