CSE 311 Lecture 06: Predicate Logic
Topics
- Key notions in predicate logic
- A brief review of Lecture 05.
- Using quantifiers
- From predicate logic to English and back.
- Negating quantified formulas
- DeMorgan’s laws for quantifiers.
- Quantifier scopes
- Bound and free variables.
- Nested quantifiers
- The quantifier order matters.
Key notions in predicate logic
A brief review of Lecture 05.
Syntax and semantics of predicate logic
- Syntax
- Predicate logic extends propositional logic with two key constructs: predicates and quantifiers ($\exists,\forall$).
- Semantics
- We define the meaning of formulas in predicate logic with respect to a domain of discourse.
Predicates
- Predicate is a function that returns a truth value.
- Cat(x) $::=$ “x is a cat”
- Prime(x) $::=$ “x is prime”
- HasTaken(x, y) $::=$ “student x has taken course y”
- LessThan(x, y) $::=$ “x < y”
- Sum(x, y, z) $::=$ “x + y = z”
- GreaterThan5(x) $::=$ “x > 5”
- HasNChars(s, n) $::=$ “string s has length n”
- Predicates can have varying arity (numbers of arguments).
- But a given predicate accepts a fixed number of arguments.
- Predicates can have any names.
- The name does not determine the meaning of the predicate.
- So we can define Cat(x) $::=$ “x is an even number”.
Domain of discourse
To give meaning to predicates in a formula, we define a set of objects that those predicates can take as input.
This set of objects is called the domain of discourse for a formula.
- For each of the following, what might the domain be?
- “x is a cat”, “x barks”, “x ruined my couch”
- “mammals” or “sentient beings” or “cats and dogs” or …
- “x is prime”, “x = 0”, “x < 0”, “x is a power of two”
- “numbers” or “integers” or “integers greater than 5” or …
- “student x has taken course y” “x is a pre-req for z”
- “students and courses” or “university entities” or …
Quantifiers
- Quantifiers let us talk about all or some objects in the domain.
- $\forall x. P(x)$
- $P(x)$ is true for every $x$ in the domain.
- Read as “for all $x$, $P(x)$”.
- Called the universal quantifier.
- $\exists x. P(x)$
- There is an $x$ in the domain for which $P(x)$ is true.
- Read as “there exists $x$, $P(x)$”.
- Called the existential quantifier.
Understanding quantifiers
The truth value of a quantified formula depends on the domain.
| $\{-3, 3\}$ | Integers | Odd Integers | |
|---|---|---|---|
| $\forall x. \text{Odd}(x)$ | True | False | True |
| $\forall x. \text{LessThan5}(x)$ | True | False | False |
| $\{-3, 3\}$ | Integers | Positive Multiples of 5 | |
|---|---|---|---|
| $\exists x. \text{Odd}(x)$ | True | True | True |
| $\exists x. \text{LessThan5}(x)$ | True | True | False |
You can think of $\forall x. P(x)$ as conjunction over all objects in the domain, and $\exists x. P(x)$ as disjunction over all objects in the domain.
- $\forall x. \text{Odd}(x)$
- over $\{-3, 3\}$ is the conjunction $\text{Odd}(-3)\wedge\text{Odd}(3)$
- over integers is the infinite conjunction $\ldots\wedge\text{Odd}(-1)\wedge\text{Odd}(0)\wedge\text{Odd}(1)\wedge\ldots$
- $\exists x. \text{Odd}(x)$
- over $\{-3, 3\}$ is the disjunction $\text{Odd}(-3)\vee\text{Odd}(3)$
- over integers is the infinite disjunction $\ldots\vee\text{Odd}(-1)\vee\text{Odd}(0)\vee\text{Odd}(1)\vee\ldots$
Using quantifiers
From predicate logic to English and back.
Statements with quantifiers
Just like with propositional logic, we need to define variables (this time predicates). And we must also now define a domain of discourse.
What is the truth value of these statements?
| $\exists x. \text{Even}(x)$ | T |
| $\forall x. \text{Odd}(x)$ | F |
| $\forall x. \text{Even}(x)\vee\text{Odd}(x)$ | T |
| $\exists x. \text{Even}(x)\wedge\text{Odd}(x)$ | F |
| $\forall x. \text{Greater}(x+1,x)$ | T |
| $\exists x. \text{Even}(x)\wedge\text{Prime}(x)$ | T |
- Domain of discourse
- Positive integers
- Predicate definitions
- Even(x) $:=$ “x is even”
- Odd(x) $:=$ “x is odd”
- Prime(x) $:=$ “x is prime”
- Greater(x, y) $:=$ “x > y”
- Equal(x, y) $:=$ “x = y”
- Sum(x, y, z) $:=$ “z = x + y”
Predicate logic to English: literal translations
- Translate the following statements to English
- $\forall x. \exists y. \text{Greater}(y, x)\wedge\text{Prime}(y)$
- For every positive integer $x$, there is a positive integer $y$, such that $y > x$ and $y$ is prime.
