Chapitre 3 : Logique du premier ordre

Logique

Chapitre 3 : Logique du premier ordre

Lucie Le Briquer

1 Theories

Remark.

s.t. means ”such that”

– ∀x, 0 +x=x

– ∀x,∀y, S(x) +y=S(x+y) – ∀x, 0×x=x

– ∀x,∀y, S(x) =S(y)⇒x=y – ∀x, x6= 0⇒ ∃y, x=S(y) – ∀x,∀y, S(x)y= (xy) +y – ∀x, ¬(S(x) = 0)

Definition 1(elementary arithmetics)

– ∀x,∀y,∀z, (xy)z=x(yz)

– ∃e,(∀x, xe=ex)∧(xx⁻¹=e)∧(x⁻¹x=e) Definition 2(group theory)

Remark.

A theory equivalent to group theory :

∀x∀y:x⁻¹×(xy) =y

2 Syntax

LetF be a set of symbols of functions f ∈F, each one having aritya(f)∈N. LetX be a set ofvariables.

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F,X given. The set of termsT(F, X) is defined by : – x∈X

– f(t1, ..., tn) where theti are terms anda(f) =n Definition 3(terms)

Example.

F ={z(0),+(2),×(3), s(1)}

3 Formulas of the first-order predicate calculus

LetP be a set of relation symbols s.t. P ∈ P has aritya(P)∈N.

CP₁(F,P) :

– an atomic formulaP(t₁, ..., t_n), wherea(P) =nandt₁, ..., t_n are terms – ifϕ, ψ∈CP1(F,P),then : ¬ϕ, ϕ⇒ψ, ϕ∧ψ, ϕ∨ψ∈CP1(F,P) Definition 4(CP1(F,P))

3.1 F -algebra

– a given non empty setD_A – for allf ∈F f_A:D^a(f)_A −→D_A Definition 5(F-algebra)

Examples.

– (N,+_N,×_N,0_N, S_N) is anF-algebra whereF ={+(2),×(2),0(0), s(1)}

– A⁰, with :

• DA⁰= Σ^∗

• +A⁰ is the concatenation

• ×A⁰ is defined by

ω×A⁰ω⁰ =ω[a7→ω⁰] fora∈Σ

• SA⁰(ω) =ω·a

– T(F, X) is aF-algebra, where functions are trivially interpreted : f_T(F,X)(t₁, ..., t_n) =f(t₁, ..., t_n) As it happens, the domain isT(F, X)

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3.2 Morphisms

A, A⁰ twoF-algebras. A morphism ofA−→A⁰ is an applicationh:DA−→DA⁰ s.t. for all f ∈F, e1, ..., en ∈DA wherea(f) =n:

h(fA(e1, ..., en)) =fA⁰(h(e1), ..., h(en)) Definition 6(morphism ofA−→A⁰)

Example.

h:

A_Σ−→N ω7→ |ω|a

– h(zA) =h(ε) = 0 =z_N

– h(ω+Aω⁰) =h(ωω⁰) =|ωω⁰|=|ω|a+|ω⁰|a =h(ω) +_Nh(ω⁰) – h(ω×Aω⁰) =h(ω[a7→ω⁰]) =|ω|a|ω⁰|a

– h(s_A(ω)) =h(ω·a) =|ω|_a+ 1 =S_N(h(ω))

Ifσ:X−→A, whereAis a F-algebra, then there exists a unique morphism : bσ:T(F, X)−→As.t. bσ(x) =σ(x) for allx∈X

Theorem 1(Birkhoff)

Proof.

bσ(t) is constructed by structural induction on t, with : – σ(x) =b σ(x)

– σ(fb (t1, ..., tn)) =fA(bσ(t1), ...,σ(tb n))

Examples.

x+s(y)

σ=







X −→AΣ

x7−→ab y 7−→b thenbσ(x+s(y)) =abba

σ⁰=







X −→N x7−→1 y7−→2 thenbσ(x+s(y)) = 4

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Remarks.

– ˆσ(t) is denotedJtKσ, A, ortσ

– σ:X −→Ais called aninterpretation – σ:X −→T(F, X) is asubstitution

– Dom(σ) ={x|xσ6=x} if Dom(σ) ={x₁, ..., x_n},σis denoted{x₁7→t₁, ..., x_n7→t_n}

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