Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment Equality Revisited Interpretation of equality Usually, we require that the equality predicate = is interpreted as same-ness. 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment Equality Revisited Interpretation of equality Usually, we require that the equality predicate = is interpreted as same-ness. Extensionality restriction This means that allowable models are restricted to those in which a = M b holds if and only if a and b are the same elements of the model’s universe. 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment Satisfaction Relation The model M satisfies φ with respect to environment l , written M | = l φ : 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment Satisfaction Relation The model M satisfies φ with respect to environment l , written M | = l φ : in case φ is of the form P ( t 1 , t 2 , . . . , t n ) , if a 1 , a 2 , . . . , a n are the results of evaluating t 1 , t 2 , . . . , t n with respect to l , and if P M ( a 1 , a 2 , . . . , a n ) = T ; 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment Satisfaction Relation The model M satisfies φ with respect to environment l , written M | = l φ : in case φ is of the form P ( t 1 , t 2 , . . . , t n ) , if a 1 , a 2 , . . . , a n are the results of evaluating t 1 , t 2 , . . . , t n with respect to l , and if P M ( a 1 , a 2 , . . . , a n ) = T ; in case φ is of the form P , if P M = T ; 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment Satisfaction Relation The model M satisfies φ with respect to environment l , written M | = l φ : in case φ is of the form P ( t 1 , t 2 , . . . , t n ) , if a 1 , a 2 , . . . , a n are the results of evaluating t 1 , t 2 , . . . , t n with respect to l , and if P M ( a 1 , a 2 , . . . , a n ) = T ; in case φ is of the form P , if P M = T ; in case φ has the form ∀ x ψ , if the M | = l [ x �→ a ] ψ holds for all a ∈ A ; 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment Satisfaction Relation The model M satisfies φ with respect to environment l , written M | = l φ : in case φ is of the form P ( t 1 , t 2 , . . . , t n ) , if a 1 , a 2 , . . . , a n are the results of evaluating t 1 , t 2 , . . . , t n with respect to l , and if P M ( a 1 , a 2 , . . . , a n ) = T ; in case φ is of the form P , if P M = T ; in case φ has the form ∀ x ψ , if the M | = l [ x �→ a ] ψ holds for all a ∈ A ; in case φ has the form ∃ x ψ , if the M | = l [ x �→ a ] ψ holds for some a ∈ A ; 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment Satisfaction Relation (continued) in case φ has the form ¬ ψ , if M | = l ψ does not hold; 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment Satisfaction Relation (continued) in case φ has the form ¬ ψ , if M | = l ψ does not hold; in case φ has the form ψ 1 ∨ ψ 2 , if M | = l ψ 1 holds or M | = l ψ 2 holds; 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment Satisfaction Relation (continued) in case φ has the form ¬ ψ , if M | = l ψ does not hold; in case φ has the form ψ 1 ∨ ψ 2 , if M | = l ψ 1 holds or M | = l ψ 2 holds; in case φ has the form ψ 1 ∧ ψ 2 , if M | = l ψ 1 holds and M | = l ψ 2 holds; and 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment Satisfaction Relation (continued) in case φ has the form ¬ ψ , if M | = l ψ does not hold; in case φ has the form ψ 1 ∨ ψ 2 , if M | = l ψ 1 holds or M | = l ψ 2 holds; in case φ has the form ψ 1 ∧ ψ 2 , if M | = l ψ 1 holds and M | = l ψ 2 holds; and in case φ has the form ψ 1 → ψ 2 , if M | = l ψ 1 holds whenever M | = l ψ 2 holds. 