8. PREDICATE LOGIC

Outline

 

James Fieser, UT Martin

updated 11/1/2023

 

The system of predicate logic emerged from the desire to construct a logically perfect language that, unlike natural languages such as English, is free of all ambiguity. Bertrand Russell, one of its main founders, describes the project here:

 

A language of that sort will be completely analytic, and will show at a glance the logical structure of the facts asserted or denied. The language which is set forth in Principia Mathematica is intended to be a language of that sort. It is a language which has only syntax and no vocabulary whatsoever. Barring the omission of a vocabulary I maintain that it is quite a nice language. It aims at being that sort of a language that, if you add a vocabulary, would be a logically perfect language. [The Philosophy of Logical Atomism, 1918]

 

Predicate logic thus aims to parallel the intricacies of natural language through a symbolic notation that reveals its underlying logic. To an extent, Aristotle's logic attempted to do the same thing, but its four forms of categorical statements were too limited to cover the full meaning of many statements. Russell addressed this limitation by creating a more complex system of notation that covers a wider range of predicates that we apply to subjects. The project that Russell started has developed into the system of logic that philosophers use today.

 

A. NON-RELATIONAL SUBJECT-PREDICATE STATEMENTS (a.k.a., monadic or one-place predicates)

 

1. Simple non-relational statements

Examples

Ms (Socrates is mortal)

Ac (Claire is an acrobat)

Ln (New York City is large)

Subject: for subjects that are individual constants (specific people, places, things), use lower case letters a-t

Predicate: for the predicate qualities, use capital letters A-Z

The predicate always appears before subject

Advantages: predicate logic allows for representing the more complex logical structures of statements, which could not be represented in propositional logic alone. For example, in propositional logic, the entire statement "Bob is a juggler" would be represented with the single term "B". With predicate logic, however, we represent this statement as "Jb", which reveals more information about the original statement.

 

2. Compound non-relational statements

Statements that use logical connectives: &, v, →, ~

Examples

Ac & Jb (Claire is an acrobat and Bob is a juggler)

Gb v Jb (Bob is either a geek or a jock)

Sj → Tj (if John is a student then John pays tuition)

Ab ↔ Sp (Beth will sing alto if and only if Pam will sing soprano)

~Vh (Hitler was not virtuous)

Advantages: again, the advantage of predicate logic is that it allows for revealing more information about the statement than would be possible in propositional logic. For example, in propositional logic, the statement "If Paul plays guitar then Quincy will play drums" would simply be "P → Q". With predicate logic, however, it would be "Gp  → Dq". So too with all of the more complex statements addressed below: there is a way in predicate logic to express the underlying logical relations symbolically.

 

B. RELATIONAL SUBJECT-PREDICATE STATEMENTS

 

Relational statements express some interaction between two parties, such as between you and another person (Bob loves Claire), you and your car (Joe dreams about his Ford truck), or your place of employment (Bob dreads his job). In these cases, there are is a subject (Bob), an object (Claire), and a predicate relating the two together (love). These are more complex than the simple non-relational statements above, which only had one thing involved, such as "Claire is an acrobat", where Claire is the subject.

 

1. Two-place and three-place relational predicates

Simple two-place relational predicates (predicate-subject-object)

Lbc (Bob loves Claire)

Abb (Bob admires himself)

Simple three-place relational predicates (predicate-subject-direct object-indirect object)

Gdbc (Donna gossiped about Bob to Claire)

Gcfb (Claire gave Fido to Bob)

Compound two-place relational predicates

Rbj → Rjb (if Bob respects Joe then Joe will respect Bob)

Compound three-place relational predicates

Gdbc & Scfb (Donna gossiped about Bob to Claire and Claire sold Fido to Bob)

 

2. Symmetry

Symmetrical relationship (relationship works in both directions)

Mbc → Mcb (if Bob is married to Claire, then Claire is married to Bob)

Note: this does not require a biconditional; the symmetrically is reflected in the reversal of the two subjects in the consequent of the conditional

Asymmetrical relationship (relationship does not work in both directions)

Fej → ~Fje (if Ed is the father of Joe, then Joe is not the father of Ed)

Nonsymmetrical relationship (relationship may or may not work in both directions)

Abc  → (Acb v ~Acb) (if Bob admires Claire then Claire may or may not admire Bob)

 