- $\forall x. \text{Prime}(x)\rightarrow(\text{Equal}(x,2)\vee\text{Odd}(x))$
- For every positive integer $x$, if $x$ is prime then $x=2$ or $x$ is odd.
- $\exists x. \exists y. \text{Sum}(x,2,y)\wedge\text{Prime}(x)\wedge\text{Prime}(y)$
- There exist positive integers $x$ and $y$ such that $x+2=y$ and $x$ and $y$ are prime.
- Domain of discourse
- Positive integers
- Predicate definitions
- Even(x) $:=$ “x is even”
- Odd(x) $:=$ “x is odd”
- Prime(x) $:=$ “x is prime”
- Greater(x, y) $:=$ “x > y”
- Equal(x, y) $:=$ “x = y”
- Sum(x, y, z) $:=$ “z = x + y”
Predicate logic to English: natural translations
- Translate the following statements to English
- $\forall x. \exists y. \text{Greater}(y, x)\wedge\text{Prime}(y)$
- For every positive integer there is a larger number that is prime.
- $\forall x. \text{Prime}(x)\rightarrow(\text{Equal}(x,2)\vee\text{Odd}(x))$
- Every prime number is 2 or odd.
- $\exists x. \exists y. \text{Sum}(x,2,y)\wedge\text{Prime}(x)\wedge\text{Prime}(y)$
- There exist prime numbers that differ by two.
- Domain of discourse
- Positive integers
- Predicate definitions
- Even(x) $:=$ “x is even”
- Odd(x) $:=$ “x is odd”
- Prime(x) $:=$ “x is prime”
- Greater(x, y) $:=$ “x > y”
- Equal(x, y) $:=$ “x = y”
- Sum(x, y, z) $:=$ “z = x + y”
English to predicate logic
- “Orange cats like lasagna.”
- $\forall x. ((\text{Orange}(x) \wedge \text{Cat}(x)) \rightarrow \text{LikesLasagna}(x))$
- “Some orange cats don’t like lasagna.”
- $\exists x. ((\text{Orange}(x) \wedge \text{Cat}(x)) \wedge \neg \text{LikesLasagna}(x))$
- Domain of discourse
- Mammals
- Predicate definitions
- Cat(x) $:=$ “x is a cat”
- Orange(x) $:=$ “x is orange”
- LikesLasagna(x) $:=$ “x likes lasagna”
English to predicate logic: translation hints
- “Orange cats like lasagna.”
- $\forall x. ((\text{Orange}(x) \wedge \text{Cat}(x)) \rightarrow \text{LikesLasagna}(x))$
- When there’s no leading quantification, it means “for all”.
- When restricting to a smaller domain in a “for all”, use implication.
- “Some orange cats don’t like lasagna.”
- $\exists x. ((\text{Orange}(x) \wedge \text{Cat}(x)) \wedge \neg \text{LikesLasagna}(x))$
- “Some” means “there exists”.
- When restricting to a smaller domain in an “exists”, use conjunction.
- When putting predicates together, like orange cats, use conjunction.
- Domain of discourse
- Mammals
- Predicate definitions
- Cat(x) $:=$ “x is a cat”
- Orange(x) $:=$ “x is orange”
- LikesLasagna(x) $:=$ “x likes lasagna”
Negating quantified formulas
DeMorgan’s laws for quantifiers.
Negations of quantifiers
- Domain of discourse
- {plum, apple, …}
- Predicate definitions
- PurpleFruit(x) $:=$ “x is a purple fruit”
Let $P$ be the formula $\forall x. \text{PurpleFruit}(x)$, “all fruits are purple.”
What is the negation of $P$?
