Satisfiability and validity of any first-order formulae A first-order formula A is: • A is satisfiable if S , v | = A for some structure S and some variable assignment v in S . • (logically) valid, denoted | = A , if S , v | = A for every structure S and every variable assignment v in S . • falsifiable, if it is not logically valid. Let A = A ( x 1 , . . . , x n ) be any first-order formula all free variables in which are amongst x 1 , . . . , x n . The sentence ∃ x 1 . . . ∃ x n A ( x 1 , . . . , x n ) is a existential closure of A ; Goranko
Satisfiability and validity of any first-order formulae A first-order formula A is: • A is satisfiable if S , v | = A for some structure S and some variable assignment v in S . • (logically) valid, denoted | = A , if S , v | = A for every structure S and every variable assignment v in S . • falsifiable, if it is not logically valid. Let A = A ( x 1 , . . . , x n ) be any first-order formula all free variables in which are amongst x 1 , . . . , x n . The sentence ∃ x 1 . . . ∃ x n A ( x 1 , . . . , x n ) is a existential closure of A ; the sentence ∀ x 1 . . . ∀ x n A ( x 1 , . . . , x n ) is a universal closure of A . Goranko
Satisfiability and validity of any first-order formulae A first-order formula A is: • A is satisfiable if S , v | = A for some structure S and some variable assignment v in S . • (logically) valid, denoted | = A , if S , v | = A for every structure S and every variable assignment v in S . • falsifiable, if it is not logically valid. Let A = A ( x 1 , . . . , x n ) be any first-order formula all free variables in which are amongst x 1 , . . . , x n . The sentence ∃ x 1 . . . ∃ x n A ( x 1 , . . . , x n ) is a existential closure of A ; the sentence ∀ x 1 . . . ∀ x n A ( x 1 , . . . , x n ) is a universal closure of A . Claim: Goranko
Satisfiability and validity of any first-order formulae A first-order formula A is: • A is satisfiable if S , v | = A for some structure S and some variable assignment v in S . • (logically) valid, denoted | = A , if S , v | = A for every structure S and every variable assignment v in S . • falsifiable, if it is not logically valid. Let A = A ( x 1 , . . . , x n ) be any first-order formula all free variables in which are amongst x 1 , . . . , x n . The sentence ∃ x 1 . . . ∃ x n A ( x 1 , . . . , x n ) is a existential closure of A ; the sentence ∀ x 1 . . . ∀ x n A ( x 1 , . . . , x n ) is a universal closure of A . Claim: • A ( x 1 , . . . , x n ) is satisfiable iff ∃ x 1 . . . ∃ x n A ( x 1 , . . . , x n ) is satisfiable. Goranko
Satisfiability and validity of any first-order formulae A first-order formula A is: • A is satisfiable if S , v | = A for some structure S and some variable assignment v in S . • (logically) valid, denoted | = A , if S , v | = A for every structure S and every variable assignment v in S . • falsifiable, if it is not logically valid. Let A = A ( x 1 , . . . , x n ) be any first-order formula all free variables in which are amongst x 1 , . . . , x n . The sentence ∃ x 1 . . . ∃ x n A ( x 1 , . . . , x n ) is a existential closure of A ; the sentence ∀ x 1 . . . ∀ x n A ( x 1 , . . . , x n ) is a universal closure of A . Claim: • A ( x 1 , . . . , x n ) is satisfiable iff ∃ x 1 . . . ∃ x n A ( x 1 , . . . , x n ) is satisfiable. • | = A ( x 1 , . . . , x n ) iff | = ∀ x 1 . . . ∀ x n A ( x 1 , . . . , x n ). Goranko
Satisfiability and validity of formulae: examples Goranko
Satisfiability and validity of formulae: examples • The formula P ( x ) is satisfiable, because its existential closure ∃ xP ( x ) is satisfiable. Goranko
Satisfiability and validity of formulae: examples • The formula P ( x ) is satisfiable, because its existential closure ∃ xP ( x ) is satisfiable. • However, P ( x ) is not valid, because its universal closure ∀ xP ( x ) is not valid. Goranko
Satisfiability and validity of formulae: examples • The formula P ( x ) is satisfiable, because its existential closure ∃ xP ( x ) is satisfiable. • However, P ( x ) is not valid, because its universal closure ∀ xP ( x ) is not valid. • The formula P ( x ) ∨ ¬ P ( x ) is valid, because ∀ x ( P ( x ) ∨ ¬ P ( x )) is valid. Goranko
