Logic and the set theory - Lecture 9: Predicate Calculus

IntroductionAbout this lectureReasoning in predicate calculusS. Choi (KAIST) Logic and set theory October 7, 2012 2 / 26

IntroductionAbout this lectureReasoning in predicate calculusInference rules for the universal quantifiersS. Choi (KAIST) Logic and set theory October 7, 2012 2 / 26

IntroductionAbout this lectureReasoning in predicate calculusInference rules for the universal quantifiersInference rules for the existential quantifiersS. Choi (KAIST) Logic and set theory October 7, 2012 2 / 26

IntroductionAbout this lectureReasoning in predicate calculusInference rules for the universal quantifiersInference rules for the existential quantifiersTheorems and quantifier equivalence rulesS. Choi (KAIST) Logic and set theory October 7, 2012 2 / 26

IntroductionAbout this lectureReasoning in predicate calculusInference rules for the universal quantifiersInference rules for the existential quantifiersTheorems and quantifier equivalence rulesInference rules for the identity predicateS. Choi (KAIST) Logic and set theory October 7, 2012 2 / 26

IntroductionAbout this lectureReasoning in predicate calculusInference rules for the universal quantifiersInference rules for the existential quantifiersTheorems and quantifier equivalence rulesInference rules for the identity predicateCourse homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://KLMS.kaist.ac.krS. Choi (KAIST) Logic and set theory October 7, 2012 2 / 26

IntroductionAbout this lectureReasoning in predicate calculusInference rules for the universal quantifiersInference rules for the existential quantifiersTheorems and quantifier equivalence rulesInference rules for the identity predicateCourse homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://KLMS.kaist.ac.krGrading and so on in the moodle. Ask questions in moodle.S. Choi (KAIST) Logic and set theory October 7, 2012 2 / 26

IntroductionSome helpful referencesSets, Logic and Categories, Peter J. Cameron, Springer. ReadChapters 3,4,5.S. Choi (KAIST) Logic and set theory October 7, 2012 3 / 26

IntroductionSome helpful referencesSets, Logic and Categories, Peter J. Cameron, Springer. ReadChapters 3,4,5.A mathematical introduction to logic, H. Enderton, AcademicPress.S. Choi (KAIST) Logic and set theory October 7, 2012 3 / 26

IntroductionSome helpful referencesSets, Logic and Categories, Peter J. Cameron, Springer. ReadChapters 3,4,5.A mathematical introduction to logic, H. Enderton, AcademicPress.http://plato.stanford.edu/contents.html has muchresource.S. Choi (KAIST) Logic and set theory October 7, 2012 3 / 26

IntroductionSome helpful referencesSets, Logic and Categories, Peter J. Cameron, Springer. ReadChapters 3,4,5.A mathematical introduction to logic, H. Enderton, AcademicPress.http://plato.stanford.edu/contents.html has muchresource.http://ocw.mit.edu/OcwWeb/Linguistics-and-Philosophy/24-241Fall-2005/CourseHome/ See "Monadic PredicateCalculus" and MPC completeness, Predicate Calculus, PredicateCalculus DeriviationsS. Choi (KAIST) Logic and set theory October 7, 2012 3 / 26

IntroductionSome helpful referencesSets, Logic and Categories, Peter J. Cameron, Springer. ReadChapters 3,4,5.A mathematical introduction to logic, H. Enderton, AcademicPress.http://plato.stanford.edu/contents.html has muchresource.http://ocw.mit.edu/OcwWeb/Linguistics-and-Philosophy/24-241Fall-2005/CourseHome/ See "Monadic PredicateCalculus" and MPC completeness, Predicate Calculus, PredicateCalculus Deriviationshttp://philosophy.hku.hk/think/pl/. See Module:Predicate Logic.S. Choi (KAIST) Logic and set theory October 7, 2012 3 / 26

IntroductionSome helpful referencesSets, Logic and Categories, Peter J. Cameron, Springer. ReadChapters 3,4,5.A mathematical introduction to logic, H. Enderton, AcademicPress.http://plato.stanford.edu/contents.html has muchresource.http://ocw.mit.edu/OcwWeb/Linguistics-and-Philosophy/24-241Fall-2005/CourseHome/ See "Monadic PredicateCalculus" and MPC completeness, Predicate Calculus, PredicateCalculus Deriviationshttp://philosophy.hku.hk/think/pl/. See Module:Predicate Logic.http://logic.philosophy.ox.ac.uk/. See "PredicateCalculus" in Tutorial.S. Choi (KAIST) Logic and set theory October 7, 2012 3 / 26