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment Semantic Entailment and Satisfiability Let Γ be a possibly infinite set of formulas in predicate logic and ψ a formula. 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment Semantic Entailment and Satisfiability Let Γ be a possibly infinite set of formulas in predicate logic and ψ a formula. Entailment Γ | = ψ iff for all models M and environments l , whenever M | = l φ holds for all φ ∈ Γ , then M | = l ψ . 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment Semantic Entailment and Satisfiability Let Γ be a possibly infinite set of formulas in predicate logic and ψ a formula. Entailment Γ | = ψ iff for all models M and environments l , whenever M | = l φ holds for all φ ∈ Γ , then M | = l ψ . Satisfiability of Formulas ψ is satisfiable iff there is some model M and some environment l such that M | = l ψ holds. 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment Semantic Entailment and Satisfiability Let Γ be a possibly infinite set of formulas in predicate logic and ψ a formula. Entailment Γ | = ψ iff for all models M and environments l , whenever M | = l φ holds for all φ ∈ Γ , then M | = l ψ . Satisfiability of Formulas ψ is satisfiable iff there is some model M and some environment l such that M | = l ψ holds. Satisfiability of Formula Sets Γ is satisfiable iff there is some model M and some environment l such that M | = l φ , for all φ ∈ Γ . 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment Semantic Entailment and Satisfiability Let Γ be a possibly infinite set of formulas in predicate logic and ψ a formula. Validity ψ is valid iff for all models M and environments l , we have M | = l ψ . 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment The Problem with Predicate Logic Entailment ranges over models Semantic entailment between sentences: φ 1 , φ 2 , . . . , φ n | = ψ requires that in all models that satisfy φ 1 , φ 2 , . . . , φ n , the sentence ψ is satisfied. 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment The Problem with Predicate Logic Entailment ranges over models Semantic entailment between sentences: φ 1 , φ 2 , . . . , φ n | = ψ requires that in all models that satisfy φ 1 , φ 2 , . . . , φ n , the sentence ψ is satisfied. How to effectively argue about all possible models? Usually the number of models is infinite; it is very hard to argue on the semantic level in predicate logic. 05—Predicate Logic II
Review: Syntax and Semantics Predicates, Functions, Terms, Formulas Proof Theory Models Equivalences and Properties Satisfaction and Entailment The Problem with Predicate Logic Entailment ranges over models Semantic entailment between sentences: φ 1 , φ 2 , . . . , φ n | = ψ requires that in all models that satisfy φ 1 , φ 2 , . . . , φ n , the sentence ψ is satisfied. How to effectively argue about all possible models? Usually the number of models is infinite; it is very hard to argue on the semantic level in predicate logic. Idea from propositional logic Can we use natural deduction for showing entailment? 