Satisfiability and validity of formulae: examples • The formula P ( x ) is satisfiable, because its existential closure ∃ xP ( x ) is satisfiable. • However, P ( x ) is not valid, because its universal closure ∀ xP ( x ) is not valid. • The formula P ( x ) ∨ ¬ P ( x ) is valid, because ∀ x ( P ( x ) ∨ ¬ P ( x )) is valid. • The formula P ( x ) ∧ ¬ P ( x ) is not satisfiable, because its existential closure ∃ x ( P ( x ) ∧ ¬ P ( x )) is not satisfiable. Goranko
First-order instances of propositional formulae Goranko
First-order instances of propositional formulae Any uniform substitution of first-order formulae for the propositional variables in a propositional formula A produces a first-order formula, called a first-order instance of A . Goranko
First-order instances of propositional formulae Any uniform substitution of first-order formulae for the propositional variables in a propositional formula A produces a first-order formula, called a first-order instance of A . Example: take the propositional formula A = ( p ∧ ¬ q ) → ( q ∨ p ) . Goranko
First-order instances of propositional formulae Any uniform substitution of first-order formulae for the propositional variables in a propositional formula A produces a first-order formula, called a first-order instance of A . Example: take the propositional formula A = ( p ∧ ¬ q ) → ( q ∨ p ) . The uniform substitution of ( 5 < x ) for p and ∃ y ( x = y 2 ) for q in A results in the first-order instance (( 5 < x ) ∧ ¬∃ y ( x = y 2 )) → ( ∃ y ( x = y 2 ) ∨ ( 5 < x )) . Goranko
Satisfiability and validity of first-order instances Goranko
Satisfiability and validity of first-order instances • If a propositional formula A is valid (tautology), then every first-order instance of A is valid. Goranko
Satisfiability and validity of first-order instances • If a propositional formula A is valid (tautology), then every first-order instance of A is valid. Thus, for instance, | = ¬¬ ( x > 0 ) → ( x > 0 ) Goranko
Satisfiability and validity of first-order instances • If a propositional formula A is valid (tautology), then every first-order instance of A is valid. Thus, for instance, | = ¬¬ ( x > 0 ) → ( x > 0 ) and | = Q ( x , y ) ∨ ¬ Q ( x , y ) . Goranko
Satisfiability and validity of first-order instances • If a propositional formula A is valid (tautology), then every first-order instance of A is valid. Thus, for instance, | = ¬¬ ( x > 0 ) → ( x > 0 ) and | = Q ( x , y ) ∨ ¬ Q ( x , y ) . What about vice versa? Goranko
Satisfiability and validity of first-order instances • If a propositional formula A is valid (tautology), then every first-order instance of A is valid. Thus, for instance, | = ¬¬ ( x > 0 ) → ( x > 0 ) and | = Q ( x , y ) ∨ ¬ Q ( x , y ) . What about vice versa? Yes. Why? Goranko
Satisfiability and validity of first-order instances • If a propositional formula A is valid (tautology), then every first-order instance of A is valid. Thus, for instance, | = ¬¬ ( x > 0 ) → ( x > 0 ) and | = Q ( x , y ) ∨ ¬ Q ( x , y ) . What about vice versa? Yes. Why? • Likewise, a propositional formula A is satisfiable if and only if some first-order instance of A is satisfiable. Goranko
Satisfiability and validity of first-order instances • If a propositional formula A is valid (tautology), then every first-order instance of A is valid. Thus, for instance, | = ¬¬ ( x > 0 ) → ( x > 0 ) and | = Q ( x , y ) ∨ ¬ Q ( x , y ) . What about vice versa? Yes. Why? • Likewise, a propositional formula A is satisfiable if and only if some first-order instance of A is satisfiable. Why? Goranko
Logical consequence in first order logic Goranko
Logical consequence in first order logic We fix an arbitrary first-order language L . Goranko
Logical consequence in first order logic We fix an arbitrary first-order language L . Given a set of L -formulae Γ, an L -structure S , and a variable assignment v in S , we write S , v | = Γ to say that S , v | = A for every A ∈ Γ. Goranko
