Section 2: Building blocks of QL
Just as sentences were the basic unit of sentential logic, predicates will be the basic unit of quantified logic. A predicate is an expression like ‘is a dog.’ This is not a sentence on its own. It is neither true nor false. In order to be true or false, we need to specify something: Who or what is it that is a dog?
The details of this will be explained in the rest of the chapter, but here is the basic idea: In QL, we will represent predicates with capital letters. For instance, we might let \(D\) stand for ‘_____ is a dog.’ We will use lower-case letters as the names of specific things. For instance, we might let \(b\) stand for Bertie. The expression \(Db\) will be a sentence in QL. It is a translation of the sentence ‘Bertie is a dog.’
In order to represent quantifier structure, we will also have symbols that represent quantifiers. For instance, ‘∃’ will mean ‘There is some _______.’ So to say that there is a dog, we can write ∃\(xDx\); that is: There is some \(x\) such that \(x\) is a dog.
That will come later. We start by defining singular terms and predicates.
Singular Terms
In English, a singular term is a word or phrase that refers to a specific person, place, or thing. The word ‘dog’ is not a singular term, because there are a great many dogs. The phrase ‘Philip’s dog Bertie’ is a singular term, because it refers to a specific little terrier.
A proper name is a singular term that picks out an individual without describing it. The name ‘Emerson’ is a proper name, and the name alone does not tell you anything about Emerson. Of course, some names are traditionally given to boys and other are traditionally given to girls. If ‘Jack Hathaway’ is used as a singular term, you might guess that it refers to a man. However, the name does not necessarily mean that the person referred to is a man— or even that the creature referred to is a person. Jack might be a giraffe for all you could tell just from the name. There is a great deal of philosophical action surrounding this issue, but the important point here is that a name is a singular term because it picks out a single, specific individual.
Other singular terms more obviously convey information about the thing to which they refer. For instance, you can tell without being told anything further that ‘Philip’s dog Bertie’ is a singular term that refers to a dog. A definite description picks out an individual by means of a unique description. In English, definite descriptions are often phrases of the form ‘the such-and-so.’ They refer to the specific thing that matches the given description. For example, ‘the tallest member of Monty Python’ and ‘the first emperor of China’ are definite descriptions. A description that does not pick out a specific individual is not a definite description. ‘A member of Monty Python’ and ‘an emperor of China’ are not definite descriptions.
We can use proper names and definite descriptions to pick out the same thing. The proper name ‘Mount Rainier’ names the location picked out by the definite description ‘the highest peak in Washington state.’ The expressions refer to the same place in different ways. You learn nothing from my saying that I am going to Mount Rainier, unless you already know some geography. You could guess that it is a mountain, perhaps, but even this is not a sure thing; for all you know it might be a college, like Mount Holyoke. Yet if I were to say that I was going to the highest peak in Washington state, you would know immediately that I was going to a mountain in Washington state.
In English, the specification of a singular term may depend on context; ‘Willard’ means a specific person and not just someone named Willard; ‘P.D. Magnus’ as a logical singular term means me and not the other P.D. Magnus. We live with this kind of ambiguity in English, but it is important to keep in mind that singular terms in QL must refer to just one specific thing.
In QL, we will symbolize singular terms with lower-case letters \(a\) through \(w\). We can add subscripts if we want to use some letter more than once. So \(a\),\( b\),\(c\),...\(w\),\(a\) 1 ,\(f\) 32 ,\(j\) 390 , and \(m\) 12 are all terms in QL.
Singular terms are called constants because they pick out specific individuals. Note that \(x\), \(y\), and \(z\) are not constants in QL. They will be variables , letters which do not stand for any specific thing. We will need them when we introduce quantifiers.
Predicates
The simplest predicates are properties of individuals. They are things you can say about an object. ‘_____ is a dog’ and ‘______ is a member of Monty Python’ are both predicates. In translating English sentences, the term will not always come at the beginning of the sentence: ‘A piano fell on ______’ is also a predicate. Predicates like these are called one-place or monadic , because there is only one blank to fill in. A one-place predicate and a singular term combine to make a sentence.
Other predicates are about the relation between two things. For instance, ‘_____ is bigger than ______’, ‘______ is to the left of ______’, and ‘______ owes money to _____.’ These are two-place or dyadic predicates, because they need to be filled in with two terms in order to make a sentence.
In general, you can think about predicates as schematic sentences that need to be filled out with some number of terms. Conversely, you can start with sentences and make predicates out of them by removing terms. Consider the sentence, ‘Vinnie borrowed the family car from Nunzio.’ By removing a singular term, we can recognize this sentence as using any of three different monadic predicates:
_____ borrowed the family car from Nunzio.
Vinnie borrowed _____ from Nunzio.
Vinnie borrowed the family car from _____.
By removing two singular terms, we can recognize three different dyadic predicates:
Vinnie borrowed _____ from _____.
_____ borrowed the family car from _____.
_____ borrowed _____ from Nunzio.
By removing all three singular terms, we can recognize one three-place or triadic predicate:
_____ borrowed _____ from _____.
If we are translating this sentence into QL, should we translate it with a one-, two-, or three-place predicate? It depends on what we want to be able to say. If the only thing that we will discuss being borrowed is the family car, then the generality of the three-place predicate is unnecessary. If the only borrowing we need to symbolize is different people borrowing the family car from Nunzio, then a one-place predicate will be enough.
In general, we can have predicates with as many places as we need. Predicates with more than one place are called polyadic . Predicates with n places, for some number n, are called n-place or n-adic .
In QL, we symbolize predicates with capital letters A through Z, with or without subscripts. When we give a symbolization key for predicates, we will not use blanks; instead, we will use variables. By convention, constants are listed at the end of the key. So we might write a key that looks like this:
Ax: \(x\) is angry.
Hx: \(x\) is happy.
T
1
xy: \(x\) is as tall or taller than \(y\).
T
2
xy: \(x\) is as tough or tougher than \(y\).
Bxyz: \(y\) is between \(x\) and \(zy\).
d: Donald
g: Gregor
m: Marybeth
We can symbolize sentences that use any combination of these predicates and terms. For example:
1. Donald is angry.
2. If Donald is angry, then so are Gregor and Marybeth.
3. Marybeth is at least as tall and as tough as Gregor.
4. Donald is shorter than Gregor.
5. Gregor is between Donald and Marybeth.
Sentence 1 is straightforward: \(Ad\). The ‘\(x\)’ in the key entry ‘\(Ax\)’ is just a placeholder; we can replace it with other terms when translating.
Sentence 2 can be paraphrased as, ‘If \(Ad\), then \(Ag\) and \(Am\).’ QL has all the truth-functional connectives of SL, so we translate this as \(Ad\) → (\(Ag\)&\(Am\)).
Sentence 3 can be translated as \(T\) 1 \(mg\) & \(T\) 2 \(mg\).
Sentence 4 might seem as if it requires a new predicate. If we only needed to symbolize this sentence, we could define a predicate like \(Sxy\) to mean ‘\(x\) is shorter than \(y\).’ However, this would ignore the logical connection between ‘shorter’ and ‘taller.’ Considered only as symbols of QL, there is no connection between \(S\) and \(T\) 1 . They might mean anything at all. Instead of introducing a new predicate, we paraphrase sentence 4 using predicates already in our key: ‘It is not the case that Donald is as tall or taller than Gregor.’ We can translate it as ¬\(T\) 1 \(dg\).
Sentence 5 requires that we pay careful attention to the order of terms in the key. It becomes \(Bdgm\).