Formalizing ﬁrst-order logic

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Formalizing first-order logic

Joris Roos

University of Wisconsin-Madison

Sommerakademie Leysin 2018 August 15, 2018

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Overview

1 Introduction

2 Syntax

3 Semantics

4 A simple theorem prover

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Introduction

Last time we considered propositional formulas:

((A∨D)→(D∨ ¬B))∧ ¬(A←(B∨C ∧D)

They consist of constants (>,⊥), literals (A,B, . . .), logical connectives and punctuation.

First-order logic (FOL) extends this in two different ways: quantifiers: ∀,∃

more complicated atomic formulas, consisting of: variables: x,y,z,· · ·

functions: f(x),g(x,y), .. predicates: P(x),x=y,· · · ,

First-order logic is powerful enough to formalize “all of mathematics”. Example. Continuity

∀x.∀ε.ε >0→(∃δ.δ >0→(∀y.|x−y|< δ→ |f(x)−f(y)|< ε))

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Introduction

((A∨D)→(D∨ ¬B))∧ ¬(A←(B∨C ∧D)

First-order logic (FOL) extends this in two different ways: quantifiers: ∀,∃

∀x.∀ε.ε >0→(∃δ.δ >0→(∀y.|x−y|< δ→ |f(x)−f(y)|< ε))

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Introduction

((A∨D)→(D∨ ¬B))∧ ¬(A←(B∨C ∧D)

First-order logic (FOL) extends this in two different ways:

quantifiers: ∀,∃

∀x.∀ε.ε >0→(∃δ.δ >0→(∀y.|x−y|< δ→ |f(x)−f(y)|< ε))

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Introduction

((A∨D)→(D∨ ¬B))∧ ¬(A←(B∨C ∧D)

more complicated atomic formulas, consisting of:

variables: x,y,z,· · · functions: f(x),g(x,y), .. predicates: P(x),x=y,· · · ,

∀x.∀ε.ε >0→(∃δ.δ >0→(∀y.|x−y|< δ→ |f(x)−f(y)|< ε))

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Introduction

((A∨D)→(D∨ ¬B))∧ ¬(A←(B∨C ∧D)

variables: x,y,z,· · ·

∀x.∀ε.ε >0→(∃δ.δ >0→(∀y.|x−y|< δ→ |f(x)−f(y)|< ε))

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Introduction

((A∨D)→(D∨ ¬B))∧ ¬(A←(B∨C ∧D)

variables: x,y,z,· · · functions: f(x),g(x,y), ..

predicates: P(x),x=y,· · · ,

∀x.∀ε.ε >0→(∃δ.δ >0→(∀y.|x−y|< δ→ |f(x)−f(y)|< ε))

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Introduction

((A∨D)→(D∨ ¬B))∧ ¬(A←(B∨C ∧D)

∀x.∀ε.ε >0→(∃δ.δ >0→(∀y.|x−y|< δ→ |f(x)−f(y)|< ε))

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Introduction

((A∨D)→(D∨ ¬B))∧ ¬(A←(B∨C ∧D)

First-order logic is powerful enough to formalize “all of mathematics”.

Example. Continuity

∀x.∀ε.ε >0→(∃δ.δ >0→(∀y.|x−y|< δ→ |f(x)−f(y)|< ε))

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Introduction

((A∨D)→(D∨ ¬B))∧ ¬(A←(B∨C ∧D)

First-order logic is powerful enough to formalize “all of mathematics”.

Example. Continuity

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1 Introduction

2 Syntax

3 Semantics

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Admissible symbols

A FOL formula is a string of symbols from a certain alphabet.

The alphabet consists of

Propositional constants: >(true),⊥(false) Logical operators: ¬,∧,∨

Punctuation: brackets ( ) and, Quantifiers: ∀,∃

Function symbols: f,g, . . . Predicate symbols: P,Q,R,. . . Variable symbols: x,y,z,. . .

We will define FOL formulas in three steps:

terms−→atomic formulas−→formulas

We often also want to fix the allowed function and predicate symbols (this is called a “language”).

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Admissible symbols

Propositional constants: >(true),⊥(false)

Logical operators: ¬,∧,∨ Punctuation: brackets ( ) and, Quantifiers: ∀,∃

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Admissible symbols

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Admissible symbols

Punctuation: brackets ( ) and,

Quantifiers: ∀,∃

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Admissible symbols

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Admissible symbols

Function symbols: f,g,. . .

Predicate symbols: P,Q,R,. . . Variable symbols: x,y,z,. . .

