9-4. DEFINITE DESCRIPTIONS
(1) The one who loves Eve is blond.
We need a predicate logic sentence which is true when (1) is true and false when it is false. If there is exactly one person who loves Eve and this person is blond, (1) is true. If this person is not blond, (1) clearly is false. But what should we say about (1) if no one loves Eve, or more than one do?
If no one, or more than one love Eve, we surely can't count (1) as true. If we insist that every sentence is true or false, and since (1) can't be true if none or more than one love Eve, we will have to count (1) as false under these conditions. Thinking about (1) in this way results in transcribing it as
(la) (3x!)(Lxe & Bx).
which is true if exactly one person loves Eve and is blond, and is false if such a person exists and is not blond or if there are none or more than one who love Eve.
From a perspective wider than predicate logic with identity we do not have to take this stand. We could, instead, suggest that there being exactly one person who loves Eve provides a precondition for, or a Presupposition of, the claim that the one who loves Eve is blond. This means that the condition that there is exactly one person who loves Eve must hold for (1) to be either true or false. If the presupposition holds-if there is exactly one person who loves Eve-then (1) is true if this unique person is blond and false if he or she is not blond. If the presupposition fails-if there is none or more than one who love Eve-then we say that (1) is neither true
151 Idmtity, Functions, and Dejinite Descriptions 9-4. Dejinite Desm'pth 155
nor false. One can design more complex systems of logic in which to formalize this idea, but predicate logic with identity does not have these resources. Hence, (la) is the best transcription we can provide.
Grammatically, 'the one who loves Eve' functions as a term. It is supposed to refer to something, and we use the expression in a sentence by attributing some property or relation to the thing purportedly referred to. We can mirror this idea in predicate logic by introducing a new kind of expression, (The u)P(u), which, when there is a unique u which is P, refers to that object. We would then like to use (The u)P(u) like a name or other constant term in combination with predicates. Thus we would transcribe
(1) as (lb) B(The x)Lxe.
Read this as the predicate 'B' applied to the "term" '(The x)Lxe'. 'The one who loves Eve' and '(The x)Lxe' are called Dejinite Descriptions, respectively in English and in logic. Traditionally, the definite description formBut what should we say in predicate logic about the transcription of (2)? We can see (2) as the negation of (1) in two very different ways. We can see (2) as the definite description '(The x)Lxe applied to the negated predicate '-B' in which case we have (2a) -B(The x)Lxe, rewritten as (3x!)(Lxe & -Bx). When we think of (1) and (2) this way, we say that the definite description has Primary Occurrence or Wide Scope. Or we can see (2) as the negation of the whole transcribed sentence: (2b) -[B(The x)Lxe], rewritten as -(3x!)(Lxe & Bx). Thinking of (1) and (2) in this second way, we say that the definite description has Secondary Occurrence or Narrow Scope. When transcribing an English sentence with a definite description into logic, you will always have to make a choice between treating the definite description as having ing operator, (The u), is written with an.upside-down Greek letter iota, primary or secondary occurrence. like this:
Here are some examples of definite descriptions transcribed into predicate logic: a) The present king of France: (The x)Kx. b) The blond son of Eve: (The x)(Bx & Sxe). c) The one who loves all who love themselves:
(The x)(Vy)(Lyy > Lxy).
But we can't treat (The x)P(x) like an ordinary term, because sometimes such "terms" don't refer. Consequently, we need a rewriting rule, just as we did for subscripted predicates and '(gx!)', to show that expressions like (lb) should be rewritten as (la):
Rule for rewriting Definite Descriptions Using '(The u)': Q[(The u)P(u)] is shorthand for (3u!)[P(u) & Q(u)], where P(u) and Q(u) are open formulas with u the only free variable.
This treatment of definite descriptions works very smoothly, given the limitations of predicate logic. It does, however, introduce an oddity about the negations of sentences which use a definite description. How should we understand
(2) The one who loves Eve is not blond.
Anyone who holds a presupposition account will have no trouble with (2): They will say that if the presupposition holds, so that there is just one person who loves Eve, then (2) is true if the person is not blond and false if he or she is blond. If the presupposition fails, then (2), just as (I), is neither true nor false.
EXERCISES Transcription Guide a: Adam Dx: x is dark-eyed e: Eve Fxy: x is a father of y c: Cain Sxy: xisasonofy Bx: x is blond Cxy: x is more clever than y Lxy: x loves y
9-12. Transcribe the following. Expressions of the form (The u) and (gu!) should not appear in your final answers. The son of Eve is blond. The son of Eve is more clever than Adam. Adam is the father of Cain. Adam loves the son of Eve. Adam loves his son. Cain loves the blond. The paternal grandfather of Adam is dark-eyed. The son of Eve is the son of Adam. The blond is more clever than the dark-eyed one. The most clever son of Adam is the father of Eve. The son of the father of Eve is more clever than the father of the son of Adam.
9-13. Transcribe the negations of the sentences of exercise 9-12,, once with the definite description having primary occurrence and once with secondary occurrence, indicating which transcription is which. Comment on how you think the notions of primary and secondary occurrence should work when a sentence has two definite descriptions.
CHAPTER SUMMARY EXERCISES This chapter has introduced the following terms and ideas. Summarize them briefly. Identity Referent Co-Referential @PI) Self-Identity Extensional Extensional Semantics Rule =I for Derivations Rule = E for Derivations Rule = for Trees Rules # for Trees Reflexive Relation Symmetric Relation Transitive Relation Equivalence Relation Function One Place Function Two and Three Place Functions Arguments of a Function Function Symbols Term Constant, or Constant Term Rules for Function Symbols in Derivations Rules for Function Symbols in Trees Presupposition Definite Description Rewrite Rule for Definite Descriptions Primary Occurrence (Wide Scope) of a Definite Description Secondary Occurrence (Narrow Scope) of a Definite Description