IntroductionSome helpful referenceshttp://en.wikipedia.org/wiki/Truth_table,S. Choi (KAIST) Logic and set theory October 7, 2012 4 / 26

IntroductionSome helpful referenceshttp://en.wikipedia.org/wiki/Truth_table,http://logik.phl.univie.ac.at/~chris/gateway/formular-uk-zentral.html, complete (i.e. has all the steps)S. Choi (KAIST) Logic and set theory October 7, 2012 4 / 26

IntroductionSome helpful referenceshttp://en.wikipedia.org/wiki/Truth_table,http://logik.phl.univie.ac.at/~chris/gateway/formular-uk-zentral.html, complete (i.e. has all the steps)http://svn.oriontransfer.org/TruthTable/index.rhtml,has xor, complete.S. Choi (KAIST) Logic and set theory October 7, 2012 4 / 26

Inference rules for the universal quantifiersInference rules for the universal quantifiersThe inferences rules of propositional calculus hold here also.S. Choi (KAIST) Logic and set theory October 7, 2012 5 / 26

Inference rules for the universal quantifiersInference rules for the universal quantifiersThe inferences rules of propositional calculus hold here also.Here we use ⊢ since this is an inference without models.S. Choi (KAIST) Logic and set theory October 7, 2012 5 / 26

Inference rules for the universal quantifiersInference rules for the universal quantifiersThe inferences rules of propositional calculus hold here also.Here we use ⊢ since this is an inference without models.Universal elimination or universal instantiation ∀E:S. Choi (KAIST) Logic and set theory October 7, 2012 5 / 26

Inference rules for the universal quantifiersInference rules for the universal quantifiersThe inferences rules of propositional calculus hold here also.Here we use ⊢ since this is an inference without models.Universal elimination or universal instantiation ∀E:Given a wff ∀βφ, we may infer φ α/β replacing each occurence of βwith some name letter α. (could be a new one withoutassumptions.)S. Choi (KAIST) Logic and set theory October 7, 2012 5 / 26

Inference rules for the universal quantifiersInference rules for the universal quantifiersThe inferences rules of propositional calculus hold here also.Here we use ⊢ since this is an inference without models.Universal elimination or universal instantiation ∀E:Given a wff ∀βφ, we may infer φ α/β replacing each occurence of βwith some name letter α. (could be a new one withoutassumptions.)Example: ∀x(M 1 x → M 2 x), M 1 S ⊢ M 2 S.S. Choi (KAIST) Logic and set theory October 7, 2012 5 / 26

Inference rules for the universal quantifiersInference rules for the universal quantifiersThe inferences rules of propositional calculus hold here also.Here we use ⊢ since this is an inference without models.Universal elimination or universal instantiation ∀E:Given a wff ∀βφ, we may infer φ α/β replacing each occurence of βwith some name letter α. (could be a new one withoutassumptions.)Example: ∀x(M 1 x → M 2 x), M 1 S ⊢ M 2 S.1. ∀x(M 1 x → M 2 x). A, 2. M 1 S A, 3. M 1 S → M 2 S. 4. M 2 S.S. Choi (KAIST) Logic and set theory October 7, 2012 5 / 26

Inference rules for the universal quantifiersExample¬Fa ⊢ ¬∀xFx.S. Choi (KAIST) Logic and set theory October 7, 2012 6 / 26

Inference rules for the universal quantifiersExample¬Fa ⊢ ¬∀xFx.1. ¬Fa A.S. Choi (KAIST) Logic and set theory October 7, 2012 6 / 26

Inference rules for the universal quantifiersExample¬Fa ⊢ ¬∀xFx.1. ¬Fa A.2.: ∀xFx H.S. Choi (KAIST) Logic and set theory October 7, 2012 6 / 26

Inference rules for the universal quantifiersExample¬Fa ⊢ ¬∀xFx.1. ¬Fa A.2.: ∀xFx H.3.: Fa. 2.S. Choi (KAIST) Logic and set theory October 7, 2012 6 / 26

Inference rules for the universal quantifiersExample¬Fa ⊢ ¬∀xFx.1. ¬Fa A.2.: ∀xFx H.3.: Fa. 2.4.: Fa ∧ ¬Fa. 1.3S. Choi (KAIST) Logic and set theory October 7, 2012 6 / 26

