6. COMPLEX PREDICATES
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1 318 Hardegree, Symbolic Logic (1) x... y...rxy (1a) x(sx & y(py Rxy)) (1b) x(px & y(sy Rxy)) (2) x... y...ryx (2a) x(sx & y(py Ryx)) (2b) x(px & y(sy Ryx)) (3) x... y...rxy (3a) x(sx y(py & Rxy)) (3b) x(px y(sy & Rxy)) (4) x... y...ryx (4a) x(sx y(py & Ryx)) (4b) x(px y(sy & Ryx)) 6. COMPLEX PREDICATES In order to further understand the translations that appear in the previous sections, and in order to be prepared for more complex translations still, we now examine the notion of complex predicate. Roughly, complex predicates stand to simple (ordinary) predicates as complex (molecular) formulas stand to simple (atomic) formulas. Like ordinary predicates, complex predicates have places; there are one-place, two-place, etc., complex predicates. However, we are going to concentrate exclusively on oneplace complex predicates. The notion of a complex one-place predicate depends on the notion of a free occurrence of a variable. This is discussed in detail in an appendix. Briefly, an occurrence of a variable in a formula is bound if it falls inside the scope of a quantifier governing that variable; otherwise, the occurrence is free. Examples (1) Fx the one and only occurrence of x is free. (2) x(fx Gx) all three occurrences of x are bound by x. (3) xrxy every occurrence of x is bound; the one and only occurrence of y is free. Next, to say that a variable (say, x ) is free in a formula F is to say that at least one occurrence of x is free in F; on the other hand, to say that x is bound in F is to say that no occurrence of x is free in F. For example, in the following formulas, x is free, but y is bound.
2 Chapter 7: Translations in Polyadic Predicate Logic 319 (f1) yrxy (f2) yrxy (f3) yryx (f4) yryx Any formula with exactly one free variable (perhaps with many occurrences) may be thought of as a complex one-place predicate. To see how this works, let us translate formulas (1)-(4) into nearly colloquial English. (e1) x (he/she) respects everyone (e2) x (he/she) respects someone (e3) x (he/she) is respected by everyone (e4) x (he/she) is respected by someone Now, if we say of someone that he(she) respects everyone, then we are attributing a complex predicate to that person. We can abbreviate this complex predicate Ax, which stands for x respects everyone. Similarly with all the other formulas above; each one corresponds to a complex predicate, which can be abbreviated by a single letter. These abbreviations may be summarized by the following schemes. Ax :: yrxy Bx :: yrxy Cx :: yryx Dx :: yryx Here, :: basically means...is short for.... Now, complex predicates can be used in sentences just like ordinary predicates. For example, we can say the following: some Freshman is A every Freshman is B no Freshman is C some Freshman is not D Recalling what A, B, C, and D are short for, these are read colloquially as follows. some Freshman respects everyone every Freshman respects someone or other no Freshman is respected by everyone some Freshman is not respected by someone (or other) These have the following as overall symbolizations. x(fx & Ax) x(fx Bx) ~ x(fx & Cx) x(fx & ~Dx)
3 320 Hardegree, Symbolic Logic But Ax, Bx, Cx, and Dx are short for more complex formulas, which when substituted yield the following formulas. x(fx & yrxy) x(fx yrxy) ~ x(fx & yryx) x(fx & ~ yryx) We can also make the following claims. every A is B every A is C every A is D Given what A, B, C, and D are short for, these read colloquially as follows. every one who respects everyone respects someone every one who respects everyone is respected by everyone every one who respects everyone is respected by someone The overall symbolizations of these sentences are given as follows. x(ax Bx) x(ax Cx) x(dx Dx) But Ax, Bx, Cx, and Dx are short for more complex formulas, which when substituted yield the following formulas. x( yrxy yrxy) x( yrxy yryx) x( yrxy yryx) Let's now consider somewhat more complicated complex predicates, given as follows. Ax: x respects every professor Bx: x is respected by every student Cx: x respects at least one professor Dx: x is respected by at least one student Given the symbolizations of the formulas to the right, we have the following abbreviations. Ax :: y(py Rxy) Bx :: y(sy Ryx) Cx :: y(py & Rxy) Dx :: y(sy & Ryx) We can combine these complex predicates with simple predicates or with each other. The following are examples.