- a. “There exists a purple fruit” ($\exists x. \text{PurpleFruit}(x)$)
- b. “There exists a non-purple fruit” ($\exists x. \neg \text{PurpleFruit}(x)$)
- c. “All fruits are not purple” ($\forall x. \neg \text{PurpleFruit}(x)$)
Key idea: think of $p$ as a conjunction over all objects in the domain, negate that conjunction, and convert back to a quantified formula.
| $p$ | $\equiv \text{PurpleFruit}(\text{plum})\wedge\text{PurpleFruit}(\text{apple})\wedge\ldots$ | $\forall$ to conjunction |
| $\neg p$ | $\equiv \neg(\text{PurpleFruit}(\text{plum})\wedge\text{PurpleFruit}(\text{apple})\wedge\ldots)$ | Negate both sides |
| $\equiv \neg\text{PurpleFruit}(\text{plum})\vee\neg\text{PurpleFruit}(\text{apple})\vee\ldots$ | DeMorgan | |
| $\equiv \exists x.\neg\text{PurpleFruit}(x)$ | Disjunction to $\exists$ |
DeMorgan’s laws for quantifiers
“There is no largest integer.”
| $\neg \exists x. \forall y. (x≥y)$ | $\equiv$ | $\forall x. \neg \forall y. (x \geq y) $ | DeMorgan |
| $\equiv$ | $\forall x. \exists y. \neg (x \geq y) $ | DeMorgan | |
| $\equiv$ | $\forall x. \exists y. (x < y) $ | Semantics of $>$ |
“For every integer there is a larger integer.”
Quantifier scopes
Bound and free variables.
Scope of quantifiers
$\exists $$x$$. (P($$x$$)\wedge Q($$x$$))$
There is an object in the domain for which both $P$ and $Q$ are true.
vs
$(\exists $$x$$. P($$x$$)) \wedge (\exists $$x$$. Q($$x$$))$
There is an object for which $P$ is true and an object for which $Q$ is true, and they may be different objects.
Understanding scope of quantifiers
The formula inside of a quantifier is called its scope.
A variable is bound if it is in the scope of some quantifier.
A variable is free if it isn’t in the scope of any quantifier.
- Example: $\forall $$y$$. ((\exists $$x$$. P($$x$$)) \rightarrow Q($$x$$, $$y$$))$
- Is $y$ in $Q(x, y)$ bound or free? Bound.
- Is $x$ in $P(x)$ bound or free? Bound.
- Is $x$ in $Q(x, y)$ bound or free? Free.
Quantifier “style”
$\forall \color{deeppink}x. (\exists \color{seagreen}y. (P(\color{deeppink}x,\color{seagreen}y)\rightarrow\forall\color{slateblue}x. Q(\color{seagreen}y,\color{slateblue}x)))$
This isn’t wrong, but it’s confusing. Help your reader by using unique names for quantified variables. Names are cheap :)
Nested quantifiers
The quantifier order matters.
When nesting quantifiers …
- Bound variable names don’t matter.
- $\forall x. \exists y. P(x, y) \equiv \forall a. \exists b. P(a, b)$
- Quantifiers can sometimes move within the enclosing formula.
- ✔ $\forall x. (Q(x) \wedge \color{seagreen}{\exists y}. P(x, y)) \equiv \forall x. \color{seagreen}{\exists y}. (Q(x) \wedge P(x, y))$
- ✘ $\forall x. (Q(x) \wedge (\neg \color{deeppink}{\exists y}. P(x, y))) \not\equiv \forall x. \color{deeppink}{\exists y}. (Q(x) \wedge \neg P(x, y))$
- But the order of quantifiers is important.
- Example: are these formulas true or false?
- $ \exists x. \forall y. \text{GreaterEq}(x, y)$ $\F$
- $ \forall y. \exists x. \text{GreaterEq}(x, y)$ $\T$
- Domain of discourse
- Integers
- Predicate definitions
- GreaterEq(x, y) $:=$ “x $\geq$ y”
Quantification with two variables
| Formula | When true | When false |
|---|---|---|
| $\forall x. \forall y. P(x, y)$ | Every pair is true. | At least one pair is false. |
| $\exists x. \exists y. P(x, y)$ | At least one pair is true. | All pairs are false. |
| $\forall x. \exists y. P(x, y)$ | Every $x$ has a corresponding $y$. | Some $x$ doesn’t have a $y$. |
| $\exists y. \forall x. P(x, y)$ | A particular $y$ works for every $x$. | Every $y$ has an $x$ that makes $P(x,y)$ false. |
- This is the form of the program synthesis query!
- $\exists P. \forall x. S(x, P(x))$
- “There is a program $P$ that satisfies the spec $S$ on every input $x$.”
Summary
- Predicate logic adds predicates and quantifiers to propositional logic.
- Predicate is a function that returns a truth value.
- Quantifiers let us talk about all ($\forall$) or some ($\exists$) objects in the domain.
- The domain of discourse is the set of objects over which the predicates and quantifiers in a formula are evaluated.
- When using quantifiers, keep in mind
- the DeMorgan’s laws for negating quantified formulas,
- which variables are free and bound, and
- the order of quantifiers.