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Review: Syntax and Semantics 1 Proof Theory 2 Equality Universal Quantification Existential Quantification Equivalences and Properties 3 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Natural Deduction for Predicate Logic Relationship between propositional and predicate logic If we consider propositions as nullary predicates, propositional logic is a sub-language of predicate logic. CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Natural Deduction for Predicate Logic Relationship between propositional and predicate logic If we consider propositions as nullary predicates, propositional logic is a sub-language of predicate logic. Inheriting natural deduction We can translate the rules for natural deduction in propositional logic directly to predicate logic. CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Natural Deduction for Predicate Logic Relationship between propositional and predicate logic If we consider propositions as nullary predicates, propositional logic is a sub-language of predicate logic. Inheriting natural deduction We can translate the rules for natural deduction in propositional logic directly to predicate logic. Example φ ψ [ ∧ i ] φ ∧ ψ CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Built-in Rules for Equality t i = t 2 [ x ⇒ t 1 ] φ [= i ] [= e ] t = t [ x ⇒ t 2 ] φ CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Properties of Equality We show: f ( x ) = g ( x ) ⊢ h ( g ( x )) = h ( f ( x )) using t 1 = t 2 [ x ⇒ t 1 ] φ [= i ] [= e ] t = t [ x ⇒ t 2 ] φ CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Properties of Equality We show: f ( x ) = g ( x ) ⊢ h ( g ( x )) = h ( f ( x )) using t 1 = t 2 [ x ⇒ t 1 ] φ [= i ] [= e ] t = t [ x ⇒ t 2 ] φ 1 f ( x ) = g ( x ) premise 2 h ( f ( x )) = h ( f ( x )) = i 3 h ( g ( x )) = h ( f ( x )) = e 1,2 CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Elimination of Universal Quantification ∀ x φ [ ∀ x e ] [ x ⇒ t ] φ CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Elimination of Universal Quantification ∀ x φ [ ∀ x e ] [ x ⇒ t ] φ Once you have proven ∀ x φ , you can replace x by any term t in φ CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Elimination of Universal Quantification ∀ x φ [ ∀ x e ] [ x ⇒ t ] φ Once you have proven ∀ x φ , you can replace x by any term t in φ , provided that t is free for x in φ . CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Example ∀ x φ [ ∀ x e ] [ x ⇒ t ] φ We prove: S ( g ( john )) , ∀ x ( S ( x ) → ¬ L ( x )) ⊢ ¬ L ( g ( john )) CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Example ∀ x φ [ ∀ x e ] [ x ⇒ t ] φ We prove: S ( g ( john )) , ∀ x ( S ( x ) → ¬ L ( x )) ⊢ ¬ L ( g ( john )) S ( g ( john )) 1 premise 2 ∀ x ( S ( x ) → ¬ L ( x )) premise 3 S ( g ( john )) → ¬ L ( g ( john )) ∀ x e 2 4 ¬ L ( g ( john )) → e 3,1 CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Introduction of Universal Quantification ✄ � x 0 . . . [ x ⇒ x 0 ] φ ✂ ✁ [ ∀ x i ] ∀ x φ CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Introduction of Universal Quantification ✄ � x 0 . . . [ x ⇒ x 0 ] φ ✂ ✁ [ ∀ x i ] ∀ x φ If we manage to establish a formula φ about a fresh variable x 0 , we can assume ∀ x φ . CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Introduction of Universal Quantification ✄ � x 0 . . . [ x ⇒ x 0 ] φ ✂ ✁ [ ∀ x i ] ∀ x φ If we manage to establish a formula φ about a fresh variable x 0 , we can assume ∀ x φ . The variable x 0 must be fresh ; we cannot introduce the same variable twice in nested boxes. CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Example ✄ � x 0 . . . [ x ⇒ x 0 ] φ ✂ ✁ ∀ x ( P ( x ) → Q ( x )) , ∀ xP ( x ) ⊢ ∀ xQ ( x ) via ∀ x φ CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Example ✄ � x 0 . . . [ x ⇒ x 0 ] φ ✂ ✁ ∀ x ( P ( x ) → Q ( x )) , ∀ xP ( x ) ⊢ ∀ xQ ( x ) via ∀ x φ 1 ∀ x ( P ( x ) → Q ( x )) premise 2 ∀ xP ( x ) premise 3 P ( x 0 ) → Q ( x 0 ) ∀ x e 1 x 0 4 P ( x 