8. PREDICATE LOGIC
Outline
James Fieser, UT Martin
updated 11/13/2025
The system of predicate logic emerged from the desire to construct a logically perfect language, which, unlike ordinary language, is free from ambiguity. Bertrand Russell, one of its founders, describes this goal:
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 seeks to represent the underlying logical form of natural language through symbolic notation. Aristotle’s categorical logic attempted something similar, but its four statement forms were too limited to express the full range of ordinary propositions. Gottlob Frege overcame this in his Begriffsschrift (1879), introducing quantifiers (“all,” “some”) and variables (x, y, z) that allowed analysis of statements like “Everyone loves someone.” Russell and Whitehead later built on Frege’s system in Principia Mathematica (1910–13). Predicate logic, then, improves on categorical logic by (1) separating quantifiers from predicates to clarify scope and meaning, (2) allowing relations between multiple subjects, not just single categories, and (3) representing individuals rather than whole classes. Today, predicate logic underlies modern analytic philosophy, mathematics, linguistics, and computer science. In this section, we will study its notation and learn to translate English sentences into symbolic form using universal and existential quantifiers.
A. NON-RELATIONAL SUBJECT-PREDICATE STATEMENTS (a.k.a., monadic or one-place predicates)
In predicate logic, we break statements into two parts—a subject (what we’re talking about) and a predicate (what is said about it). This lets us represent the internal structure of ordinary statements more precisely. For instance, instead of treating “Socrates is mortal” as a single unit, we can display its logical form as Ms, showing both the predicate is mortal (M) and the subject Socrates (s).
1. Simple non-relational statements: these are the simplest type of subject-predicate statements since they involve only a single subject and predicate, such as “Socrates is mortal”.
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: these are statements that still involve only single subjects and predicates but connect two or more of these statements with logical operators such as “and,” “or,” “if…then,” or “not,”, such as in “Socrates is Mortal and Zeus is immortal.”
Statements that use logical connectives: &, v, →, ~
Examples
Ms & Iz (Socrates is Mortal and Zeus is immortal)
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 an interaction or connection between two or more individuals, 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 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 involved only one thing, such as “Claire is an acrobat,” where Claire is simply the subject of a one-place predicate. Relational statements, by contrast, introduce relations that can connect two, three, or even more subjects.
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. Additional-place relational predicates (Predicate Arity)
Relational predicates vary by the number of terms they require, where each additional variable expands relational complexity:
One-place (monadic): Mx (x is mortal)
Two-place (dyadic): Lxy (x loves y)
Three-place (triadic): Gxyz (x gave y to z)
Four-place (tetradic): Sxyzt (x sold y to z for t -- e.g. for ten dollars)
Five-place (pentadic): Sxyztu (x sold y to z for t in u -- e.g. for $10 in Ukraine)
3. 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)
4. 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. EXISTENTIAL AND UNIVERSAL QUANTIFIERS
Quantifiers are terms that tell us how many things there are, and in predicate logic we use two quantifiers: a universal one of the form “for all x”, and an existential one of the form “there exists some x”. Quantifiers allow us to make broad claims, such as “everything is finite” or “something is a frog”, without having to name any specific object.
1. Quantification basics
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 “there exists some x”)
Simple quantification statements
∀x (Fx): for all x, x is F
Example: “Everything is finite” (for all x, x is fine)
Ǝx (Fx): there exists some x such that x is F
Example: “some frog exists” (there exists some x such that x is a frog)
Note: The convention of using letters like “F” for predicates comes from Frege, who treated predicates as mathematical functions and followed the mathematical tradition of using the Greek letter φ (phi) to stand for an arbitrary function. Early analytic logicians such Russell and Whitehead later replaced φ with the Roman letter F, partly because it was easier to print and partly because “F” could be read as standing for “function.” This convention has remained standard in modern predicatelogic textbooks.