4 Chapter 7: Translations in Polyadic Predicate Logic 321 (1) some S is A (2) some P is B (3) every S is C (4) every P is D The colloquial readings are: (r1) there is a student who respects every professor (r2) there is a professor who is respected by every student (r3) every student respects at least one professor (some professor or other) (r4) every professor is respected by at least one student (some student or other) And the overall symbolizations are given as follows. (o1) x(sx & Ax) (o2) x(px & Bx) (o3) x(sx Cx) (o4) x(px Dx) But Ax, Bx, Cx, Dx are short for more complex formulas, which when substituted yield the following formulas. (f1) x(sx & y(py Rxy)) (f2) x(px & y(sy Ryx)) (f3) x(sx y(py & Rxy)) (f4) x(px y(sy & Ryx)) These correspond to the formulas obtained by the technique of quantifier specification, presented in the previous section. The advantage of understanding complex predicates is that it allows us to combine the complex predicates into the same formula. The following are examples. Ax: x respects every professor Bx: x is respected by every student Cx: x is respected by at least one professor no A is B every B is C These may be read colloquially as no one who respects every professor is respected by every student everyone who is respected by every student is respected by at least one professor The overall symbolizations are, respectively,
5 322 Hardegree, Symbolic Logic ~ x(ax & Bx) x(bx Cx) but Ax, Bx, and Cx stand for more complex formulas, which when substituted yield the following formulas. ~ x( y(py Rxy) & y(sy Ryx)) x( y(sy Ryx) y(py & Ryx)) 7. THREE-PLACE PREDICATES So far, we have concentrated on two-place predicates. In the present section, we look at examples that involve quantification over formulas based on three-place predicates. As mentioned in the previous chapter, there are numerous three place predicate expressions in English. The most common, perhaps, are constructed from verbs that take a subject, a direct object, and an indirect object. For example, in the sentence Kay loaned her car to Jay may be grammatically analyzed thus: subject: verb: direct object: indirect object: Kay loaned her car Jay The remaining word, to, marks Jay as the indirect object of the verb. In general, prepositions such as to and from, as well as others, are used to mark indirect objects. The following sentence uses from to mark the indirect object. Jay borrowed Kay's car (from Kay) Letting c name the particular individual car in question, the above sentences can be symbolized as follows. Lkcj Bjck The convention is to write subject first, direct object second, and indirect object last. As usual, variables (pronouns) may replace one or more of the constants (proper nouns) in above formulas, and as usual, the resulting formulas can be quantified, either universally or existentially. The following are examples.
6 Chapter 7: Translations in Polyadic Predicate Logic 323 Kay loaned her car to him(her) Kay loaned her car to someone Kay loaned her car to everyone Jay borrowed it from Kay Jay borrowed something from Kay Jay borrowed everything from Kay Lkcx xlkcx xlkcx Bjxk xbjxk xbjxk As before, we can also further specify the quantifiers. Rather than saying someone or everyone, we can say some student or every student ; rather than saying something or everything, we can say some car or every car. Quantifier specification works the same as before. Kay loaned her car to some student Kay loaned her car to every student Jay borrowed some car from Kay Jay borrowed every car from Kay x(sx & Lkcx) x(sx Lkcx) x(cx & Bjxk) x(cx Bjxk) These are examples of single-quantification; we can quantify over every place in a predicate, so in the predicates we are considering, we can quantify over three places. Two quantifiers first; let's change our example slightly. First note the following: For example, x rents y to z z rents y from x Avis rents this car to Jay iff Jay rents this car from Avis. Letting Rxyz stand for x rents y to z, consider the following. Example 1 Example 2 every student has rented a car from Avis x(sx y(cy & Rayx)) for any x, if x is a student, then there is a y such that, y is a car and Avis has rented y to x there is at least one car that Avis has rented to every student x(cx & y(sy Raxy))
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