0 ) ∀ x e 2 5 Q ( x 0 ) → e 3,4 6 ∀ xQ ( x ) ∀ x i 3–5 CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Introduction of Existential Quantification [ x ⇒ t ] φ [ ∃ x i ] ∃ x φ CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Introduction of Existential Quantification [ x ⇒ t ] φ [ ∃ x i ] ∃ x φ In order to prove ∃ x φ , it suffices to find a term t as “witness” CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Introduction of Existential Quantification [ x ⇒ t ] φ [ ∃ x i ] ∃ x φ In order to prove ∃ x φ , it suffices to find a term t as “witness”, provided that t is free for x in φ . CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Example ∀ x φ ⊢ ∃ x φ CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Example ∀ x φ ⊢ ∃ x φ Recall: Definition of Models A model M for ( F , P ) consists of: A non-empty set U , the universe ; 1 ... 2 CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Example ∀ x φ ⊢ ∃ x φ Recall: Definition of Models A model M for ( F , P ) consists of: A non-empty set U , the universe ; 1 ... 2 Remark Compare this with Traditional Logic (Coq Quiz 1). CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Example ∀ x φ ⊢ ∃ x φ Recall: Definition of Models A model M for ( F , P ) consists of: A non-empty set U , the universe ; 1 ... 2 Remark Compare this with Traditional Logic (Coq Quiz 1). Because U must not be empty, we should be able to prove the sequent above. CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Example (continued) ∀ x φ ⊢ ∃ x φ CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Example (continued) ∀ x φ ⊢ ∃ x φ 1 ∀ x φ premise 2 [ x ⇒ x ] φ ∀ x e 1 3 ∃ x φ ∃ x i 2 CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Elimination of Existential Quantification ✞ ☎ x 0 [ x ⇒ x 0 ] φ . . ∃ x φ . χ [ x ⇒ x 0 ] φ ✝ ✆ [ ∃ e ] χ CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Elimination of Existential Quantification ✞ ☎ x 0 [ x ⇒ x 0 ] φ . . ∃ x φ . χ [ x ⇒ x 0 ] φ ✝ ✆ [ ∃ e ] χ Making use of ∃ If we know ∃ x φ , we know that there exist at least one object x for which φ holds. CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Elimination of Existential Quantification ✞ ☎ x 0 [ x ⇒ x 0 ] φ . . ∃ x φ . χ [ x ⇒ x 0 ] φ ✝ ✆ [ ∃ e ] χ Making use of ∃ If we know ∃ x φ , we know that there exist at least one object x for which φ holds. We call that element x 0 , and assume [ x ⇒ x 0 ] φ . CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Elimination of Existential Quantification ✞ ☎ x 0 [ x ⇒ x 0 ] φ . . ∃ x φ . χ [ x ⇒ x 0 ] φ ✝ ✆ [ ∃ e ] χ Making use of ∃ If we know ∃ x φ , we know that there exist at least one object x for which φ holds. We call that element x 0 , and assume [ x ⇒ x 0 ] φ . Without assumptions on x 0 , we prove χ ( x 0 not in χ ). CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Example ∀ x ( P ( x ) → Q ( x )) , ∃ xP ( x ) ⊢ ∃ xQ ( x ) 1 ∀ x ( P ( x ) → Q ( x )) premise 2 ∃ xP ( x ) premise 3 P ( x 0 ) assumption x 0 4 P ( x 0 ) → Q ( x 0 ) ∀ x e 1 5 Q ( x 0 ) → e 4,3 6 ∃ xQ ( x ) ∃ x i 5 7 ∃ xQ ( x ) ∃ x e 2,3–6 CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Example ∀ x ( P ( x ) → Q ( x )) , ∃ xP ( x ) ⊢ ∃ xQ ( x ) 1 ∀ x ( P ( x ) → Q ( x )) premise 2 ∃ xP ( x ) premise 3 P ( x 0 ) assumption x 0 4 P ( x 0 ) → Q ( x 0 ) ∀ x e 1 5 Q ( x 0 ) → e 4,3 6 ∃ xQ ( x ) ∃ x i 5 7 ∃ xQ ( x ) ∃ x e 2,3–6 Note that ∃ xQ ( x ) within the box does not contain x 0 , and therefore can be “exported” from the box. CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Another Example 1 ∀ x ( Q ( x ) → R ( x )) premise 2 ∃ x ( P ( x ) ∧ Q ( x )) premise 3 P ( x 0 ) ∧ Q ( x 0 ) assumption x 0 4 Q ( x 0 ) → R ( x 0 ) ∀ x e 1 5 Q ( x 0 ) ∧ e 2 3 6 R ( x 0 ) → e 4,5 7 P ( x 0 ) ∧ e 1 3 8 P ( x 0 ) ∧ R ( x 0 ) ∧ i 7, 6 ∃ x ( P ( x ) ∧ R ( x ) ∃ x i 8 9 10 ∃ x ( P ( x ) ∧ R ( x )) ∃ x e 2,3–9 CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Equality Proof Theory Universal Quantification Equivalences and Properties Existential Quantification Variables must be fresh! This is not a proof! 