Historical note on existential quantifiers: In the Critique of Pure Reason, Kant famously argued that existence is not a real predicate. For him, “exists” is still a predicate of a thing, but not a normal one, because saying something exists does not add any descriptive content to the concept of that thing. Rather, existence simply “posits” the object as actual, without telling us anything new about its qualities. Although Kant never developed a formal system of quantification, his idea anticipates the modern existential quantifier pioneered by Frege. Frege argued that existence is not a predicate of objects at all. For, it makes no sense to treat “exists” as a property of an individual in the way that “is red” or “is tall” are properties. Modern logicians follow Frege on this point and do not treat existence as a predicate in formal logic. Frege also maintained that existential claims in quantification logic are really about concepts, meaning that a statement like Ǝx(Fx) says the concept F has at least one instance. Accordingly, in this view, existence is a second-level notion that applies to concepts, not to individual objects. Most contemporary philosophers, however, do not accept this second-level interpretation and instead take the existential quantifier to be about objects within the domain of things we are talking about, whether those things are real, abstract, or even merely fictional. They read Ǝx(Fx) as saying simply that there is some object satisfying F.
2. Quantification and Aristotle’s Four statement forms:
One of the major achievements of quantification logic was its ability to represent Aristotle’s four categorical forms (A, E, I, and O statements) in a more precise and versatile way. By introducing quantifiers (“all,” “some”) and variables (x, y, z), predicate logic could express not only the traditional categorical relations but also a much wider range of statements about individuals and their properties. This advancement made Aristotle’s system permanently obsolete, since every categorical form can now be captured and expanded upon within a single unified logical system.
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”. Stated more generally, here are the two contrasting statements:
Correct: ∀x(Sx → Px): For all x, if x is S then x is P (i.e., “Everything that is an S is also a P.”)
Incorrect: ∀x(Sx & Px): For all x, x is S and x is P (i.e., “Everything is both an S and a P.”)
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". Again, stated more generally, here are the two contrasting statements:
Correct: ∀x(Sx → ~Px): “For all x, if x is an S, then x is not a P” (i.e., No S are P)
Incorrect: ∀x(Sx & ~Px): “For all x, x is an S and x is not a P” (i.e., everything is both an S and not a P)
I: some S is P (e.g., some students are pilots)
Ǝx(Sx & Px)
There exists 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)
There exists some x, x is S and it is not the case that x is P
3. Existential Import
When learning to represent A, E, I, and O statements in predicate logic, it’s easy to confuse which take the form of conjunctions (&) and which take the form of conditionals (→). The distinction rests on existential import. As discussed in the chapter on categorical logic, particular statements (I, O) have existential import, while universal statements (A, E) do not. That is, particulars assume that something exists, whereas universals make no such assumption. This explains why A and E statements use conditionals, while I and O statements use conjunctions. For example, let U mean “is a unicorn” and H mean “has a single horn”:
A: ∀x(Ux → Hx): For all x, if x is a unicorn, then x has a single horn.
I: Ǝx(Ux & Hx): There exists some x such that x is a unicorn and x has a single horn.
In the real world, unicorns don’t exist. Still, we understand that IF something were a unicorn, THEN it would have a single horn. But in a fantasy world where a unicorn does exist, there is an x that both is a unicorn AND has a single horn. In short: universals (A, E), which lack existential import, are best expressed as “if–then” statements, while particulars (I, O), which presuppose existence, are best expressed as “and” statements. It was precisely the issue of existential import that first led Frege and Russel to construe A, E, I and O propositions this way within predicate logic.
4. Examples of quantified non-relational statements
Once we introduce quantifiers, we can express simple subject–predicate statements with much greater precision, even when no relational predicates are involved.
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)
5. Examples of quantified relational statements
Quantifiers also apply to relational predicates, allowing us to express statements involving two-place relations such as “admires,” “loves,” or “is taller than,” in a clear and systematic way.
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
Identity predicates allow us to state when two names refer to the very same object, and they give us a precise way to express uniqueness, exclusivity, and superlatives in predicate logic. By combining the identity symbol (=) with quantifiers, we can translate ordinary sentences about “the only,” “no one except,” or “exactly one” into precise logical form.
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
“The present King of France is Bald” (Bertrand Russell)
Ǝx[Kx & ∀y(Ky → y=x) & Bx]
Translation: there exists some x such that x is king and, for all y, if y is the king of France, then y=x (i.e., y is the same person as x), AND x is bald.
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
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: (Ǝx)(x is a sailboat & I want to own x);
De dicto interpretation: "I want it that (Ǝx)(x is a sailboat & I own x)".
