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First-order logic syntax and semantics First-order logic is also known as predicate logic and predicate calculus Signature Signature n ≥ 1. Σ is a family of sets ΣFn , for n ≥ 0 and sets ΣRn , for Signature Signature n ≥ 1. Σ Elements of is a family of sets ΣFn are symbols of ΣFn , for n ≥ 0 and sets ΣRn , for n-argument operations. Signature Signature n ≥ 1. Σ is a family of sets ΣFn ΣFn , for n ≥ 0 and sets ΣRn , for are symbols of n-argument operations. Elements of are symbols of n-argument relations. Equality does not belong to Elements of ΣR n sign = Σ. Signature Signature n ≥ 1. Σ is a family of sets ΣFn ΣFn , for n ≥ 0 and sets ΣRn , for are symbols of n-argument operations. Elements of are symbols of n-argument relations. Equality does not belong to Elements of ΣR n sign = Σ. If the signature is nite and the arities are known, it is often presented as a sequence of symbols, e.g., +, ·, 0, 1 Variables and terms We x a countably innite set X of individual variables. Variables and terms We x a countably innite set The set of terms TΣ (X ) X of individual variables. over signature Σ and variable set X: Variables and terms We x a countably innite set The set of terms TΣ (X ) X of individual variables. over signature Individual variables are terms Σ and variable set X: Variables and terms We x a countably innite set The set of terms TΣ (X ) X of individual variables. over signature Individual variables are terms Σ and variable set X: n ≥ 0 and every symbol f ∈ ΣFn , if t1 , . . . , tn are terms, then f (t1 , . . . , tn ) is also a term For every Variables and terms We x a countably innite set The set of terms TΣ (X ) X of individual variables. over signature Individual variables are terms Σ and variable set X: n ≥ 0 and every symbol f ∈ ΣFn , if t1 , . . . , tn are terms, then f (t1 , . . . , tn ) is also a term The set FV (t ) of variables occurring in t : For every Variables and terms We x a countably innite set The set of terms TΣ (X ) X of individual variables. over signature Individual variables are terms Σ and variable set X: n ≥ 0 and every symbol f ∈ ΣFn , if t1 , . . . , tn are terms, then f (t1 , . . . , tn ) is also a term The set FV (t ) of variables occurring in t : FV (x ) = {x }. For every Variables and terms We x a countably innite set The set of terms TΣ (X ) X of individual variables. over signature Individual variables are terms Σ and variable set X: n ≥ 0 and every symbol f ∈ ΣFn , if t1 , . . . , tn are terms, then f (t1 , . . . , tn ) is also a term The set FV (t ) of variables occurring in t : FV (x ) = {x }. S FV (f (t1 , . . . , tn )) = ni=1 FV (ti ). For every Atomic formulas The set of atomic formulas over Σ and X: Atomic formulas The set of atomic formulas over Falsity ⊥ Σ is an atomic formula and X: Atomic formulas The set of atomic formulas over Falsity ⊥ Σ is an atomic formula and X: n ≥ 1, each symbol r ∈ ΣRn , and each n terms t1 , . . . , tn ∈ TΣ (X ), the expression r (t1 , . . . , tn ) is an atomic For each formula Atomic formulas The set of atomic formulas over Falsity ⊥ Σ and X: is an atomic formula n ≥ 1, each symbol r ∈ ΣRn , and each n terms t1 , . . . , tn ∈ TΣ (X ), the expression r (t1 , . . . , tn ) is an atomic For each formula For each two terms formula t1 , t2 , the expression (t1 = t2 ) is an atomic Formulas The set of formulas over Σ and X: Formulas The set of formulas