1 ∃ xP ( x ) premise 2 ∀ x ( P ( x ) → Q ( x )) premise 3 x 0 4 P ( x 0 ) assumption x 0 5 P ( x 0 ) → Q ( x 0 ) ∀ x e 2 6 Q ( x 0 ) → e 5,4 7 Q ( x 0 ) ∃ x e 1, 4–6 8 ∀ yQ ( y ) ∀ y i 3–7 CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness Review: Syntax and Semantics 1 Proof Theory 2 Equivalences and Properties 3 Quantifier Equivalences Soundness and Completeness Undecidability, Compactness CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness Equivalences Two-way-provable We write φ ⊣⊢ ψ iff φ ⊢ ψ and also ψ ⊢ φ . CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness Equivalences Two-way-provable We write φ ⊣⊢ ψ iff φ ⊢ ψ and also ψ ⊢ φ . Some simple equivalences ¬∀ x φ ⊣⊢ ∃ x ¬ φ CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness Equivalences Two-way-provable We write φ ⊣⊢ ψ iff φ ⊢ ψ and also ψ ⊢ φ . Some simple equivalences ¬∀ x φ ⊣⊢ ∃ x ¬ φ ¬∃ x φ ⊣⊢ ∀ x ¬ φ CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness Equivalences Two-way-provable We write φ ⊣⊢ ψ iff φ ⊢ ψ and also ψ ⊢ φ . Some simple equivalences ¬∀ x φ ⊣⊢ ∃ x ¬ φ ¬∃ x φ ⊣⊢ ∀ x ¬ φ ∀ x ∀ y φ ⊣⊢ ∀ y ∀ x φ CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness Equivalences Two-way-provable We write φ ⊣⊢ ψ iff φ ⊢ ψ and also ψ ⊢ φ . Some simple equivalences ¬∀ x φ ⊣⊢ ∃ x ¬ φ ¬∃ x φ ⊣⊢ ∀ x ¬ φ ∀ x ∀ y φ ⊣⊢ ∀ y ∀ x φ ∃ x ∃ y φ ⊣⊢ ∃ y ∃ x φ CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness Equivalences Two-way-provable We write φ ⊣⊢ ψ iff φ ⊢ ψ and also ψ ⊢ φ . Some simple equivalences ¬∀ x φ ⊣⊢ ∃ x ¬ φ ¬∃ x φ ⊣⊢ ∀ x ¬ φ ∀ x ∀ y φ ⊣⊢ ∀ y ∀ x φ ∃ x ∃ y φ ⊣⊢ ∃ y ∃ x φ ∀ x φ ∧ ∀ x ψ ⊣⊢ ∀ x ( φ ∧ ψ ) CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness Equivalences Two-way-provable We write φ ⊣⊢ ψ iff φ ⊢ ψ and also ψ ⊢ φ . Some simple equivalences ¬∀ x φ ⊣⊢ ∃ x ¬ φ ¬∃ x φ ⊣⊢ ∀ x ¬ φ ∀ x ∀ y φ ⊣⊢ ∀ y ∀ x φ ∃ x ∃ y φ ⊣⊢ ∃ y ∃ x φ ∀ x φ ∧ ∀ x ψ ⊣⊢ ∀ x ( φ ∧ ψ ) ∃ x φ ∨ ∃ x ψ ⊣⊢ ∃ x ( φ ∨ ψ ) CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness ¬∀ x φ ⊢ ∃ x ¬ φ 1 ¬∀ x φ premise 2 ¬∃ x ¬ φ assumption 3 x 0 4 ¬ [ x ⇒ x 0 ] φ assumption 5 ∃ x ¬ φ ∃ x i 4 6 ⊥ ¬ e 5, 2 7 [ x ⇒ x 0 ] φ PBC 4–6 8 ∀ x φ ∀ x i 3–7 9 ⊥ ¬ e 8, 1 ∃ x ¬ φ 10 PBC 2–9 CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness ∃ x ∃ y φ ⊢ ∃ y ∃ x φ Assume that x and y are different variables. CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness ∃ x ∃ y φ ⊢ ∃ y ∃ x φ Assume that x and y are different variables. 1 ∃ x ∃ y φ premise 2 [ x ⇒ x 0 ]( ∃ y φ ) assumption x 0 3 ∃ y ([ x ⇒ x 0 ] φ def of subst ( x , y different) 4 [ y ⇒ y 0 ][ x ⇒ x 0 ] φ assumption y 0 5 [ x ⇒ x 0 ][ y ⇒ y 0 ] φ def of subst ( x , y , x 0 , y 0 different) 6 ∃ x [ y → y 0 ] φ ∃ x i 5 7 ∃ y ∃ x φ ∃ y i 6 8 ∃ y ∃ x φ ∃ y e 3, 4–7 9 ∃ y ∃ x φ ∃ x e 1, 2–8 CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness More Equivalences Assume that x is not free in ψ ∀ x φ ∧ ψ ⊣⊢ ∀ x ( φ ∧ ψ ) ∀ x φ ∨ ψ ⊣⊢ ∀ x ( φ ∨ ψ ) ∃ x φ ∧ ψ ⊣⊢ ∃ x ( φ ∧ ψ ) ∃ x φ ∨ ψ ⊣⊢ ∃ x ( φ ∨ ψ ) CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness Central Result of Natural Deduction