E. EXAMPLES OF PREDICATE LOGIC STATEMENTS USED IN PHILOSOPHY
Predicate logic is used throughout philosophy to express complex ideas with clarity and precision. The following examples show how quantifiers and predicates can capture statements from ethics, epistemology, metaphysics, and language in a fully formal way.
1. “There is exactly one God” (philosophy of religion)
Ǝx [Gx & ∀y (Gy → y = x)]
Translation: There exists some x such that x is God, and for all y, if y is God, then y = x. (Classic uniqueness form, like Russell’s “King of France.”)
2. “Nothing exists that is both square and round”
~Ǝx (Sx & Rx)
Translation: It is not the case that there exists an x such that x is square and x is round. (Simple negated existential; good for impossible combinations.)
3. “If an action is morally right, then it ought to be done” (ethics)
∀x (Rx → Ox)
Translation: For all actions x, if x is right, then x ought to be done.
4. “Some actions are wrong but nevertheless performed” (ethics)
Ǝx (Wx & Px)
Translation: There exists some x such that x is wrong and x is performed.
5. “If S knows that p, then p is true” (truth is a necessary condition of knowledge)
∀s ∀p(Ksp → Tp)
Translation: For all subjects s and all propositions p, if s knows p, then p is true.
6. “Every effect has a cause” (causal principle)
∀x(Ex → ƎyCyx)
Translation: For all x, if x is an effect, then there exists some y such that y causes x. (Illustrates a universal conditional plus embedded existential quantifier.)
7. “Every mind is aware of itself” (philosophy of mind)
∀x (Mx → Axx)
Translation: For all x, if x is a mind, then x is aware of itself. (Illustrates reflexive relation Axx.)
8. “Someone fears everyone”
Ǝx ∀y(Fxy)
Translation: There exists some x such that for all y, x fears y. (Inverted quantifier order compared with “Everyone fears someone.”)
9. “Everyone loves someone” (illustrates quantifier scope ambiguity)
(a) ∀x Ǝy(Lxy)
Translation: For every person x, there exists at least one person y such that x loves y. (Each person loves at least one person, possibly different for each.)
(b) Ǝy ∀x(Lxy)
Translation: There exists at least one person y such that, for every person x, x loves y. (There is a single person whom everyone loves.)
10. “Some sentences are about themselves” (Linguistic paradox)
Ǝx (Sx & Axx)
Translation: There exists some x such that x is a sentence and x is about itself. (A self-reference schema; ties into the Liar Paradox.)
11. “No statement can be both true and false” (Linguistic paradox)
~Ǝx (Tx & Fx)
Translation: It is not the case that there exists any x such that x is both true and false.
12. “Everyone admires someone who admires them” (quantifier scope puzzle)
∀x Ǝy (Axy & Ayx)
Translation: For each person x, there exists some person y such that x admires y and y admires x.
13. “Someone loves everyone who loves someone” (quantifier scope puzzle)
Ǝx ∀y [(Ǝz Lyz) → Lxy]
Translation: There exists a person x such that, for every y, if y loves someone, then x loves.
14. “Jones buttered the toast slowly with a knife in the bathroom at midnight” (Donald Davidson)
Ǝe [Butter(e, Jones, toast) & Slowly(e) & With(e, knife) & In(e, bathroom) & At(e, midnight)]
Translation: there exists some event e such that e is Jones buttering the toast, and e was slow, and e was done with the knife, and e occurred in the bathroom, and e happened at midnight.
15. “For some possible world, there exists a person in that world and a person in another world such that the first person is a counterpart of the second person” (David Lewis)
Ǝw1 Ǝw2 [w1≠w2 & Ǝp1 Ǝp2 (Mp1w1 & Mp2w2 & Cp1p2)]
Translation: There exist some worlds w1 and w2 such that w1 is not identical to w2, and there exists some person p1 and person p2 such that p1 is a member of w1, and p2 is a member of w2, and p1 is a counterpart of p2.
Explanation:
w1, w2 = w1 and w2 are worlds.
Mp1w1, Mp2w2 = p1 is a member of w1; p2 is a member of w2.
Cp1p2 = p1 is a counterpart of p2.
w1≠w2 ensures the worlds are distinct.