over Σ and X: Each atomic formula is a formula Formulas The set of formulas over Σ and X: Each atomic formula is a formula If ϕ, ψ are formulas, then (ϕ → ψ) is a formula Formulas The set of formulas over Σ and X: Each atomic formula is a formula If ϕ, ψ If ϕ are formulas, then is a formula and formula (ϕ → ψ) x ∈X is a formula is a variable, then (∀x ϕ) is a Free variables of a formula The set of free variables FV (ϕ) of a formula ϕ: Free variables of a formula The set of free variables FV (⊥) = ∅; FV (ϕ) of a formula ϕ: Free variables of a formula The set of free variables FV (ϕ) of a formula ϕ: FV (⊥) = ∅; S FV (r (t1 , . . . , tn )) = ni=1 FV (ti ); Free variables of a formula The set of free variables FV (ϕ) of a formula ϕ: FV (⊥) = ∅; S FV (r (t1 , . . . , tn )) = ni=1 FV (ti ); FV (t1 = t2 ) = FV (t1 ) ∪ FV (t2 ); Free variables of a formula The set of free variables FV (ϕ) of a formula ϕ: FV (⊥) = ∅; S FV (r (t1 , . . . , tn )) = ni=1 FV (ti ); FV (t1 = t2 ) = FV (t1 ) ∪ FV (t2 ); FV (ϕ → ψ) = FV (ϕ) ∪ FV (ψ); Free variables of a formula The set of free variables FV (ϕ) of a formula ϕ: FV (⊥) = ∅; S FV (r (t1 , . . . , tn )) = ni=1 FV (ti ); FV (t1 = t2 ) = FV (t1 ) ∪ FV (t2 ); FV (ϕ → ψ) = FV (ϕ) ∪ FV (ψ); FV (∀x ϕ) = FV (ϕ) − {x }. Free variables of a formula The set of free variables FV (ϕ) of a formula ϕ: FV (⊥) = ∅; S FV (r (t1 , . . . , tn )) = ni=1 FV (ti ); FV (t1 = t2 ) = FV (t1 ) ∪ FV (t2 ); FV (ϕ → ψ) = FV (ϕ) ∪ FV (ψ); FV (∀x ϕ) = FV (ϕ) − {x }. A formula without quantiers is an open formula. A formula without free variables is a sentence, or a closed formula. Syntax abbreviations Additional propositional connectives are abbreviations: (¬ϕ) ϕ→⊥ for (ϕ ∨ ψ) for ((¬ϕ) → ψ) (ϕ ∧ ψ) for (¬((¬ϕ) ∨ (¬ψ))) (ϕ ↔ ψ) for ((ϕ → ψ) ∧ (ψ → ϕ)) The existential quantier is an abbreviation, too: (∃x ϕ) means (¬(∀x ¬ϕ)). Free vs. bound occurrences of a variable Each variable occurrence in an atomic formula is a free one Free vs. bound occurrences of a variable Each variable occurrence in an atomic formula is a free one Free (bound) occurrences in formula ϕ → ψ. ϕ and ψ remain free (bound) in the Free vs. bound occurrences of a variable Each variable occurrence in an atomic formula is a free one Free (bound) occurrences in formula ϕ and ψ remain free (bound) in the ϕ → ψ. Free occurrences of x in ϕ become bound in ∀x ϕ. Occurrences of other variables in this formula do not change their status Free vs. bound occurrences of a variable Each variable occurrence in an atomic formula is a free one Free (bound) occurrences in formula ϕ and ψ remain free (bound) in the ϕ → ψ. Free occurrences of x in ϕ become bound in ∀x ϕ. Occurrences of other variables in this formula do not change their status The distinction between free and bound variables resembles the distinction between local and global variables in a procedure Semantics of formulas A structure A over Σ consists of Semantics of formulas A structure a A over Σ consists of nonempty set A: the carrier or universe of A Semantics of formulas A structure a A over Σ consists of nonempty set A: the carrier or universe of an interpretation of each symbol f A : An → A f ∈ ΣFn A as an n-ary function Semantics of formulas A structure a A over Σ consists of nonempty set A: the carrier or universe of A an interpretation of each symbol f ∈ ΣFn as an n-ary function an interpretation of each symbol r ∈ ΣRn as an n-ary relation f A : An → A r A ⊆ An Semantics of formulas A structure a A over Σ consists of nonempty set A: the carrier or universe of A an interpretation of each symbol f ∈ ΣFn as an n-ary function an interpretation of each symbol r ∈ ΣRn as an n-ary relation f A : An → A r A ⊆ An A i, A = hA, f1A , . . . , fnA , r1A , . . . , rm f1 , . . . , fn , r1 , . . . , rm are the symbols in the Notation: where signature Valuation A valuation in a Σ-structure A is a function %:X →A Valuation A valuation in a is a function %:X →A x ∈ X and an element a valuation %x : X → A: ( %(y ) y 6= x a %x (y ) = For a valuation a modied Σ-structure A %, a variable a otherwise a ∈ A we dene Values of terms The value of a term % is denoted [[t ]]A % or t ∈ TΣ (X ) in a Σ-structure A under valuation [[t ]]% . Values of terms The value of a term % is denoted [[t ]]A % or [[x ]]A % = %(x ). t ∈ TΣ (X ) in a Σ-structure A under valuation [[t ]]% . Values of terms The value of a term % is denoted [[t ]]A % or [[x ]]A % = %(x ). t ∈ TΣ (X ) in a Σ-structure A under valuation [[t ]]% . A A A [[f (t1 , . . . , tn )]]A % = f ([[t1 ]]% , . . . , [[tn ]]% ). Formula satisfaction (A, %) |= ϕ is read: The formula valuation ϕ is satised in the structure A under the %. The formula The formula ϕ is true in the structure A under the valuation %. ϕ holds in the structure A under the valuation %. Formula satisfaction (A, %) |= ϕ is read: The formula valuation ϕ is satised in the structure A under the %. The formula The formula (A, %) |= ⊥ ϕ is true in the structure A under the valuation %. ϕ holds in the structure A under the valuation %. does not hold Formula satisfaction (A, %) |= ϕ is read: The formula valuation ϕ is satised in the structure A under the %. The formula The formula (A, %) |= ⊥ ϕ is true in the structure A under the valuation %. ϕ holds in the structure A under the valuation %. does not hold n ≥ 1, r ∈ ΣRn and terms t1 , . . . , tn A A (A, %) |= r (t1 , . . . , tn ) i h[[t1 ]]A % , . . . [[tn ]]% i ∈ r . For Formula satisfaction (A, %) |= ϕ is read: The formula valuation ϕ is satised in the structure A under the %. The formula The formula (A, %) |= ⊥ ϕ is true in the structure A under the valuation %. ϕ holds in the structure A under the valuation %. does not hold n ≥ 1, r ∈ ΣRn and terms t1 , . . . , tn A A (A, %) |= r (t1 , . . . , tn ) i h[[t1 ]]A % , . . . [[tn ]]% i ∈ r . A (A, %) |= t1 = t2 , i [[t1 ]]A % = [[t2 ]]% . For Formula satisfaction (A, %) |= ϕ is read: The formula valuation ϕ is satised in the structure A under the %. The formula The formula (A, %) |= ⊥ ϕ is true in the structure A under the valuation %. ϕ holds in the structure A under the valuation %. does not hold n ≥ 1, r ∈ ΣRn and terms t1 , . . . , tn A A (A, %) |= r (t1 , . . . , tn ) i h[[t1 ]]A % , . . . [[tn ]]% i ∈ r . A (A, %) |= t1 = t2 , i [[t1 ]]A % = [[t2 ]]% . For (A, %) |= (ϕ → ψ), holds if (A, %) |= ϕ does not hold or (A, %) |= ψ Formula satisfaction (A, %) |= ϕ is read: The formula valuation ϕ is satised in the structure A under the %. The formula The formula (A, %) |= ⊥ ϕ is true in the structure A under the valuation %. ϕ holds in the structure A under the valuation %. does not hold n ≥ 1, r ∈ ΣRn and terms t1 , . . . , tn A A (A, %) |= r (t1 , . . . , tn ) i h[[t1 ]]A % , . . . [[tn ]]% i ∈ r . A (A, %) |= t1 = t2 , i [[t1 ]]A % = [[t2 ]]% . For (A, %) |= (ϕ → ψ), if (A, %) |= ϕ holds (A, %) |= (∀x ϕ) i for every does not hold