Logical consequence in first order logic We fix an arbitrary first-order language L . Given a set of L -formulae Γ, an L -structure S , and a variable assignment v in S , we write S , v | = Γ to say that S , v | = A for every A ∈ Γ. A formula A follows logically from a set of formulae Γ, denoted Γ | = A , if for every structure S and a variable assignment v : VAR →S : S , v | = Γ implies S , v | = A . Goranko
Logical consequence in first order logic We fix an arbitrary first-order language L . Given a set of L -formulae Γ, an L -structure S , and a variable assignment v in S , we write S , v | = Γ to say that S , v | = A for every A ∈ Γ. A formula A follows logically from a set of formulae Γ, denoted Γ | = A , if for every structure S and a variable assignment v : VAR →S : S , v | = Γ implies S , v | = A . Note that ∅ | = A iff | = A . Why? Goranko
Logical consequence: examples Goranko
Logical consequence: examples • If A 1 , . . . , A n , B are prop. formulae such that A 1 , . . . , A n | = B , Goranko
Logical consequence: examples • If A 1 , . . . , A n , B are prop. formulae such that A 1 , . . . , A n | = B , and n , B ′ are first-order instances of A 1 , . . . , A n , B obtained by A ′ 1 , . . . , A ′ the same substitution, Goranko
Logical consequence: examples • If A 1 , . . . , A n , B are prop. formulae such that A 1 , . . . , A n | = B , and n , B ′ are first-order instances of A 1 , . . . , A n , B obtained by A ′ 1 , . . . , A ′ the same substitution, then A ′ 1 , . . . , A ′ = B ′ . n | Goranko
Logical consequence: examples • If A 1 , . . . , A n , B are prop. formulae such that A 1 , . . . , A n | = B , and n , B ′ are first-order instances of A 1 , . . . , A n , B obtained by A ′ 1 , . . . , A ′ the same substitution, then A ′ 1 , . . . , A ′ = B ′ . n | For example: Goranko
Logical consequence: examples • If A 1 , . . . , A n , B are prop. formulae such that A 1 , . . . , A n | = B , and n , B ′ are first-order instances of A 1 , . . . , A n , B obtained by A ′ 1 , . . . , A ′ the same substitution, then A ′ 1 , . . . , A ′ = B ′ . n | For example: ∃ xA , ∃ xA → ∀ yB | = ∀ yB . Goranko
Logical consequence: examples • If A 1 , . . . , A n , B are prop. formulae such that A 1 , . . . , A n | = B , and n , B ′ are first-order instances of A 1 , . . . , A n , B obtained by A ′ 1 , . . . , A ′ the same substitution, then A ′ 1 , . . . , A ′ = B ′ . n | For example: ∃ xA , ∃ xA → ∀ yB | = ∀ yB . • ∀ xP ( x ) , ∀ x ( P ( x ) → Q ( x )) | = ∀ xQ ( x ). Goranko
Logical consequence: examples • If A 1 , . . . , A n , B are prop. formulae such that A 1 , . . . , A n | = B , and n , B ′ are first-order instances of A 1 , . . . , A n , B obtained by A ′ 1 , . . . , A ′ the same substitution, then A ′ 1 , . . . , A ′ = B ′ . n | For example: ∃ xA , ∃ xA → ∀ yB | = ∀ yB . • ∀ xP ( x ) , ∀ x ( P ( x ) → Q ( x )) | = ∀ xQ ( x ). NB: this is not an instance of a propositional logical consequence. Goranko
Logical consequence: examples • If A 1 , . . . , A n , B are prop. formulae such that A 1 , . . . , A n | = B , and n , B ′ are first-order instances of A 1 , . . . , A n , B obtained by A ′ 1 , . . . , A ′ the same substitution, then A ′ 1 , . . . , A ′ = B ′ . n | For example: ∃ xA , ∃ xA → ∀ yB | = ∀ yB . • ∀ xP ( x ) , ∀ x ( P ( x ) → Q ( x )) | = ∀ xQ ( x ). NB: this is not an instance of a propositional logical consequence. • ∃ xP ( x ) ∧ ∃ xQ ( x ) �| = ∃ x ( P ( x ) ∧ Q ( x )). Goranko
Logical consequence: examples • If A 1 , . . . , A n , B are prop. formulae such that A 1 , . . . , A n | = B , and n , B ′ are first-order instances of A 1 , . . . , A n , B obtained by A ′ 1 , . . . , A ′ the same substitution, then A ′ 1 , . . . , A ′ = B ′ . n | For example: ∃ xA , ∃ xA → ∀ yB | = ∀ yB . • ∀ xP ( x ) , ∀ x ( P ( x ) → Q ( x )) | = ∀ xQ ( x ). NB: this is not an instance of a propositional logical consequence. • ∃ xP ( x ) ∧ ∃ xQ ( x ) �| = ∃ x ( P ( x ) ∧ Q ( x )). Indeed, the structure N ′ obtained from N where P ( x ) is interpreted as ‘ x is even ’ and Q ( x ) is interpreted as ‘ x is odd ’ is a counter-model: N ′ | = ∃ xP ( x ) ∧ ∃ xQ ( x ) Goranko