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Admissible symbols

Function symbols: f,g,. . . Predicate symbols: P,Q,R,. . .

Variable symbols: x,y,z,. . .

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Admissible symbols

Function symbols: f,g,. . . Predicate symbols: P,Q,R,. . . Variable symbols: x,y,z,. . .

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Admissible symbols

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Admissible symbols

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FOL language

Definition (Language)

A first-order language(or signature)L is a pair (F,P) where

F is a set of pairs (f,n) with f a function symbol and n a natural number.

P is a set of pairs (P,n) with P a predicate symbol and n a natural number.

The number n is called arityof the predicate or function symbol. A 0-ary function symbol is also called a constant symbol.

Example. Thelanguage of arithmetic is

L_arith= ({(0,0),(1,0),(+,2),(∗,2)},{(=,2)}) The language of set theoryis

L_set= ({},{(=,2),(∈,2)})

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FOL language

L_set= ({},{(=,2),(∈,2)})

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FOL language

L_set= ({},{(=,2),(∈,2)})

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FOL language

L_set= ({},{(=,2),(∈,2)})

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FOL language

L_arith= ({(0,0),(1,0),(+,2),(∗,2)},{(=,2)})

The language of set theoryis

L_set= ({},{(=,2),(∈,2)})

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FOL language

L_set= ({},{(=,2),(∈,2)})

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Terms

Definition (Term)

The set of terms of a FOL language L= (F,P) is the smallest set (of admissible strings) such that

1 Every variable symbol is a term.

2 IfT1, . . . ,Tn are terms and (f,n)∈F, then f(T1, . . . ,Tn) is a term.

(Ifn= 0 in (2), then the assumption is vacuous, so all constant symbols are terms.)

This is powerful enough to express all algebraic terms commonly used in mathematics.

Example.

(x+y)(x−y) +y∗y is a term of L_arith (up to syntax sugar).

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Terms

Definition (Term)

Example.

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Terms

Definition (Term)

Example.

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Terms

Definition (Term)

Example.

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Terms

Definition (Term)

Example.

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Terms

Definition (Term)

Example.

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Atomic formulas

Definition (Atomic formula)

An atomic formula of a FOL language L= (F,P) is a string of the form P(T₁, . . . ,T_n)

where T₁, . . . ,T_n are terms of Land (P,n)∈P.

The propositional constants>,⊥ are also atomic formulas.

(Ifn= 0 thenP(T₁, . . . ,T_n) is meant to be just the predicate symbolP.)

Example.

(x+y)(x−y) +y∗y = 0 is an atomic formula of Larith (up to syntax sugar). Example.

x ∈y is an atomic formula of L_set.

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Atomic formulas

Example.

(x+y)(x−y) +y∗y = 0 is an atomic formula of Larith (up to syntax sugar). Example.

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Atomic formulas

Example.

(x+y)(x−y) +y∗y = 0 is an atomic formula of Larith (up to syntax sugar).

Example.

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Atomic formulas

Example.

(x+y)(x−y) +y∗y = 0 is an atomic formula of Larith (up to syntax sugar).

Example.

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Formulas

Definition (Formula)

The set of formulas of a FOL languageL is the smallest set (of strings of admissible symbols) such that

1 Atomic formulas ofL are formulas.

2 IfF is a formula, then¬F is a formula.

3 IfF,F⁰ are formulas, then∨(F,F⁰) and∧(F,F⁰) are formulas.

4 Ifx is a variable symbol andF is a formula, then∀(x,F) and∃(x,F) are formulas.

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Formulas

1 Atomic formulas ofLare formulas.

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Formulas

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Formulas

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Formulas

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Syntactic sugar

Example. This is a formula (of a suitable FOL language):

∃(x,∀(y,∧(Q(x),¬P(x,y))))

For readability we prefer to write the formula as follows:

∃x.∀y.Q(x)∧ ¬P(x,y)

This is just syntax sugar! We always keep in mind that the formula

“actually” looks as above.

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Syntactic sugar

∃(x,∀(y,∧(Q(x),¬P(x,y)))) For readability we prefer to write the formula as follows:

∃x.∀y.Q(x)∧ ¬P(x,y)

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Syntactic sugar

∃(x,∀(y,∧(Q(x),¬P(x,y)))) For readability we prefer to write the formula as follows:

∃x.∀y.Q(x)∧ ¬P(x,y)

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1 Introduction

2 Syntax

3 Semantics

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Interpretation

To give meaning to a first-order formula we first need to give meaning to its atomic formulas. This requires the following data:

Definition (Interpretation)

An interpretation M of a FOL language L= (F,P) consists of:

1 a nonempty set D called thedomain,

2 for eachn-ary function (f,n)∈F with n≥1, a function f_M :Dⁿ→D,

3 for eachn-ary predicate (P,n)∈P, a function PM :Dⁿ→ {true,false}.

