First Order Syntax

ﬁrst-order logic: ingredients

There is a domain of interpretation, and

I variables: x, y, z,... ranging over the domain

I constant symbols (names): 0 1 π e Albert Einstein

I function symbols: + ∗ − / sin(−) exp(−) I predicates: = > 6 male(−) parent(−, −) I propositional connectives: ∧ ∨ q −→ ←→ O ↑ ↓ I quantiﬁers: ∀ ∃.

Tibor Beke ﬁrst order syntax False.

sentences

Let the domain of interpretation be the set of real numbers.

∀a ∀b ∀c ∃x(ax2 + bx + c = 0)

This formula is an example of a sentence from ﬁrst-order logic. It has no free variables: every one of the variables a, b, c, x is in the scope of a quantiﬁer.

A sentence has a well-deﬁned truth value that can be discovered (at least, in principle). The sentence above (interpreted over the real numbers) is

Tibor Beke ﬁrst order syntax sentences

Let the domain of interpretation be the set of real numbers.

∀a ∀b ∀c ∃x(ax2 + bx + c = 0)

This formula is an example of a sentence from ﬁrst-order logic. It has no free variables: every one of the variables a, b, c, x is in the scope of a quantiﬁer.

A sentence has a well-deﬁned truth value that can be discovered (at least, in principle). The sentence above (interpreted over the real numbers) is False.

Tibor Beke ﬁrst order syntax “Every woman has two children ”

but is, in fact, logically equivalent to every woman having at least one child. That child could play the role of both y and z.

sentences

Let the domain of discourse be a set of people, let F (x) mean “x is female” and let P(x, y) mean “x is a parent of y”. ∀x F (x) −→ ∃y∃zP(x, y) ∧ P(x, z)

This sentence seems to be saying

Tibor Beke ﬁrst order syntax sentences

Let the domain of discourse be a set of people, let F (x) mean “x is female” and let P(x, y) mean “x is a parent of y”. ∀x F (x) −→ ∃y∃zP(x, y) ∧ P(x, z)

This sentence seems to be saying

“Every woman has two children ”

but is, in fact, logically equivalent to every woman having at least one child. That child could play the role of both y and z.

Tibor Beke ﬁrst order syntax Suppose the domain of discourse is the natural numbers 0, 1, 2, 3,... . What is the meaning of the following formula? ∀p∀q pq = n −→ (p = 1 O q = 1) “n is a prime number”

open formulas

A ﬁrst-order formula that contains free variables is said to be open. You cannot associate a truth value to it unless the free variables are specialized to some element of the domain: ∀x f (x) 6 f (x0) An open formula can be thought of as deﬁning a property (“predicate”) of its free variable(s).

Tibor Beke ﬁrst order syntax “n is a prime number”

open formulas

Suppose the domain of discourse is the natural numbers 0, 1, 2, 3,... . What is the meaning of the following formula? ∀p∀q pq = n −→ (p = 1 O q = 1)

Tibor Beke ﬁrst order syntax open formulas

Suppose the domain of discourse is the natural numbers 0, 1, 2, 3,... . What is the meaning of the following formula? ∀p∀q pq = n −→ (p = 1 O q = 1) “n is a prime number”

Tibor Beke ﬁrst order syntax “ x is a grandparent of y ”

∃w∃z∃uP(w, z) ∧ P(z, x) ∧ P(w, u) ∧ P(u, y)

“ x and y are cousins (or siblings, or the same people!) ”

open formulas

F (x): x is female M(x): x is male P(x, y): x is a parent of y

∃zP(x, z) ∧ P(z, y)

Tibor Beke ﬁrst order syntax ∃w∃z∃uP(w, z) ∧ P(z, x) ∧ P(w, u) ∧ P(u, y)

“ x and y are cousins (or siblings, or the same people!) ”

open formulas

F (x): x is female M(x): x is male P(x, y): x is a parent of y

∃zP(x, z) ∧ P(z, y)

“ x is a grandparent of y ”

Tibor Beke ﬁrst order syntax “ x and y are cousins (or siblings, or the same people!) ”

open formulas

F (x): x is female M(x): x is male P(x, y): x is a parent of y

∃zP(x, z) ∧ P(z, y)

“ x is a grandparent of y ”

∃w∃z∃uP(w, z) ∧ P(z, x) ∧ P(w, u) ∧ P(u, y)

Tibor Beke ﬁrst order syntax open formulas

F (x): x is female M(x): x is male P(x, y): x is a parent of y

∃zP(x, z) ∧ P(z, y)

“ x is a grandparent of y ”

∃w∃z∃uP(w, z) ∧ P(z, x) ∧ P(w, u) ∧ P(u, y)

“ x and y are cousins (or siblings, or the same people!) ”

Tibor Beke ﬁrst order syntax ﬁrst-order logic: syntax

We will deﬁne the notion of well-formed formula. The deﬁnition will be “step-by-step” (inductive).