Logical consequence: examples • If A 1 , . . . , A n , B are prop. formulae such that A 1 , . . . , A n | = B , and n , B ′ are first-order instances of A 1 , . . . , A n , B obtained by A ′ 1 , . . . , A ′ the same substitution, then A ′ 1 , . . . , A ′ = B ′ . n | For example: ∃ xA , ∃ xA → ∀ yB | = ∀ yB . • ∀ xP ( x ) , ∀ x ( P ( x ) → Q ( x )) | = ∀ xQ ( x ). NB: this is not an instance of a propositional logical consequence. • ∃ xP ( x ) ∧ ∃ xQ ( x ) �| = ∃ x ( P ( x ) ∧ Q ( x )). Indeed, the structure N ′ obtained from N where P ( x ) is interpreted as ‘ x is even ’ and Q ( x ) is interpreted as ‘ x is odd ’ is a counter-model: N ′ | = ∃ xP ( x ) ∧ ∃ xQ ( x ), while N ′ �| = ∃ x ( P ( x ) ∧ Q ( x )). Goranko
Logical consequence: some basic properties Goranko
Logical consequence: some basic properties Logical equivalence in first-order logic satisfies all basic properties of propositional logical consequence. In particular, the following are equivalent: Goranko
Logical consequence: some basic properties Logical equivalence in first-order logic satisfies all basic properties of propositional logical consequence. In particular, the following are equivalent: 1. A 1 , . . . , A n | = B . Goranko
Logical consequence: some basic properties Logical equivalence in first-order logic satisfies all basic properties of propositional logical consequence. In particular, the following are equivalent: 1. A 1 , . . . , A n | = B . 2. A 1 ∧ · · · ∧ A n | = B . Goranko
Logical consequence: some basic properties Logical equivalence in first-order logic satisfies all basic properties of propositional logical consequence. In particular, the following are equivalent: 1. A 1 , . . . , A n | = B . 2. A 1 ∧ · · · ∧ A n | = B . 3. | = A 1 ∧ · · · ∧ A n → B . Goranko
Logical consequence: some basic properties Logical equivalence in first-order logic satisfies all basic properties of propositional logical consequence. In particular, the following are equivalent: 1. A 1 , . . . , A n | = B . 2. A 1 ∧ · · · ∧ A n | = B . 3. | = A 1 ∧ · · · ∧ A n → B . 4. | = A 1 → ( A 2 → · · · ( A n → B ) . . . ). Goranko
Logical consequence: some basic properties Logical equivalence in first-order logic satisfies all basic properties of propositional logical consequence. In particular, the following are equivalent: 1. A 1 , . . . , A n | = B . 2. A 1 ∧ · · · ∧ A n | = B . 3. | = A 1 ∧ · · · ∧ A n → B . 4. | = A 1 → ( A 2 → · · · ( A n → B ) . . . ). Furthermore, for any first-order formula A and a term t that is free for substitution for x in A : Goranko
Logical consequence: some basic properties Logical equivalence in first-order logic satisfies all basic properties of propositional logical consequence. In particular, the following are equivalent: 1. A 1 , . . . , A n | = B . 2. A 1 ∧ · · · ∧ A n | = B . 3. | = A 1 ∧ · · · ∧ A n → B . 4. | = A 1 → ( A 2 → · · · ( A n → B ) . . . ). Furthermore, for any first-order formula A and a term t that is free for substitution for x in A : 1. ∀ xA | = A [ t / x ]. Goranko
Logical consequence: some basic properties Logical equivalence in first-order logic satisfies all basic properties of propositional logical consequence. In particular, the following are equivalent: 1. A 1 , . . . , A n | = B . 2. A 1 ∧ · · · ∧ A n | = B . 3. | = A 1 ∧ · · · ∧ A n → B . 4. | = A 1 → ( A 2 → · · · ( A n → B ) . . . ). Furthermore, for any first-order formula A and a term t that is free for substitution for x in A : 1. ∀ xA | = A [ t / x ]. 2. A [ t / x ] | = ∃ xA . Goranko
First-order logical consequence: more basic properties Goranko
First-order logical consequence: more basic properties 1. If A 1 , . . . , A n | = B then ∀ xA 1 , . . . , ∀ xA n | = ∀ xB . Goranko
First-order logical consequence: more basic properties 1. If A 1 , . . . , A n | = B then ∀ xA 1 , . . . , ∀ xA n | = ∀ xB . 2. If A 1 , . . . , A n | = B and x does not occur free in A 1 , . . . , A n then A 1 , . . . , A n | = ∀ xB . Goranko