or (A, %) |= ψ a ∈ A holds (A, %ax ) |= ϕ. Satisfaction does not depend on non-free variables Fact For any Σ-structure A and any formula ϕ, assign equal values to all free variables od (A, %) |= ϕ i if valuations ϕ, then (A, %0 ) |= ϕ. % and %0 Satisfaction does not depend on non-free variables Fact For any Σ-structure A and any formula ϕ, assign equal values to all free variables od (A, %) |= ϕ i if valuations ϕ, % and %0 then (A, %0 ) |= ϕ. Hence simplied notation: (A, x : a, y : b) |= ϕ instead of (A, %) |= ϕ, when free variables in If ϕ %(x ) = a and %(y ) = b, and there are no other ϕ is a sentence, then the valuation can be disregarded. Satisfaction does not depend on non-free variables Fact For any Σ-structure A and any formula ϕ, assign equal values to all free variables od (A, %) |= ϕ i if valuations ϕ, % and %0 then (A, %0 ) |= ϕ. Hence simplied notation: (A, x : a, y : b) |= ϕ instead of (A, %) |= ϕ, when free variables in If ϕ %(x ) = a and %(y ) = b, and there are no other ϕ is a sentence, then the valuation can be disregarded. Hence notation A |= ϕ Isomorphism of structures Given are two structures A = hA, . . .i and B = hB , . . .i over Σ Isomorphism of structures Given are two structures h:A→B h:A∼ = B) if: Function A = hA, . . .i and is an isomorphism of B = hB , . . .i Σ-structures over Σ (denoted Isomorphism of structures Given are two structures A = hA, . . .i and B = hB , . . .i over Σ h : A → B is an isomorphism of Σ-structures (denoted h:A∼ = B) if: h is a bijection (onto and 1-1) Function Isomorphism of structures Given are two structures A = hA, . . .i and B = hB , . . .i over Σ h : A → B is an isomorphism of Σ-structures (denoted h:A∼ = B) if: h is a bijection (onto and 1-1) F For n ≥ 0, f ∈ Σn and a1 , . . . , an ∈ A Function h(f A (a1 , . . . , an )) = f B (h(a1 ), . . . , h(an )) Isomorphism of structures Given are two structures A = hA, . . .i and B = hB , . . .i over Σ h : A → B is an isomorphism of Σ-structures (denoted h:A∼ = B) if: h is a bijection (onto and 1-1) F For n ≥ 0, f ∈ Σn and a1 , . . . , an ∈ A Function h(f A (a1 , . . . , an )) = f B (h(a1 ), . . . , h(an )) For n ≥ 1, r ∈ ΣRn and a1 , . . . , an ∈ A r A (a1 , . . . , an ) i r B (h(a1 ), . . . , h(an )) Properties of isomorphisms Composition of two isomorphisms is an isomorphism Properties of isomorphisms Composition of two isomorphisms is an isomorphism The reverse function of an isomorphism is an isomorphism Properties of isomorphisms Composition of two isomorphisms is an isomorphism The reverse function of an isomorphism is an isomorphism Identity idA : A → A is an isomorphism idA : A ∼ =A Isomorphic structures A onto B A∼ =B If there exists an isomorphism from are said to be isomorphic, denoted then these structures Isomorphic structures A onto B A∼ =B If there exists an isomorphism from are said to be isomorphic, denoted The relation of isomorphism is transitive then these structures Isomorphic structures A onto B A∼ =B If there exists an isomorphism from are said to be isomorphic, denoted The relation of isomorphism is transitive symmetrical then these structures Isomorphic structures A onto B A∼ =B If there exists an isomorphism from are said to be isomorphic, denoted The relation of isomorphism is transitive symmetrical reexive then these structures Isomorphisms and logic Theorem If h:A∼ = B then for every formula ϕ (A, %) |= ϕ i (B, h ◦ %) |= ϕ Isomorphisms and logic