(In the casen= 0,f_Mis simply a constant inD; similarly,P_Mis a constant truth value.)

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Interpretation

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Interpretation

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Interpretation

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Interpretation

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Interpretation

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Examples

An interpretation of L_arith is given by arithmetic of natural numbers:

1 D :=N

2 0M := 0, 1M := 1, +M(x,y) :=x+y,∗_M(x,y) :=x∗y

3 =_M (x,y) :=true iffx =y We call this one M_N.

Another interpretation of L_arith is given by boolean arithmetic:

1 D :={0,1}

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y(mod2),

∗_M(x,y) :=x·y(mod2))

3 =_M (x,y) :=true iffx =y We call this one Mbool.

The following is also an interpretation:

1 D :=Z,

2 0_M := 7, 1_M := 0, +_M(x,y) :=x−3y²,∗_M(x,y) :=x+y,

3 =M (x,y) :=true iffx 6=y Let us call this one Mnonsense.

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Examples

1 D :=N

2 0M := 0, 1M := 1, +M(x,y) :=x+y,∗_M(x,y) :=x∗y

1 D :={0,1}

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y(mod2),

∗_M(x,y) :=x·y(mod2))

1 D :=Z,

2 0_M := 7, 1_M := 0, +_M(x,y) :=x−3y²,∗_M(x,y) :=x+y,

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Examples

1 D :=N

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y,∗_M(x,y) :=x∗y

1 D :={0,1}

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y(mod2),

∗_M(x,y) :=x·y(mod2))

1 D :=Z,

2 0_M := 7, 1_M := 0, +_M(x,y) :=x−3y²,∗_M(x,y) :=x+y,

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Examples

1 D :=N

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y,∗_M(x,y) :=x∗y

3 =_M (x,y) :=trueiff x =y

We call this one M_N.

1 D :={0,1}

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y(mod2),

∗_M(x,y) :=x·y(mod2))

1 D :=Z,

2 0_M := 7, 1_M := 0, +_M(x,y) :=x−3y²,∗_M(x,y) :=x+y,

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Examples

1 D :=N

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y,∗_M(x,y) :=x∗y

3 =_M (x,y) :=trueiff x =y We call this one M_N.

1 D :={0,1}

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y(mod2),

∗_M(x,y) :=x·y(mod2))

1 D :=Z,

2 0_M := 7, 1_M := 0, +_M(x,y) :=x−3y²,∗_M(x,y) :=x+y,

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Examples

1 D :=N

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y,∗_M(x,y) :=x∗y

1 D :={0,1}

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y(mod2),

∗_M(x,y) :=x·y(mod2))

1 D :=Z,

2 0_M := 7, 1_M := 0, +_M(x,y) :=x−3y²,∗_M(x,y) :=x+y,

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Examples

1 D :=N

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y,∗_M(x,y) :=x∗y

1 D :={0,1}

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y(mod2),

∗_M(x,y) :=x·y(mod2))

1 D :=Z,

2 0_M := 7, 1_M := 0, +_M(x,y) :=x−3y²,∗_M(x,y) :=x+y,

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Examples

1 D :=N

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y,∗_M(x,y) :=x∗y

1 D :={0,1}

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y(mod2),

∗_M(x,y) :=x·y(mod2))

1 D :=Z,

2 0_M := 7, 1_M := 0, +_M(x,y) :=x−3y²,∗_M(x,y) :=x+y,

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Examples

1 D :=N

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y,∗_M(x,y) :=x∗y

1 D :={0,1}

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y(mod2),

∗_M(x,y) :=x·y(mod2))

3 =_M (x,y) :=trueiff x =y We call this one Mbool.