First-order logical consequence: more basic properties 1. If A 1 , . . . , A n | = B then ∀ xA 1 , . . . , ∀ xA n | = ∀ xB . 2. If A 1 , . . . , A n | = B and x does not occur free in A 1 , . . . , A n then A 1 , . . . , A n | = ∀ xB . 3. If A 1 , . . . , A n | = B and A 1 , . . . , A n are sentences, then A 1 , . . . , A n | = ∀ xB , and hence A 1 , . . . , A n | = B , where B is any universal closure of B . Goranko
First-order logical consequence: more basic properties 1. If A 1 , . . . , A n | = B then ∀ xA 1 , . . . , ∀ xA n | = ∀ xB . 2. If A 1 , . . . , A n | = B and x does not occur free in A 1 , . . . , A n then A 1 , . . . , A n | = ∀ xB . 3. If A 1 , . . . , A n | = B and A 1 , . . . , A n are sentences, then A 1 , . . . , A n | = ∀ xB , and hence A 1 , . . . , A n | = B , where B is any universal closure of B . 4. If A 1 , . . . , A n | = B [ c / x ], where c is a constant symbol not occurring in A 1 , . . . , A n , B , then A 1 , . . . , A n | = ∀ xB ( x ). Goranko
First-order logical consequence: more basic properties 1. If A 1 , . . . , A n | = B then ∀ xA 1 , . . . , ∀ xA n | = ∀ xB . 2. If A 1 , . . . , A n | = B and x does not occur free in A 1 , . . . , A n then A 1 , . . . , A n | = ∀ xB . 3. If A 1 , . . . , A n | = B and A 1 , . . . , A n are sentences, then A 1 , . . . , A n | = ∀ xB , and hence A 1 , . . . , A n | = B , where B is any universal closure of B . 4. If A 1 , . . . , A n | = B [ c / x ], where c is a constant symbol not occurring in A 1 , . . . , A n , B , then A 1 , . . . , A n | = ∀ xB ( x ). 5. If A 1 , . . . , A n , A [ c / x ] | = B , where c is a constant symbol not occurring in A 1 , . . . , A n , A , or B , then A 1 , . . . , A n , ∃ xA | = B . Goranko
Equality in first-order logic Goranko
Equality in first-order logic Goranko
Equality in first-order logic The equality symbol =, sometimes also called identity, is a special binary relational symbol, always meant to be interpreted as the identity of objects in the domain of discourse. Goranko
Equality in first-order logic The equality symbol =, sometimes also called identity, is a special binary relational symbol, always meant to be interpreted as the identity of objects in the domain of discourse. The equality is very useful to specify constraints on the size of the model: Goranko
Equality in first-order logic The equality symbol =, sometimes also called identity, is a special binary relational symbol, always meant to be interpreted as the identity of objects in the domain of discourse. The equality is very useful to specify constraints on the size of the model: 1. The sentence � λ n = ∃ x 1 · · · ∃ x n ( ¬ x i = x j ) 1 ≤ i � = j ≤ n states that the domain has at least n elements. Goranko
Equality in first-order logic The equality symbol =, sometimes also called identity, is a special binary relational symbol, always meant to be interpreted as the identity of objects in the domain of discourse. The equality is very useful to specify constraints on the size of the model: 1. The sentence � λ n = ∃ x 1 · · · ∃ x n ( ¬ x i = x j ) 1 ≤ i � = j ≤ n states that the domain has at least n elements. 2. The sentence µ n = ¬ λ n +1 or, equivalently, � µ n = ∀ x 1 · · · ∀ x n +1 ( x i = x j ) 1 ≤ i � = j ≤ n +1 states that the domain has at most n elements. Goranko
Equality in first-order logic The equality symbol =, sometimes also called identity, is a special binary relational symbol, always meant to be interpreted as the identity of objects in the domain of discourse. The equality is very useful to specify constraints on the size of the model: 1. The sentence � λ n = ∃ x 1 · · · ∃ x n ( ¬ x i = x j ) 1 ≤ i � = j ≤ n states that the domain has at least n elements. 2. The sentence µ n = ¬ λ n +1 or, equivalently, � µ n = ∀ x 1 · · · ∀ x n +1 ( x i = x j ) 1 ≤ i � = j ≤ n +1 states that the domain has at most n elements. 3. The sentence σ n = λ n ∧ µ n states that the domain has exactly n elements. Goranko