φ 1 , . . . , φ n | = ψ iff φ 1 , . . . , φ n ⊢ ψ proven by Kurt G¨ odel, in 1929 in his doctoral dissertation CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness Recall: Decidability Decision problems A decision problem is a question in some formal system with a yes-or-no answer. CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness Recall: Decidability Decision problems A decision problem is a question in some formal system with a yes-or-no answer. Decidability Decision problems for which there is an algorithm that returns “yes” whenever the answer to the problem is “yes”, and that returns “no” whenever the answer to the problem is “no”, are called decidable . CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness Recall: Decidability Decision problems A decision problem is a question in some formal system with a yes-or-no answer. Decidability Decision problems for which there is an algorithm that returns “yes” whenever the answer to the problem is “yes”, and that returns “no” whenever the answer to the problem is “no”, are called decidable . Decidability of satisfiability The question, whether a given propositional formula is satisifiable, is decidable. CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness Undecidability of Predicate Logic Theorem The decision problem of validity in predicate logic is undecidable: no program exists which, given any language in predicate logic and any formula φ in that language, decides whether | = φ . CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness Undecidability of Predicate Logic Theorem The decision problem of validity in predicate logic is undecidable: no program exists which, given any language in predicate logic and any formula φ in that language, decides whether | = φ . Proof sketch Establish that the Post Correspondence Problem (PCP) is undecidable CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness Undecidability of Predicate Logic Theorem The decision problem of validity in predicate logic is undecidable: no program exists which, given any language in predicate logic and any formula φ in that language, decides whether | = φ . Proof sketch Establish that the Post Correspondence Problem (PCP) is undecidable Translate an arbitrary PCP , say C , to a formula φ . CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness Undecidability of Predicate Logic Theorem The decision problem of validity in predicate logic is undecidable: no program exists which, given any language in predicate logic and any formula φ in that language, decides whether | = φ . Proof sketch Establish that the Post Correspondence Problem (PCP) is undecidable Translate an arbitrary PCP , say C , to a formula φ . Establish that | = φ holds if and only if C has a solution. CS 3234: Logic and Formal Systems 05—Predicate Logic II
Review: Syntax and Semantics Quantifier Equivalences Proof Theory Soundness and Completeness Equivalences and Properties Undecidability, Compactness Undecidability of Predicate Logic Theorem The decision problem of validity in predicate logic is undecidable: no program exists which, given any language in predicate logic and any formula φ in that language, decides whether | = φ . Proof sketch Establish that the Post Correspondence Problem (PCP) is undecidable Translate an arbitrary PCP , say C , to a formula φ . Establish that | = φ holds if and only if C has a solution. Conclude that validity of predicate logic formulas is undecidable. CS 3234: Logic and Formal Systems 05—Predicate Logic II
Recommend
More recommend
