In Volume I you gained a firm foundation in sentence logic. But there must be more to logic, as you can see from the next examples. Consider the following two English arguments and their transcriptions into sentence logic:
(1 ) Everyone lwes Adam. A (2) Eve loves Adam. B
Eve loves Adam. B Someone loves Adam. C
In sentence logic, we can only transcribe the sentences in these arguments as atomic sentence letters. But represented with sentence letters, both natural deduction and truth trees tell us that these arguments are invalid. No derivation will allow us to derive 'B' from 'A' or 'C' from 'B'. A&~B is a counterexample to the first argument, and B&~C is a counter-example to the second. An argument is valid only if it has no counterexamples.
Something has gone terribly wrong. Clearly, if everyone loves Adam, then so does Eve. If the premise is true, without fail the conclusion will be true also. In the same way, if Eve loves Adam, then someone loves Adam. Once again, there is no way in which the premise could be true and the conclusion false. But to say that if the premises are true, then without fail the conclusion will be true also is just what we intend when we say that an argument is valid. Since sentence logic describes these arguments as invalid, it looks like something has to be wrong with sentence logic.
Sentence logic is fine as far as it goes. The trouble is that it does not go far enough. These two arguments owe their validity to the internal logical structure of the sentences appearing in the arguments, and sentence logic does not describe this internal logical structure. To deal with this shortcoming, we must extend sentence logic in a way which will display the needed logical structure and show how to use this structure in testing arguments for validity. We will keep the sentence logic we have learned in Volume I. But we will extend it to what logicians call Predicate Logic (also sometimes called Quantificational Logic).
Predicate logic deals with sentences which say something about someone or something. Consider the sentence 'Adam is blond.' This sentence attributes the property of being blond to the person named 'Adam'. The sentence does this by applying the predicate (the word) 'blond' to the name 'Adam'. A sentence of predicate logic does the same thing but in a simplified way.
We will put capital letters to a new use. Let us use the capital letter 'B', not now as a sentence letter, but to transcribe the English word 'blond'. And let us use 'a' to transcribe the name 'Adam'. For 'Adam is blond.', predicate logic simply writes 'Ba', which you should understand as the predicate 'B' being applied to the name 'a'. This, in turn, you should understand as stating that the person named by 'a' (namely, Adam) has the property indicated by 'B' (namely, the property of being blond).
Of course, on a different occasion, we could use 'B' to transcribe a different English predicate, such as 'bachelor', 'short', or 'funny'. And we could use 'a' as a name for different people or things. It is only important to stick to the same transcription use throughout one problem or example.
Predicate logic can also express relations which hold between things or people. Let's consider the simple statement that Eve loves Adam. This tells us that there is something holding true of Eve and Adam together, namely, that the first loves the second. To express this in predicate logic we will again use our name for Adam, 'a'. We will use a name for Eve, say, the letter 'e'. And we will need a capital letter to stand for the relation of loving, say, the letter 'L'. Predicate logic writes the sentence 'Eve loves Adam.' as 'Lea'. This is to be read as saying that the relation indicated by 'L' holds between the two things named by the lowercase letters 'e' and 'a'. Once again, in a different example or problem, 'L', 'a', and 'e' could be used for different relations, people, or things.
You might be a little surprised by the order in which the letters occur in 'Lea'. But don't let that bother you. It's just the convention most often used in logic: To write a sentence which says that a relation holds between' two things, first write the letter which indicates the relation and then write the names of the things between which the relation is supposed to hold. Some logicians write 'Lea' as 'L(e,a)', but we will not use this notation. Note, also, the order in which the names 'e' and 'a' appear in 'Lea'. 'Lea' is a different sentence from 'Lae'. 'Lea' says that Eve loves Adam. 'Lae' says that Adam loves Eve. One of these sentences might be true while the other one is false! Think of 'L' as expressing the relation, which holds just in case the first thing named loves the second thing named.