Inference rules for the universal quantifiersExample¬Fa ⊢ ¬∀xFx.1. ¬Fa A.2.: ∀xFx H.3.: Fa. 2.4.: Fa ∧ ¬Fa. 1.35. ¬∀xFx. 1-4S. Choi (KAIST) Logic and set theory October 7, 2012 6 / 26

Inference rules for the universal quantifiersUniversal Introductionφ containing a name letter α without any conditions on it. Wereplace by ∀βφ β/α .S. Choi (KAIST) Logic and set theory October 7, 2012 7 / 26

Inference rules for the universal quantifiersUniversal Introductionφ containing a name letter α without any conditions on it. Wereplace by ∀βφ β/α .Here φ β/α is the result of replacing all occurance of α with β.S. Choi (KAIST) Logic and set theory October 7, 2012 7 / 26

Inference rules for the universal quantifiersUniversal Introductionφ containing a name letter α without any conditions on it. Wereplace by ∀βφ β/α .Here φ β/α is the result of replacing all occurance of α with β.Some restrictions:S. Choi (KAIST) Logic and set theory October 7, 2012 7 / 26

Inference rules for the universal quantifiersUniversal Introductionφ containing a name letter α without any conditions on it. Wereplace by ∀βφ β/α .Here φ β/α is the result of replacing all occurance of α with β.Some restrictions:◮ The name letter α may not appear in any assumptions.S. Choi (KAIST) Logic and set theory October 7, 2012 7 / 26

Inference rules for the universal quantifiersUniversal Introductionφ containing a name letter α without any conditions on it. Wereplace by ∀βφ β/α .Here φ β/α is the result of replacing all occurance of α with β.Some restrictions:◮ The name letter α may not appear in any assumptions.◮ The name letter α may not appear in any hypothesis in effect at theline where φ occurs.S. Choi (KAIST) Logic and set theory October 7, 2012 7 / 26

Inference rules for the universal quantifiersUniversal Introductionφ containing a name letter α without any conditions on it. Wereplace by ∀βφ β/α .Here φ β/α is the result of replacing all occurance of α with β.Some restrictions:◮ The name letter α may not appear in any assumptions.◮ The name letter α may not appear in any hypothesis in effect at theline where φ occurs.◮ φ β/α here is the result of replacing every occurance of α with β.S. Choi (KAIST) Logic and set theory October 7, 2012 7 / 26

Inference rules for the universal quantifiersUniversal Introductionφ containing a name letter α without any conditions on it. Wereplace by ∀βφ β/α .Here φ β/α is the result of replacing all occurance of α with β.Some restrictions:◮ The name letter α may not appear in any assumptions.◮ The name letter α may not appear in any hypothesis in effect at theline where φ occurs.◮ φ β/α here is the result of replacing every occurance of α with β.◮ We can introduce one quantifier at a time.S. Choi (KAIST) Logic and set theory October 7, 2012 7 / 26

Inference rules for the universal quantifiersExamplesFa A. ⊢ ∀xAx. This is incorrect.S. Choi (KAIST) Logic and set theory October 7, 2012 8 / 26

Inference rules for the universal quantifiersExamplesFa A. ⊢ ∀xAx. This is incorrect.∀x(Fx → Gx), Fa → Ga, : Fa : Ga, : ∀xGx, Fa → ∀xGx. This isincorrect.S. Choi (KAIST) Logic and set theory October 7, 2012 8 / 26

Inference rules for the universal quantifiersExamplesFa A. ⊢ ∀xAx. This is incorrect.∀x(Fx → Gx), Fa → Ga, : Fa : Ga, : ∀xGx, Fa → ∀xGx. This isincorrect.∀xLxx A., Laa, ∀zLza. This is incorrect.S. Choi (KAIST) Logic and set theory October 7, 2012 8 / 26

Inference rules for the universal quantifiersExamplesFa A. ⊢ ∀xAx. This is incorrect.∀x(Fx → Gx), Fa → Ga, : Fa : Ga, : ∀xGx, Fa → ∀xGx. This isincorrect.∀xLxx A., Laa, ∀zLza. This is incorrect.∀x(Fx ∧ Gx) ⊢ ∀xFx ∧ ∀xGx.S. Choi (KAIST) Logic and set theory October 7, 2012 8 / 26