• A term is a composite of function symbols, variables and constants. Terms denote elements of the domain — speciﬁc or unspeciﬁed elements. Examples of terms x π Pope Francis 23 · 24 sin2(kx) + 23 a · x2 + b · x + c

Tibor Beke ﬁrst order syntax ﬁrst-order logic: atomic formulas

• Suppose t1, t2,..., tn are terms, and R(−, −,..., −) is an n-ary relation symbol. The expression

R(t1, t2,..., tn)

is called an atomic formula.

Atomic formulas contain neither propositional connectives nor quantiﬁers. They are the simplest types of expressions that can be evaluated to True or False — if their free variables are specialized to some element of the domain, that is. They are the same, essentially, as the “atomic propositions” of propositional logic.

Tibor Beke ﬁrst order syntax examples of atomic formulas

π > e x = y friends(Big Bird, Cookie Monster) spouse(Pope Francis, x) collinear(p, q, r) x2 + 1 > sin(z) + 23

Tibor Beke ﬁrst order syntax ﬁrst-order logic: well-formed formulas

We will deﬁne well-formed formulas the following way:

I Any atomic formula is well-formed.

I If α and β are well-formed and ? is any binary propositional connective, then (α ? β)

is well-formed. (‘?’ could be any of ∧ ∨ −→ O ←→ ↑ ↓ .) I If α is well-formed, so is qα. I If α is well-formed and x is any variable then the expressions

∀xα

∃xα are both well-formed.

Tibor Beke ﬁrst order syntax not well-formed (if we’re sticklers!)

wﬀ

not wﬀ

wﬀ

wﬀ (strange but true!)

which of the following is a well-formed formula?

P(x) ∧ Q(y) ∧ R(z)

∃x P(x, y) ∧ Q(x) ∨ R(y)

∃x ∧ ∃y P(x)OQ(y) q q∀x∃yR(x) q ∀xQ(x, y) −→q∃yqqQ(z, z) q ∀yq ∃x P(z) ←→ ∃xQ(x)

q ∀zq ∀z∃zQ(x, y)

Tibor Beke ﬁrst order syntax wﬀ

not wﬀ

wﬀ

wﬀ (strange but true!)

which of the following is a well-formed formula?

P(x) ∧ Q(y) ∧ R(z) not well-formed (if we’re sticklers!)

∃x P(x, y) ∧ Q(x) ∨ R(y)

∃x ∧ ∃y P(x)OQ(y) q q∀x∃yR(x) q ∀xQ(x, y) −→q∃yqqQ(z, z) q ∀yq ∃x P(z) ←→ ∃xQ(x)

q ∀zq ∀z∃zQ(x, y)

Tibor Beke ﬁrst order syntax not wﬀ

not wﬀ

wﬀ

wﬀ (strange but true!)

which of the following is a well-formed formula?

P(x) ∧ Q(y) ∧ R(z) not well-formed (if we’re sticklers!)

∃x P(x, y) ∧ Q(x) ∨ R(y) wﬀ

∃x ∧ ∃y P(x)OQ(y) q q∀x∃yR(x) q ∀xQ(x, y) −→q∃yqqQ(z, z) q ∀yq ∃x P(z) ←→ ∃xQ(x)

q ∀zq ∀z∃zQ(x, y)

Tibor Beke ﬁrst order syntax not wﬀ

wﬀ

wﬀ (strange but true!)

which of the following is a well-formed formula?

P(x) ∧ Q(y) ∧ R(z) not well-formed (if we’re sticklers!)

∃x P(x, y) ∧ Q(x) ∨ R(y) wﬀ

∃x ∧ ∃y P(x)OQ(y) not wﬀ q q∀x∃yR(x) q ∀xQ(x, y) −→q∃yqqQ(z, z) q ∀yq ∃x P(z) ←→ ∃xQ(x)

q ∀zq ∀z∃zQ(x, y)

Tibor Beke ﬁrst order syntax wﬀ

wﬀ

wﬀ (strange but true!)

which of the following is a well-formed formula?

P(x) ∧ Q(y) ∧ R(z) not well-formed (if we’re sticklers!)

∃x P(x, y) ∧ Q(x) ∨ R(y) wﬀ

∃x ∧ ∃y P(x)OQ(y) not wﬀ q q∀x∃yR(x) not wﬀ q ∀xQ(x, y) −→q∃yqqQ(z, z) q ∀yq ∃x P(z) ←→ ∃xQ(x)

q ∀zq ∀z∃zQ(x, y)

Tibor Beke ﬁrst order syntax wﬀ

wﬀ (strange but true!)

which of the following is a well-formed formula?

P(x) ∧ Q(y) ∧ R(z) not well-formed (if we’re sticklers!)

∃x P(x, y) ∧ Q(x) ∨ R(y) wﬀ

∃x ∧ ∃y P(x)OQ(y) not wff q q∀x∃yR(x) not wff q ∀xQ(x, y) −→q∃yqqQ(z, z) wff q ∀yq ∃x P(z) ←→ ∃xQ(x)

q ∀zq ∀z∃zQ(x, y)

Tibor Beke ﬁrst order syntax wﬀ (strange but true!)

which of the following is a well-formed formula?

