Quantifiers, Predicates, and Names

1. Quantifiers, Predicates, and Names Kareem Khalifa Department of Philosophy Middlebury College Universals, by Dena Shottenkirk

2. Quick thing about typing hypothetical derivations

3. Overview • What is quantification/predicate logic? • Why this matters • Singular Propositions • Universal and Existential Propositions • Exercises

4. What is predicate logic? • To this point, we’ve been studying propositional logic. • This means that our smallest units of logical analysis are whole sentences. • Predicate logic studies the relations between names, predicates, and quantifiers. • Names: Refer to individual people, places and things. Ex. ‘Khalifa,’‘Middlebury,’‘This computer’ • Predicates: Refer to properties of and relations between people, places, and things. Ex. ‘…is a professor.’‘…teaches at…,’ etc. • Quantifiers: Logical operators that reflect relations between subjects and predicates. Ex. “All” and “Some”

5. Example • Khalifa is a professor. Khalifa works at Middlebury. Everybody who is a professor and works at Middlebury teaches at Middlebury. So Khalifa teaches at Middlebury. Predicate Quantifier Name

6. Why does predicate logic matter? • There are many inferences whose validity cannot be captured by propositional logic. • All men are mortal. Socrates is a man. So Socrates is mortal. • A, S ├ M is clearly invalid! • Predicate logic gives us a way of tying together names, predicates, and quantifiers so that we can discern the validity of these inferences. • In this example, it tells us that there are special ways of using “All” that will make this inference valid. More on this later…

7. Singular Propositions • The most basic proposition in predicate logic is one in which something is predicated of an individual name(s). • Examples. Socrates is a man. Khalifa is a professor. Khalifa teaches at Middlebury. • Names are represented as lowercase letters a through t. • s = Socrates; k = Khalifa; m = Middlebury • Predicates are represented with capital letters. • Some predicates are one-place predicates: M = is a man; P = is a professor. • Others are n-place predicates: T = teaches at.

8. Proper notation for singular propositions • PREDICATE, then NAME(S) = PROPOSITION. • Socrates is a man: Ms • Khalifa is a professor: Pk • Khalifa teaches at Middlebury: Tkm • With n-place predicates, order of names matters. • Tmk means “Middlebury teaches at Khalifa.” This is nonsense! • Furthermore, it’s ungrammatical to write a name followed by a predicate. • kP Professor, Khalifa is.

9. Since these are propositions… • You can apply all of the logical connectives from propositional logic to them. • Khalifa is a professor andteaches at Middlebury. • Pk & Tkm • Either Khalifa is a professor or he is a god. • Pk v Gk • Etc.

10. Propositional functions • We use the letters u through z to denote variables for names. • When we have a predicate followed by variables, we have a propositional function. • Mx, Px, Txy • The best way to understand variables is as equivalent to the English word “thing.” • Without quantifiers, these are notgrammatical. • Thing is mortal, Thing is professor, Thing teaches at other thing.

11. Quantifiers • Two kinds: • Universal: represented either as (x) or as x: “For all x…” • Existential: x: “There is at least one x such that…” • These are also not propositions by themselves. • However, quantifiers plus propositional functions are propositions. • xBx= Everything is beautiful. • xPx = Someone is a professor. • xRxk = Somebody respects Khalifa.

12. How to interpret “is/are” in predicate logic:  • We make many universal statements using “is” and “are” () • Every student is happy. • All dogs are mammals. • We represent “is” and “are” using  • Every student is happy = x(Sx  Hx) • All dogs are mammals = x(Dx  Mx) • Literally, this says • For all x, if x is a student, then x is happy. • For all x, if x is a dog, then x is a mammal.

13. How to interpret “is/are” in predicate logic:  • We also use “is” and “are” with existential quantifiers () • At least one student is happy. • Some dogs are beagles. • Here “is” and “are” are represented by “&” • At least one student is happy = x(Sx&Hx) • Some dogs are beagles = x(Dx&Bx) • Literally: • There is at least one x such that x is a student and x is happy. • There is at least one x such that x is a dog and x is happy.