Inference rules for the universal quantifiersExamplesFa A. ⊢ ∀xAx. This is incorrect.∀x(Fx → Gx), Fa → Ga, : Fa : Ga, : ∀xGx, Fa → ∀xGx. This isincorrect.∀xLxx A., Laa, ∀zLza. This is incorrect.∀x(Fx ∧ Gx) ⊢ ∀xFx ∧ ∀xGx.Proof: 1. ∀x(Gx ∧ Fx) A.S. Choi (KAIST) Logic and set theory October 7, 2012 8 / 26

Inference rules for the universal quantifiersExamplesFa A. ⊢ ∀xAx. This is incorrect.∀x(Fx → Gx), Fa → Ga, : Fa : Ga, : ∀xGx, Fa → ∀xGx. This isincorrect.∀xLxx A., Laa, ∀zLza. This is incorrect.∀x(Fx ∧ Gx) ⊢ ∀xFx ∧ ∀xGx.Proof: 1. ∀x(Gx ∧ Fx) A.2. Fa ∧ Ga from 1.S. Choi (KAIST) Logic and set theory October 7, 2012 8 / 26

Inference rules for the universal quantifiersExamplesFa A. ⊢ ∀xAx. This is incorrect.∀x(Fx → Gx), Fa → Ga, : Fa : Ga, : ∀xGx, Fa → ∀xGx. This isincorrect.∀xLxx A., Laa, ∀zLza. This is incorrect.∀x(Fx ∧ Gx) ⊢ ∀xFx ∧ ∀xGx.Proof: 1. ∀x(Gx ∧ Fx) A.2. Fa ∧ Ga from 1.3. Fa.S. Choi (KAIST) Logic and set theory October 7, 2012 8 / 26

Inference rules for the universal quantifiersExamplesFa A. ⊢ ∀xAx. This is incorrect.∀x(Fx → Gx), Fa → Ga, : Fa : Ga, : ∀xGx, Fa → ∀xGx. This isincorrect.∀xLxx A., Laa, ∀zLza. This is incorrect.∀x(Fx ∧ Gx) ⊢ ∀xFx ∧ ∀xGx.Proof: 1. ∀x(Gx ∧ Fx) A.2. Fa ∧ Ga from 1.3. Fa.4. Gb.S. Choi (KAIST) Logic and set theory October 7, 2012 8 / 26

Inference rules for the universal quantifiersExamplesFa A. ⊢ ∀xAx. This is incorrect.∀x(Fx → Gx), Fa → Ga, : Fa : Ga, : ∀xGx, Fa → ∀xGx. This isincorrect.∀xLxx A., Laa, ∀zLza. This is incorrect.∀x(Fx ∧ Gx) ⊢ ∀xFx ∧ ∀xGx.Proof: 1. ∀x(Gx ∧ Fx) A.2. Fa ∧ Ga from 1.3. Fa.4. Gb.5. ∀xFx.S. Choi (KAIST) Logic and set theory October 7, 2012 8 / 26

Inference rules for the universal quantifiersExamplesFa A. ⊢ ∀xAx. This is incorrect.∀x(Fx → Gx), Fa → Ga, : Fa : Ga, : ∀xGx, Fa → ∀xGx. This isincorrect.∀xLxx A., Laa, ∀zLza. This is incorrect.∀x(Fx ∧ Gx) ⊢ ∀xFx ∧ ∀xGx.Proof: 1. ∀x(Gx ∧ Fx) A.2. Fa ∧ Ga from 1.3. Fa.4. Gb.5. ∀xFx.6. ∀xGx.S. Choi (KAIST) Logic and set theory October 7, 2012 8 / 26

Inference rules for the universal quantifiersExamplesFa A. ⊢ ∀xAx. This is incorrect.∀x(Fx → Gx), Fa → Ga, : Fa : Ga, : ∀xGx, Fa → ∀xGx. This isincorrect.∀xLxx A., Laa, ∀zLza. This is incorrect.∀x(Fx ∧ Gx) ⊢ ∀xFx ∧ ∀xGx.Proof: 1. ∀x(Gx ∧ Fx) A.2. Fa ∧ Ga from 1.3. Fa.4. Gb.5. ∀xFx.6. ∀xGx.7. ∀xFx ∧ ∀xGx.S. Choi (KAIST) Logic and set theory October 7, 2012 8 / 26