3. Transitivity

Transitive relationship (relationship is like hypothetical syllogism)

(Tje & Tea) → Tja (if Jill is taller than Ella, and Ella is taller than Agnus, then Jill is taller than Agnus)

Note: this does not require a hypothetical syllogism argument form; the relationship is reflected in the order of the three subjects

Intransitive relationship (relationship is not like hypothetical syllogism)

(Fej & Fja) → ~Fea (if Ed is the father of Joe, and Joe is the father of Al, then Ed is not the father of Al)

Nontransitive relationship (relationship may or may not be like hypothetical syllogism)

(Abc & Acj) → (Abj v ~Abj) (If Bob admires Claire, and Claire admires Joe, then Bob may or may not admire Joe)

 

C. QUANTIFIERS AND VARIABLES

 

1. Symbols used

Variables: use letters u-z

Different variables do not necessarily designate different objects

Choice of variables makes no difference to meaning

Two quantifiers

: universal quantifier (“x” means “for all x”)

Ǝ: existential quantifier (“Ǝx” means “for some x”)

 

2. Four quantification statement forms:

A: all S is P (e.g., all students are people)

x(Sx → Px)

For all x, if x is S then x is P

Warning: do not use a conjunction with A statements, since it will not mean the same thing. E.g. x(Sx & Px) means “everything is a student and a person”

E: no S is P (e.g, no student is a pelican)

x(Sx  → ~Px)

For all x, if x is S then it is not the case that x is P

Warning: again, do not use a conjunction with E statements, which, in this case x(Sx  & ~Px) means "everything is a student and not a pelican"

I: some S is P (e.g., some students are pilots)

Ǝx(Sx & Px)

For some x, x is S and x is P

O: some S is not P (e.g., some students are not partiers)

Ǝx(Sx & ~Px)

For some x, x is S and it is not the case that x is P

 

3. Examples of quantified non-relational statements

Frogs are green: x(Fx → Gx)

There is at least one green frog: Ǝx(Fx & Gx)

Green frogs exist: Ǝx(Fx & Gx)

Some frogs are not green: Ǝx(Fx & ~Gx)

Everything is a frog: x(Fx)

Something is a frog: Ǝx(Fx)

Not everything is a frog: ~x(Fx)

Nothing is a frog: x~(Fx)

Alternatively, ~Ǝx(Fx), as per a quantifier equivalence rule in the next chapter.

Everything is a green frog: x(Fx & Gx)

 

4. Examples of quantified relational statements

Bob admires nothing: x~(Abx)

Nothing admires Bob: x~(Axb)

There is something which both Bob and Claire admire: Ǝx(Abx & Acx)

If Bob admires himself, then he admires something: Abb → Ǝx(Abx)

A musician admires Bob: Ǝx(Mx & Axb)

Every musician admires Bob x(Mx → Axb)

Everything is a musician that admires Bob: x(Mx & Axb)

 

D. IDENTITY PREDICATES

 

1. Identity predicate: the symbol =, which means “identical to”

Mark Twain is Samuel Clemens

t=c

Translation: Twain is identical to Clemens

George Eliot is not Samuel Clemens

~e=c (alternatively e≠c)

Translation: it is not the case that Eliot is identical to Clemens

 

2. The only

Joe is the only guitarist in town

Gj & x(Gx → x=j)

Translation: Joe is a Guitarist, and, for all x, IF x is a Guitarist THEN x is identical to Joe

Alternative: Gj & x(~x=j  → ~Gx)

Translation: Joe is a Guitarist, and, for all x, IF it is not the case that x is identical to Joe THEN it is not the case that x is a Guitarist

Joe is the only one who finished

Fj & x(Fx → x=j)

Translation: Joe Finished and, for all x, IF x Finished THEN x is identical to Joe

 

3. Only

Only Joe survived

Sj & x(Sx  → x=j)

Translation: Joe Survived and, for all x, IF x Survived THEN x is identical to Joe

Only Mark Twain wrote Huckleberry Finn

Wth & x(Wxh → x=t)

Translation: Twain Wrote Huckleberry Finn, and, for all x, IF x Wrote Huckleberry Finn then x is identical to Twain

 

4. No . . . except

No pilots survived except Joe

Pj & Sj & x[(Px & Sx)x=j]

Translation: Joe is a Pilot and Joe Survied, and, for all x, IF x is a Pilot and x Survived, THEN x is Joe