Theorem If h:A∼ = B then for every formula ϕ (A, %) |= ϕ If i (B, h ◦ %) |= ϕ x1 , . . . , xn are the free variables of ϕ, then (A, x1 : a1 , . . . , xn : an ) |= ϕ i (B, x1 : h(a1 ), . . . , xn : h(an )) |= ϕ Isomorphisms and logic cntd. Theorem If A∼ =B then for every sentence A |= ϕ ϕ i B |= ϕ Elementary equivalence A and B sentence are elementary equivalent (denoted ϕ A ≡ B), i for every of rst-order logic over their common signature, if and only if B |= ϕ A |= ϕ Elementary equivalence A and B sentence are elementary equivalent (denoted ϕ B |= ϕ Corollary A∼ =B i for every of rst-order logic over their common signature, if and only if If A ≡ B), to A ≡ B. A |= ϕ Elementary equivalence A and B sentence are elementary equivalent (denoted ϕ A ≡ B), i for every of rst-order logic over their common signature, if and only if A |= ϕ B |= ϕ Corollary If A∼ =B to A ≡ B. Intuitively, isomorphic structures are logically indistinguishable Validity and satisability of formulas A formula such that ϕ is satisable (A, %) |= ϕ. in A, if there exists a valuation % in A Validity and satisability of formulas A formula such that A formula ϕ is satisable (A, %) |= ϕ. ϕ is satisable in A, if there exists a valuation is satisable, if there exists a structure A, % in A in which ϕ Validity and satisability of formulas A formula such that A formula ϕ is satisable (A, %) |= ϕ. ϕ in A, if there exists a valuation is satisable, if there exists a structure A, % in in which is satisable ϕ is true (satised, valid) in valuation % in A A, if (A, %) |= ϕ A holds for every ϕ Validity and satisability of sentences A sentence is valid ϕ is satisable if there exists a structure A, in which ϕ Validity and satisability of sentences A sentence ϕ is satisable if there exists a structure is valid A is then said to be a model of ϕ (denoted A |= ϕ) A, in which ϕ Validity and satisability of sentences A sentence ϕ is satisable if there exists a structure A, in which ϕ is valid A is then said to be a model of ϕ Σ-structure A is a model of a set A |= ϕ holds for every ϕ ∈ Γ. if (denoted A |= ϕ) of sentences Γ (denoted A |= Γ), Validity and satisability of sentences A sentence ϕ is satisable if there exists a structure A, in which ϕ is valid A is then said to be a model of ϕ Σ-structure A is a model of a set A |= ϕ holds for every ϕ ∈ Γ. (denoted A |= ϕ) of sentences Γ (denoted A |= Γ), if Sentence ϕ Σ-structure is a tautology (denoted |= ϕ), if it is valid in every The rst proof: theorem If h:A∼ = B then for every formula ϕ (A, %) |= ϕ i (B, h ◦ %) |= ϕ The rst proof: Lemma If h:A∼ = B then for every term t h([[t ]]A% ) = [[t ]]Bh◦% Induction: The rst proof: Lemma If h:A∼ = B then for every term t h([[t ]]A% ) = [[t ]]Bh◦% Induction: If t is x , then the thesis h(%(x )) = (h ◦ %)(x ) holds The rst proof: Lemma If h:A∼ = B then for every term t h([[t ]]A% ) = [[t ]]Bh◦% Induction: x , then the thesis h(%(x )) = (h ◦ %)(x ) holds If t to f (t1 , . . . , tn ) to If t is h([[f (t1 , . . . , tn )]]A% = h(f A ([[t1 ]]A% , . . . , [[tn ]]A% )) A = f B (h([[t1 ]]A % ), . . . , h([[tn ]]% )) B = f B ([[t1 ]]B h◦% , . . . , [[tn ]]h◦% ) = [[f (t1 , . . . , tn )]]B h◦% The rst proof: atomic formulas The rst proof: atomic formulas (A, %) 6|= ⊥ and (B, h ◦ %) 6|= ⊥ The rst proof: atomic formulas (A, %) 6|= ⊥ and (B, h ◦ %) 6|= ⊥ (A, %) |= r (t1 , . . . , tn ) i i i i A A h[[t1 ]]A % , . . . , [[tn ]]% i ∈ r A B hh([[t1 ]]A % ), . . . , h([[tn ]]% )i ∈ r B B h[[t1 ]]B h◦% , . . . , [[tn ]]h◦% i ∈ r (B, h ◦ %) |= r (t1 , . . . , tn ) The rst proof: atomic formulas (A, %) 6|= ⊥ and (B, h ◦ %) 6|= ⊥ (A, %) |= r (t1 , . . . , tn ) i i i i (A, %) |= t1 = t2 A A h[[t1 ]]A % , . . . , [[tn ]]% i ∈ r A B hh([[t1 ]]A % ), . . . , h([[tn ]]% )i ∈ r B B h[[t1 ]]B h◦% , . . . , [[tn ]]h◦% i ∈ r (B, h ◦ %) |= r (t1 , . . . , tn ) i A [[t1 ]]A % = [[t2 ]]% h([[t1 ]]A% ) = h([[t2 ]]A% ) B B i [[t1 ]]h◦% = [[t2 ]]h◦% i (B, h ◦ %) |= t1 = t2 i The rst proof: compound formulas The rst proof: compound formulas (A, %) |= (ϕ → ψ) i (A, %) 6|= ϕ i (B, h ◦ %) 6|= ϕ i or (A, %) |= ψ or (B, h ◦ %) |= ψ (B, h ◦ %) |= (ϕ → ψ) The rst proof: compound formulas (A, %) |= (ϕ → ψ) i (A, %) 6|= ϕ i (B, h ◦ %) 6|= ϕ i (A, %) |= (∀x ϕ) or (A, %) |= ψ or (B, h ◦ %) |= ψ (B, h ◦ %) |= (ϕ → ψ) a ∈ A holds (A, %ax ) |= ϕ a i for all a ∈ A holds (B, h ◦ (%x )) |= ϕ h(a) i for all h (a) ∈ B holds (B, (h ◦ %)x ) |= ϕ b i for all b ∈ B holds (B, (h ◦ %)x ) |= ϕ i (B, h ◦ %) |= (∀x ϕ) i for all Term substitution ϕ(t /x ) variable is the result of substituting x in ϕ. t for every free occurrence of a Term substitution ϕ(t /x ) variable is the result of substituting x Example: in ϕ. Formulas ∀y (y ≤ x ) ∀z (z ≤ x ) express the same property t for every free occurrence of a Term substitution ϕ(t /x ) variable is the result of substituting x Example: in t for every free occurrence of a ϕ. Formulas ∀y (y ≤ x ) ∀z (z ≤ x ) express the same property y for x ∀y (y ≤ y ) ∀z (z ≤ y ). Substituting which are dierent in those formulas yields Permissible substitution ⊥(t /x ) = ⊥; Permissible substitution ⊥(t /x ) = ⊥; r (t1 , . . . , tn )(t /x ) = r (t1 (t /x ), . . . , tn (t /x )); Permissible substitution ⊥(t /x ) = ⊥; r (t1 , . . . , tn )(t /x ) = r (t1 (t /x ), . . . , tn (t /x )); (t1 = t2 )(t /x ) = (t1 (t /x ) = t2 (t /x )); Permissible substitution ⊥(t /x ) = ⊥; r (t1 , . . . , tn )(t /x ) = r (t1 (t /x ), . . . , tn (t /x )); (t1 = t2 )(t /x ) = (t1 (t /x ) = t2 (t /x )); (ϕ → ψ)(t /x ) = ϕ(t /x ) → ψ(t /x ); Permissible substitution ⊥(t /x ) = ⊥; r (t1 , . . . , tn )(t /x ) = r (t1 (t /x ), . . . , tn (t /x )); (t1 = t2 )(t /x ) = (t1 (t /x ) = t2 (t /x )); (ϕ → ψ)(t /x ) = ϕ(t /x ) → ψ(t /x ); (∀x ϕ)(t /x ) = ∀x ϕ; Permissible substitution ⊥(t /x ) = ⊥; r (t1 , . . . , tn )(t /x ) = r (t1 (t /x ), . . . , tn (t /x )); (t1 = t2 )(t /x ) = (t1 (t /x ) = t2 (t /x )); (ϕ → ψ)(t /x ) = ϕ(t /x ) → ψ(t /x ); (∀x ϕ)(t /x ) = ∀x ϕ; (∀y ϕ)(t /x ) = ∀y ϕ(t /x ), when y 6= x , and y 6∈ FV (t ); Permissible substitution ⊥(t /x ) = ⊥; r (t1 , . . . , tn )(t /x ) = r (t1 (t /x ), . . . , tn (t /x )); (t1 = t2 )(t /x ) = (t1 (t /x ) = t2 (t /x )); (ϕ → ψ)(t /x ) = ϕ(t /x ) → ψ(t /x ); (∀x ϕ)(t /x ) = ∀x ϕ; (∀y ϕ)(t /x ) = ∀y ϕ(t /x ), when y 6= x , and y 6∈ FV (t ); otherwise the substitution is not permissible Substitution lemma Let A t be any structure be any term %:X →A be any valuation in A Substitution lemma Let A t be any structure %:X →A be any valuation in be any term Then: For any term s and any variable x A [[s (t /x )]]A % = [[s ]]%ax where a = [[t ]]A% . A Substitution lemma Let A t be any structure %:X →A be any valuation in A be any term Then: For any term s and any variable x A [[s (t /x )]]A % = [[s ]]%ax where a = [[t ]]A% . For any formula ϕ, if term t is permissible for (A, %) |= ϕ(t /x ) where a = [[t ]]A% . i x (A, %ax ) |= ϕ, in ϕ, then