P(x) ∧ Q(y) ∧ R(z) not well-formed (if we’re sticklers!)

∃x P(x, y) ∧ Q(x) ∨ R(y) wﬀ

∃x ∧ ∃y P(x)OQ(y) not wff q q∀x∃yR(x) not wff q ∀xQ(x, y) −→q∃yqqQ(z, z) wff q ∀yq ∃x P(z) ←→ ∃xQ(x) wff

q ∀zq ∀z∃zQ(x, y)

Tibor Beke ﬁrst order syntax which of the following is a well-formed formula?

P(x) ∧ Q(y) ∧ R(z) not well-formed (if we’re sticklers!)

∃x P(x, y) ∧ Q(x) ∨ R(y) wﬀ

∃x ∧ ∃y P(x)OQ(y) not wff q q∀x∃yR(x) not wff q ∀xQ(x, y) −→q∃yqqQ(z, z) wff q ∀yq ∃x P(z) ←→ ∃xQ(x) wff

q ∀zq ∀z∃zQ(x, y) wﬀ (strange but true!)

Tibor Beke ﬁrst order syntax unique parsing

If α is a wﬀ but not atomic then it must arise from smaller wﬀ (“subformulas”) using one of the three rules of formation: (1) If α and β are well-formed and ? is any propositional connective, then (α ? β) is well-formed. (2) If α is well-formed, so is qα. (3) If α is well-formed and x is any variable then the expressions ∀xα and ∃xα are both well-formed. The following is true (and requires proof, which will not be given here):

If a well-formed formula is not atomic then it can be generated from well-formed subformulas using one of rules (1), (2) or (3). There is no ambiguity which rule applied, and the subformulas involved are uniquely determined too.

Tibor Beke ﬁrst order syntax unique parsing: example

n o ∀xP(x, y) ∧ ∃yQ(z) O qq∀zR(z, x) ∨ ∀xP(y) is well-formed. It comes from ∀xP(x, y) ∧ ∃yQ(z) and qq∀zR(z, x) ∨ ∀xP(y) , both of which are well-formed, using an instance of rule (1). It cannot be broken up into well-formed subformulas any other way.

Its subformulas can be broken up into yet smaller subformulas, generating what is called the “parse tree” of the formula. This process in unambiguous. A well-formed formula cannot be “read” two diﬀerent ways.

Tibor Beke ﬁrst order syntax why this is important

I using very few rules of “grammar”, we were able to specify all the formulas we consider meaningful; no need for elaborate rules of parenthesization; no chance for ambiguity like

p ∧ q ∨ r

I this method of specifying the syntax of well-formed expressions has become inﬂuential both for programming languages (ALGOL 60, C) and for natural language processing

I wﬀ are great for logic and computer algorithms; for human consumption, we will drop the outermost parentheses and continue to write formulas like

x > 0 ∨ x = 0 ∨ x < 0

rather than ((x > 0 ∨ x = 0) ∨ x < 0) etc.

Tibor Beke ﬁrst order syntax scope and binding

In a well-formed formula, a variable may appear free, or it may be in the scope of a quantiﬁer. One also says that the quantiﬁer binds an occurrence of a variable. A variable can appear several times in a formula, free at some locations and bound at others. can appear confusing!

∀x(q ∃y∀z(P(x, y, z) ↓ ∀xQ(x)) ∧ R(x, z)) ∨ ∃yS(x, y) That initial for every x, just which of the other x’s is it referring to? And the y? z?

Tibor Beke ﬁrst order syntax scope and binding

I In an atomic formula R(t1, t2,..., tn), any variables that appear are free.

I Let x be some variable. In the formula ∀xα, the free occurrences of x in α become bound by the initial ∀x. Those occurrences of x in α that are not free (that are already bound by some quantiﬁer) will not fall under the scope of the initial ∀x. Similarly for ∃xα.

I Iterating this rule (as one builds up a formula from subformulas) one can determine unambiguously(!) which quantiﬁer binds which variable.

Tibor Beke ﬁrst order syntax Homework Problem 19

Let our variables range over the natural numbers 0, 1, 2, 3,... with +, ·, =, > having their usual meaning. Write down a first-order formula that expresses the following: There are infinitely many prime numbers. Hint: you cannot directly say “infinitely many” in first order logic — you only have “for every ... ” and “there is some ... ”. But you could express the following: For any number n, there is a number bigger than n that is prime. Or, if you want, you could express the following: It is not true that there is a biggest prime number. Due Tue, Oct 30, in class.

Tibor Beke ﬁrst order syntax Homework Problem 20

Suppose your variables range over all people. Let the binary predicate R(x, y) mean “person x respects person y”.

Formalize the following statements in ﬁrst order logic, using the predicates R(−, −) and ‘ = ’ only:

(a) Everyone is respected by someone. (b) Everyone respects at least two people. (c) No one is respected by everyone. (d) If you respect everyone you’ll ﬁnd that everyone respects you. (e) If you respect no one you’ll ﬁnd that no one respects you. (f) People that you respect, will respect you too.

Due Tue, Oct 30, in class.

Tibor Beke ﬁrst order syntax