No people except Joe love Michelle

Pj & Ljm & x[(Px & Lxm → x=j]

Translation: Joe is a person, and Joe Loves Michelle, and, for all x, IF x is a person and x Loves Michelle, THEN x is Joe

 

5. All . . . except

All pilots survived except Joe

Pj & ~Sj & x[(Px & ~x=j) → Sx]

Translation: Joe is a Pilot, and it is not the case that Joe Survived, and for all x, IF x is a Pilot and x is not Joe, THEN x Survived

All people except Joe love Michelle:

Pj & ~Ljm & x[(Px & ~x=j) → Lxm]

Translation: Joe is a Person, and it is not the case that Joe Loves Michelle, and for all x, IF x is a Person and it is not the case that x is Joe, THEN x Loves Michelle

 

6. Superlatives

Joe is the greatest trombonist

Tj & x[(Tx & ~x=j) → Gjx]

Translation: Joe is a Trombonist, and for all x, IF x is a Trombonist and it is not the case that x is Joe, THEN joe is Greater than x

 

7. At most

At most one thing exists

x y(x=y)

Translation: for all x and for all y, x is identical to y (i.e., for any two things in the cosmos, they are the same thing)

At most one student failed

(Sx & Sy) & x y[(Fx & Fy) → x=y]

Translation: x is a Student and y is a Student, and, for all x and for all y, IF x Failed and y Failed, THEN x is identical to y (i.e., x and y are the same object)

 

8. At least

At least two things exist

Ǝx Ǝy(~x=y)

Translation: there exist some x and there exists some y such that it is not the case that x is identical to y

There are at least two giraffes in the zoo

Ǝx Ǝy[(Gx & Gy) & ~x=y]

Translation: there exist some x and there exists some y such that, x is a Giraffe and y is a Giraffe, and it is not the case that x is identical to y (i.e., x and y are not the same object)

Comment: while we might normally assume that x and y are different objects, in logic the possibility is open that they might be the same. Recall, for example, the statement "Mark Twain is Samuel Clemens", symbolized as "t=c", where the two are the same. That is why, in this problem about the two giraffes, we need to indicate "~x=y".

 

9. Exactly

Exactly one thing exists

Ǝx y(x=y)

Translation: there exists some x, and for all y, x is identical to y (i.e., we begin asserting that some thing x exists, and we continue saying that, for all things y, they are that one thing)

There is exactly one giraffe in the zoo

Ǝx Ǝy[(Gx & Gy) & x=y]

Translation: there exist some x and there exists some y such that x and y are both giraffes in the zoo, and x is identical to y (i.e., x and y are the same object)

 

10. Relational identity predicates

If Mark Twain is Samuel Clemens, then Samuel Clemens wrote Huckleberry Finn

t=c → Wch

Translation: IF Twain is identical to Clemens, then Clemens wrote Huckleberry Finn

No American author is better than Mark Twain

x(Ax  → ~Bxt)

Translation: for all x, IF x is an author, THEN it is not the case that x is Better than Twain

 

11. Definite Description

Comment: A definite description is a description we give of somebody that uniquely defines that person, and separates him from anything else in the cosmos at any period in time.

Homer is the author of the Iliad

Ǝx[Axi & y(Ayi → y=x) & x=h]

Translation: there exists some x such that, x is the Author of the Iliad and, for all y, IF y is the Author of the Iliad, THEN y=x (i.e., y is the same person as x) AND x is Homer (i.e., we begin by narrowing down the author of the Iliad to a single person, then identifying him as Homer)

The inventor of the toothbrush was British

Ǝx[Ixt & y(Iyt → y=x) & Bx]

Translation: there exists some x such that x is the Inventor of the toothbrush and, for all y, IF y is the inventor of the toothbrush THEN x and y are the same, and x is British (i.e., we begin by narrowing down the inventor of the toothbrush to a single person, then identifying him as being British)

 

12. De Re and De Dicto Interpretations

This involves a contrast between two ways of interpreting ambiguous sentences. "De re" interpretations relate me to a thing, whereas "de dicto" interpretations relate me to a statement (where "de re" is Latin for "of the thing itself", and "de dicto" for "of the word")

I want a sailboat

De re interpretation: (x)(x is a sailboat & I want to own x);

De dicto interpretation: "I want it that (x)(x is a sailboat & I own x)".