Here is a nasty piece of terminology which I have to give you because it is traditional and you will run into it if you continue your study of logic. Logicians use the word Argument for a letter which occurs after a predicate or a relation symbol. The letter 'a' in 'Ba' is the argument of the predicate 'B'. The letters 'e' and 'a' in 'Lea' are the arguments of the relation symbol 'L'. This use of the word 'argument' has nothing to do with the use in which we talk about an argument from premises to a conclusion.
At this point you might be perplexed by the following question. I have now used capital letters for three different things. I have used them to indicate atomic sentences. I have used them as predicates. And I have used them as relation symbols. Suppose you encounter a capital letter in a sentence of predicate logic. How are you supposed to know whether it is an atomic sentence letter, a predicate, or a relation symbol?
Easy. If the capital letter is followed by two lowercase letters, as in 'Lea', you know the capital letter is a relation symbol. If the capital letter is followed by one lowercase letter, as in 'Ba', you know the capital letter is a predicate. And if the capital letter is followed by no lowercase letters at all, as in 'A', you know it is an atomic sentence letter.
There is an advantage to listing the arguments of a relation symbol after the relation symbol, as in 'Lea'. We can see that there is something important in common between relation symbols and predicates. To attribute a relation as holding between two things is to say that something is true about the two things taken together and in the order specified. To attribute a property as holding of one thing is to say that something is true about that one thing. In the one case we attribute something to one thing, and in the other we attribute something to two things.
We can call attention to this similarity between predicates and relations in a way which also makes our terminology a bit smoother. We can indicate the connection by calling a relation symbol a Two Place Predicate, that is, a symbol which is very like an ordinary predicate except that it has two argument places instead of one. In fact, we may sometimes want to talk about three place predicates (equally well called 'three place relation symbols'). For example, to transcribe 'Eve is between Adam and Cid', I introduce 'c' as a name for Cid and the three place predicate 'K' to indicate the three place relation of being between. My transcription is 'Keac', which you can think of as saying that the three place relation of being between holds among Eve, Adam, and Cid, with the first being between the second and the third.
This is why our new logic is called 'predicate logic': It involves predicates of one place, two places, three places, or indeed, any number of places. As I mentioned, logicians also refer to these symbols as one place, two place, or many place relation symbols. But logicians never call the resulting system of logic 'relation logic'. I have no idea why not.
Our familiar sentence logic built up all sentences from atomic sentence letters. Predicate logic likewise builds up compound sentences from atomic sentences. But we have expanded our list of what counts as an atomic sentence. In addition to atomic sentence letters, we will include sentences such as 'Ba' and 'Lea'. Indeed, any one place predicate followed by one name, any two place predicate followed by two names, and so on, will now also count as an atomic sentence. We can use our expanded stock of atomic sentences to build up compound sentences with the help of the connectives, just as before.
How would you say, for example, 'Either Eve loves Adam or Adam is not blond.'? 'Lea v ~Ba'. Try 'Adam loves himself and if he is blond then he loves Eve too.': 'Laa & (Ba ⊃ Lae)'.
In summarizing this section, we say
In predicate logic, a capital letter without a following lowercase letter is (as in sentence logic) an atomic sentence. Predicate logic also includes predicates applied to names among its atomic sentences. A capital letter followed by one name is a One Place Predicate applied to one name. A capital letter followed by two names is a Two Place Predicate applied to two names, where the order of the names is important. Predicates with three or more places are used similarly.
In the following exercises, use this transcription guide:
Bx: x is blond
Cx: x is a cat
Lxy: x loves y
Txy: x is taller than y
1-1. Transcribe the following predicate logic sentences into English:
- Tce ⊃ Lce
- Lce v Lcc
- ~(Lce & Lca)
- Bc ≡ (Lce v Lcc)
1-2. Transcribe the following English sentences into sentences of predicate logic;
- Cid is a cat.
- Cid is taller than Adam.
- Either Cid is a cat or he is taller than Adam.
- If Cid is taller than Eve then he loves her.
- Cid loves Eve if he is taller than she is.
- Eve loves both Adam and Cid.
- Eve loves either Adam or Cid.
- Either Adam loves Eve or Eve loves Adam, but both love Cid.
- Only if Cid is a cat does Eve love him.
- Eve is taller than but does not love Cid.