Inference rules for the universal quantifiersExamplesFa A. ⊢ ∀xAx. This is incorrect.∀x(Fx → Gx), Fa → Ga, : Fa : Ga, : ∀xGx, Fa → ∀xGx. This isincorrect.∀xLxx A., Laa, ∀zLza. This is incorrect.∀x(Fx ∧ Gx) ⊢ ∀xFx ∧ ∀xGx.Proof: 1. ∀x(Gx ∧ Fx) A.2. Fa ∧ Ga from 1.3. Fa.4. Gb.5. ∀xFx.6. ∀xGx.7. ∀xFx ∧ ∀xGx.Is the converse true also. How does one prove it?S. Choi (KAIST) Logic and set theory October 7, 2012 8 / 26

Inference rules for the universal quantifiersExamples∀x(Fx → (Gx ∨ Hx)), ∀x¬Gx ⊢ ∀xFx → ∀xHx.S. Choi (KAIST) Logic and set theory October 7, 2012 9 / 26

Inference rules for the universal quantifiersExamples∀x(Fx → (Gx ∨ Hx)), ∀x¬Gx ⊢ ∀xFx → ∀xHx.1. ∀x(Fx → (Gx ∨ Hx)) A.S. Choi (KAIST) Logic and set theory October 7, 2012 9 / 26

Inference rules for the universal quantifiersExamples∀x(Fx → (Gx ∨ Hx)), ∀x¬Gx ⊢ ∀xFx → ∀xHx.1. ∀x(Fx → (Gx ∨ Hx)) A.2. ∀x¬Gx A.S. Choi (KAIST) Logic and set theory October 7, 2012 9 / 26

Inference rules for the universal quantifiersExamples∀x(Fx → (Gx ∨ Hx)), ∀x¬Gx ⊢ ∀xFx → ∀xHx.1. ∀x(Fx → (Gx ∨ Hx)) A.2. ∀x¬Gx A.3. Fa → (Ga ∨ Ha).S. Choi (KAIST) Logic and set theory October 7, 2012 9 / 26

Inference rules for the universal quantifiersExamples∀x(Fx → (Gx ∨ Hx)), ∀x¬Gx ⊢ ∀xFx → ∀xHx.1. ∀x(Fx → (Gx ∨ Hx)) A.2. ∀x¬Gx A.3. Fa → (Ga ∨ Ha).4. ¬Ga.S. Choi (KAIST) Logic and set theory October 7, 2012 9 / 26

Inference rules for the universal quantifiersExamples∀x(Fx → (Gx ∨ Hx)), ∀x¬Gx ⊢ ∀xFx → ∀xHx.1. ∀x(Fx → (Gx ∨ Hx)) A.2. ∀x¬Gx A.3. Fa → (Ga ∨ Ha).4. ¬Ga.5.: ∀xFx H.S. Choi (KAIST) Logic and set theory October 7, 2012 9 / 26

Inference rules for the universal quantifiersExamples∀x(Fx → (Gx ∨ Hx)), ∀x¬Gx ⊢ ∀xFx → ∀xHx.1. ∀x(Fx → (Gx ∨ Hx)) A.2. ∀x¬Gx A.3. Fa → (Ga ∨ Ha).4. ¬Ga.5.: ∀xFx H.6.: Fa.S. Choi (KAIST) Logic and set theory October 7, 2012 9 / 26

Inference rules for the universal quantifiersExamples∀x(Fx → (Gx ∨ Hx)), ∀x¬Gx ⊢ ∀xFx → ∀xHx.1. ∀x(Fx → (Gx ∨ Hx)) A.2. ∀x¬Gx A.3. Fa → (Ga ∨ Ha).4. ¬Ga.5.: ∀xFx H.6.: Fa.7.: Ga ∨ Ha. from 3.S. Choi (KAIST) Logic and set theory October 7, 2012 9 / 26

Inference rules for the universal quantifiersExamples∀x(Fx → (Gx ∨ Hx)), ∀x¬Gx ⊢ ∀xFx → ∀xHx.1. ∀x(Fx → (Gx ∨ Hx)) A.2. ∀x¬Gx A.3. Fa → (Ga ∨ Ha).4. ¬Ga.5.: ∀xFx H.6.: Fa.7.: Ga ∨ Ha. from 3.8.: Ha. 4 7. Disjunctive SyllogismS. Choi (KAIST) Logic and set theory October 7, 2012 9 / 26