14. Some important English expressions formalized • Nothing is an F = Everything is a non-F. • ~xFx  x~Fx • Something is a non-F = Not everything is F. • x~Fx  ~xFx • No F’s are G’s = Every F is a non-G. • ~x(Fx&Gx)  x(Fx~Gx) • Some F’s are non-G’s = Not all F’s are G’s. • x(Fx&~Gx)  ~x(Fx Gx)

15. Something that doesn’t track with ordinary language • x(FxGx) • There is at least one x such that if x is F, then x is G. • Example: • There exists a thing such that if it is an angel, then it is beautiful. • This might be true of nearly anything. • If my hand is an angel, then it is beautiful. • If beer is an angel, then it is beautiful. • Etc.

16. Similarly… • x(Fx&Gx) says something very different than x(Fx Gx) • The first statement says that everything is F&G. The second only says that everything that’s already an F is a G. • Compare: • Everything’s funky and good. • Everything funky is good.

17. n-place predicates • As we’ve already seen, there are predicates that involve multiple names. • Khalifa teaches at Middlebury = Tkm • Quantifiers also apply to these, e.g. • xTxm = Someone teaches at Middlebury. • xTkx = Khalifa teaches somewhere • x yTxy = Someone teaches somewhere • xyTxy = Everyone teaches somewhere

18. A few nuances with n-place predicates • If you use one variable for an n-place predicate, you often get a reflexive relationship, e.g. • xLxx = Someone loves him/herself. • Using two variables is compatible with, but does not entail, a reflexive relationship, e.g. • xyLxy = Someone loves someone, but x and ycould refer to the same person.

19. Be mindful of the scope of the quantifier • There’s a big difference between the following: • xyLxy & xy~Lxy • Imagine four people (a-d); a is to the left of b, and c is not to the left of d • xy(Lxy & ~Lxy) • Imagine two people; a is both to the left and not to the left of b • So the second sentence is a contradiction.

20. Mixing existential and universal quantifiers • Always keep quantifiers as close to the variables they govern. • Compare the following: • x(yLxyHx) = All lovers are happy. • For all x, if x loves some y, then x is happy. • xy(LxyHx)  Everyone has something that if they loved it, it would make them happy. • For all x, there exists some y, such that if x loves y, then x is happy.

21. Example of the difference • x(yLxyHx). • Jack loves Jill, so Jack is happy. • xy(LxyHx) • Jack doesn’t love Jill, but if he did, he would be happy.

22. With n-place predicates, order of quantifiers matters • Consider two expressions: • xyLxy = There’s some x such that, for all y, x loathes y • yxLxy = For all y, there’s some x such that x loathes y • The first requires a single person that loathes everything. • The second requires that everything is loathed by at least one person, but this need not be the same person.

23. Exercise 6.1.1 • Beth is fortunate. So is Carl. Therefore both Carl and Beth are fortunate. • Fb, Fc |- Fc & Fb

24. 6.1.7 • Everything good is praiseworthy. Healing is good. Therefore healing is praiseworthy. • x(Gx→Px), Gh ├ Ph

25. 6.1.14 • Not all acts are just. Therefore there are acts that are not just. • ~x(Ax→Jx) ├ x(Ax & ~Jx)

26. 6.1.21 • If Beth loves and respects Al, then Al is fortunate. But then Beth does not love Al, since Al is not fortunate, though Beth respects him. • (Lba & Rba)→Fa, ~Fa & Rba ├ ~Lba

27. 6.1.28 • Al loves anything that Beth loves. Beth loves Al. Therefore Al loves something. • x(Lbx → Lax), Lba ├ xLax

28. 6.1.35 • Al respects a thing if and only if it does not respect itself. Ergo, happiness is maximized. • x(Rax ↔ ~Rxx) ├ H