Quantifiers, Predicates, and Names

Kareem Khalifa

Department of Philosophy

Middlebury College

Universals, by Dena Shottenkirk

Quick thing about typing hypothetical derivations

Overview

• What is quantification/predicate logic?

• Why this matters

• Singular Propositions

• Universal and Existential Propositions

• Exercises

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”

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.

Name Predicate Quantifier

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…

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.

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.

Since these are propositions…

• You can apply all of the logical connectives from propositional logic to them.

• Khalifa is a professor and teaches at Middlebury.– Pk & Tkm

• Either Khalifa is a professor or he is a god.– Pk v Gk

• Etc.

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 not grammatical.– Thing is mortal, Thing is professor, Thing teaches at

other thing.

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.

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.

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.

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)

Something that doesn’t track with ordinary language

x(FxGx)– 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.

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.

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 xyTxy = Everyone teaches somewhere

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. xyLxy = Someone loves someone, but x

and y could refer to the same person.

Be mindful of the scope of the quantifier

• There’s a big difference between the following: xyLxy & xy~Lxy

• Imagine four people (a-d); a is to the left of b, and c is not to the left of d

xy(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.

Mixing existential and universal quantifiers

• Always keep quantifiers as close to the variables they govern.

• Compare the following: x(yLxyHx) = All lovers are happy.

• For all x, if x loves some y, then x is happy.

xy(LxyHx) 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.

Example of the difference

x(yLxyHx).– Jack loves Jill, so Jack is

happy.

xy(LxyHx)– Jack doesn’t love Jill, but if

he did, he would be happy.

With n-place predicates, order of quantifiers matters

• Consider two expressions: xyLxy = There’s some x such that, for all y, x

loathes y yxLxy = 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.

Exercise 6.1.1

• Beth is fortunate. So is Carl. Therefore both Carl and Beth are fortunate.

• Fb, Fc |- Fc & Fb

6.1.7

• Everything good is praiseworthy. Healing is good. Therefore healing is praiseworthy.

x(Gx→Px), Gh ├ Ph

6.1.14

• Not all acts are just. Therefore there are acts that are not just.

• ~x(Ax→Jx) ├ x(Ax & ~Jx)

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

6.1.28

• Al loves anything that Beth loves. Beth loves Al. Therefore Al loves something.

x(Lbx → Lax), Lba ├ xLax

6.1.35

• Al respects a thing if and only if it does not respect itself. Ergo, happiness is maximized.

x(Rax ↔ ~Rxx) ├ H