Equality in first-order logic The equality symbol =, sometimes also called identity, is a special binary relational symbol, always meant to be interpreted as the identity of objects in the domain of discourse. The equality is very useful to specify constraints on the size of the model: 1. The sentence � λ n = ∃ x 1 · · · ∃ x n ( ¬ x i = x j ) 1 ≤ i � = j ≤ n states that the domain has at least n elements. 2. The sentence µ n = ¬ λ n +1 or, equivalently, � µ n = ∀ x 1 · · · ∀ x n +1 ( x i = x j ) 1 ≤ i � = j ≤ n +1 states that the domain has at most n elements. 3. The sentence σ n = λ n ∧ µ n states that the domain has exactly n elements. Proof: exercises. Goranko
Using equality for counting The sentences λ n , µ n , σ n are easily relativised for every formula A ( x ) containing (possibly amongst others) the free variable x to the subset of the domain consisting of the elements satisfying A . Goranko
Using equality for counting The sentences λ n , µ n , σ n are easily relativised for every formula A ( x ) containing (possibly amongst others) the free variable x to the subset of the domain consisting of the elements satisfying A . For instance: • “ At least two students scored distinction in the exam ” can be formalised in the domain of all humans as ∃ x 1 ∃ x 2 ( Student ( x 1 ) ∧ Student ( x 2 ) ∧ ¬ x 1 = x 2 ∧ Dist ( x 1 ) ∧ Dist ( x 2 )) Goranko
Using equality for counting The sentences λ n , µ n , σ n are easily relativised for every formula A ( x ) containing (possibly amongst others) the free variable x to the subset of the domain consisting of the elements satisfying A . For instance: • “ At least two students scored distinction in the exam ” can be formalised in the domain of all humans as ∃ x 1 ∃ x 2 ( Student ( x 1 ) ∧ Student ( x 2 ) ∧ ¬ x 1 = x 2 ∧ Dist ( x 1 ) ∧ Dist ( x 2 )) • “ Exactly two students scored distinction in the exam ” can likewise be formalised as: ∃ x 1 ∃ x 2 ( Student ( x 1 ) ∧ Student ( x 2 ) ∧ x 1 � = x 2 ∧ Dist ( x 1 ) ∧ Dist ( x 2 )) ∧ ∀ y (( Student ( y ) ∧ y � = x 1 ∧ y � = x 2 ) → ¬ Dist ( y )) (where � = stands for ¬ =) Goranko
Using equality to express uniqueness, functionality, etc. 1. For any formula A ( x ) (which may also contain other free variables), the formula ∃ ! xA ( x ) := ( A ( x ) ∧ ∀ y ( A ( y ) → x = y )) states Goranko
Using equality to express uniqueness, functionality, etc. 1. For any formula A ( x ) (which may also contain other free variables), the formula ∃ ! xA ( x ) := ( A ( x ) ∧ ∀ y ( A ( y ) → x = y )) states that there is a unique element in the domain of discourse satisfying A , for the current values of the parameters ¯ z . Goranko
Using equality to express uniqueness, functionality, etc. 1. For any formula A ( x ) (which may also contain other free variables), the formula ∃ ! xA ( x ) := ( A ( x ) ∧ ∀ y ( A ( y ) → x = y )) states that there is a unique element in the domain of discourse satisfying A , for the current values of the parameters ¯ z . 2. Thus, the sentence ∀ x ∃ ! yR ( x , y ) in the language with = and a binary relational symbol R states that the relation R is functional, i.e. every element of the domain is R -related to a unique element. Goranko
Using equality to express uniqueness, functionality, etc. 1. For any formula A ( x ) (which may also contain other free variables), the formula ∃ ! xA ( x ) := ( A ( x ) ∧ ∀ y ( A ( y ) → x = y )) states that there is a unique element in the domain of discourse satisfying A , for the current values of the parameters ¯ z . 2. Thus, the sentence ∀ x ∃ ! yR ( x , y ) in the language with = and a binary relational symbol R states that the relation R is functional, i.e. every element of the domain is R -related to a unique element. 3. The sentence ∀ x ∀ y ( f ( x ) = f ( y ) → x = y )) in the language with = and a unary functional symbol f states that the function f is injective, that is (read by contraposition), assigns different values to different arguments. Goranko
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