1 D :=Z,

2 0_M := 7, 1_M := 0, +_M(x,y) :=x−3y²,∗_M(x,y) :=x+y,

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Examples

1 D :=N

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y,∗_M(x,y) :=x∗y

1 D :={0,1}

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y(mod2),

∗_M(x,y) :=x·y(mod2))

1 D :=Z,

2 0_M := 7, 1_M := 0, +_M(x,y) :=x−3y²,∗_M(x,y) :=x+y,

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Examples

1 D :=N

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y,∗_M(x,y) :=x∗y

1 D :={0,1}

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y(mod2),

∗_M(x,y) :=x·y(mod2))

1 D :=Z,

2 0_M := 7, 1_M := 0, +_M(x,y) :=x−3y²,∗_M(x,y) :=x+y,

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Examples

1 D :=N

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y,∗_M(x,y) :=x∗y

1 D :={0,1}

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y(mod2),

∗_M(x,y) :=x·y(mod2))

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Examples

1 D :=N

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y,∗_M(x,y) :=x∗y

1 D :={0,1}

2 0_M := 0, 1_M := 1, +_M(x,y) :=x+y(mod2),

∗_M(x,y) :=x·y(mod2))

1 D :=Z,

2 0_M := 7, 1_M := 0, +_M(x,y) :=x−3y²,∗_M(x,y) :=x+y,

3 =M (x,y) :=trueiff x 6=y

Let us call this one^{Joris Roos} Mnonsense. First-order logic August 15, 2018 13 / 30

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Term evaluation

Definition

A valuation v for an interpretationM is a map from the set of variable symbols to the domain D: v(x)∈D for every variable symbolx.

Let M be an interpretation andv a valuation. Then to each termt we assign a value evM,v(t)∈D such that

1 ev_M,v(x) =v(x) ifx is a variable symbol,

2

evM,v(f(t1, . . . ,tn)) =fM(evM,v(t1), . . . ,evM,v(tn)) for every n-ary function symbol f.

By structural induction this defines a unique map from the set of terms to the domain D.

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Term evaluation

Definition

A valuation v for an interpretationM is a map from the set of variable symbols to the domain D: v(x)∈D for every variable symbolx. Let M be an interpretation andv a valuation. Then to each termt we assign a value evM,v(t)∈D such that

2

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Term evaluation

Definition

2

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Term evaluation

Definition

2

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Term evaluation

Definition

2

By structural induction this defines a unique map from the set of terms to

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Evaluation of formulas

For each formula F, interpretation M and valuationv we want to define a meaning ev_M,v(F)∈ {true,false}.

The idea is again structural induction, but there is a catch!

The semantics of the quantifiers require us to modify the assignmentv for the variables bound by a quantifier.

For every variable symbolx and every a∈D we define v[x7→a]

to be the assignment that maps x toa and every y 6=x to v(y).

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Evaluation of formulas

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Evaluation of formulas

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Evaluation of formulas

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Evaluation of formulas

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Evaluation of formulas

Now we can run the structural induction:

1 evM,v(P(t1, . . . ,tn)) :=PM(evM,v(t1), . . . ,evM,v(tn)) for every n-ary predicate symbol P.

2 evM,v(¬F) istrue iffevM,v(F) isfalse.

3 ev_M,v(F ∨F⁰) is true iff at least one ofev_M_,v(F),ev_M,v(F⁰) is true.

4 ev_M,v(F ∧F⁰) is true iff both,ev_M,v(F) andev_M,v(F⁰) are true.

5 ev_M,v(∃x.F) is trueiff there exists a∈D such that ev_M,v[x7→a](F) is true.

6 ev_M,v(∀x.F) is trueiff for all a∈D,ev_M,v[x7→a](F) istrue.

Beware: the domainD is nonempty, but otherwise arbitrary (e.g. could be uncountably infinite) and the functions f_M,P_M need not be computable. To enable a straightforward computer implementation of the semantics we need to restrict to finite D.

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Evaluation of formulas

2 evM,v(¬F) is true iffevM,v(F) isfalse.

3 ev_M,v(F ∨F⁰) is true iff at least one ofev_M_,v(F),ev_M,v(F⁰) is true.

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Evaluation of formulas

3 ev_M,v(F ∨F⁰) is trueiff at least one of ev_M,v(F),ev_M,v(F⁰) is true.

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Evaluation of formulas

4 ev_M,v(F ∧F⁰) is trueiff both, ev_M,v(F) andev_M,v(F⁰) are true.

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Evaluation of formulas

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Evaluation of formulas

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Evaluation of formulas

Beware: the domainD is nonempty, but otherwise arbitrary (e.g. could be uncountably infinite) and the functions f_M,P_M need not be computable.

To enable a straightforward computer implementation of the semantics we need to restrict to finite D.

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Evaluation of formulas

Beware: the domainD is nonempty, but otherwise arbitrary (e.g. could be uncountably infinite) and the functions f_M,P_M need not be computable.

To enable a straightforward computer implementation of the semantics we need to restrict to finite D.