Inference rules for the universal quantifiersExamples∀x(Fx → (Gx ∨ Hx)), ∀x¬Gx ⊢ ∀xFx → ∀xHx.1. ∀x(Fx → (Gx ∨ Hx)) A.2. ∀x¬Gx A.3. Fa → (Ga ∨ Ha).4. ¬Ga.5.: ∀xFx H.6.: Fa.7.: Ga ∨ Ha. from 3.8.: Ha. 4 7. Disjunctive Syllogism9.: ∀xHx.S. Choi (KAIST) Logic and set theory October 7, 2012 9 / 26

Inference rules for the universal quantifiersExamples∀x(Fx → (Gx ∨ Hx)), ∀x¬Gx ⊢ ∀xFx → ∀xHx.1. ∀x(Fx → (Gx ∨ Hx)) A.2. ∀x¬Gx A.3. Fa → (Ga ∨ Ha).4. ¬Ga.5.: ∀xFx H.6.: Fa.7.: Ga ∨ Ha. from 3.8.: Ha. 4 7. Disjunctive Syllogism9.: ∀xHx.10. ∀xFx → ∀xHx.S. Choi (KAIST) Logic and set theory October 7, 2012 9 / 26

Inference rules for the universal quantifiersInterchangiblility proof:∀x∀y interchangible to ∀y∀x.S. Choi (KAIST) Logic and set theory October 7, 2012 10 / 26

Inference rules for the universal quantifiersInterchangiblility proof:∀x∀y interchangible to ∀y∀x.∃x∃y interchangible to ∃y∃x.S. Choi (KAIST) Logic and set theory October 7, 2012 10 / 26

Inference rules for the universal quantifiersInterchangiblility proof:∀x∀y interchangible to ∀y∀x.∃x∃y interchangible to ∃y∃x.The first one can be done.S. Choi (KAIST) Logic and set theory October 7, 2012 10 / 26

Inference rules for the universal quantifiersInterchangiblility proof:∀x∀y interchangible to ∀y∀x.∃x∃y interchangible to ∃y∃x.The first one can be done.∀x∀yφ(x, y) ↔ ∀y∀xφ(x, y).S. Choi (KAIST) Logic and set theory October 7, 2012 10 / 26

Inference rules for the universal quantifiersInterchangiblility proof:∀x∀y interchangible to ∀y∀x.∃x∃y interchangible to ∃y∃x.The first one can be done.∀x∀yφ(x, y) ↔ ∀y∀xφ(x, y).Proof follows from the universal instantiations and introductions.S. Choi (KAIST) Logic and set theory October 7, 2012 10 / 26

Inference rules for the universal quantifiersInterchangiblility proof:∀x∀y interchangible to ∀y∀x.∃x∃y interchangible to ∃y∃x.The first one can be done.∀x∀yφ(x, y) ↔ ∀y∀xφ(x, y).Proof follows from the universal instantiations and introductions.The second one is similar and proved after the inference rules forexistential quantifiers.S. Choi (KAIST) Logic and set theory October 7, 2012 10 / 26

Inference rules for existential quantifiersInference rules for existential quantifiersExistential introduction ∃I.S. Choi (KAIST) Logic and set theory October 7, 2012 11 / 26

Inference rules for existential quantifiersInference rules for existential quantifiersExistential introduction ∃I.φ contains α. Replace with ∃βφ β/α . Here we replace someoccurrences of α with β.S. Choi (KAIST) Logic and set theory October 7, 2012 11 / 26

Inference rules for existential quantifiersInference rules for existential quantifiersExistential introduction ∃I.φ contains α. Replace with ∃βφ β/α . Here we replace someoccurrences of α with β.The previous occurrence of α does not matter.S. Choi (KAIST) Logic and set theory October 7, 2012 11 / 26

Inference rules for existential quantifiersInference rules for existential quantifiersExistential introduction ∃I.φ contains α. Replace with ∃βφ β/α . Here we replace someoccurrences of α with β.The previous occurrence of α does not matter.We can introduce one quantifier at a time. (This is also true in ∀I.)S. Choi (KAIST) Logic and set theory October 7, 2012 11 / 26

Inference rules for existential quantifiersInference rules for existential quantifiersExample: ¬∃xFx ⊢ ∀x¬Fx.S. Choi (KAIST) Logic and set theory October 7, 2012 12 / 26

Inference rules for existential quantifiersInference rules for existential quantifiersExample: ¬∃xFx ⊢ ∀x¬Fx.1. ¬∃xFx. A.S. Choi (KAIST) Logic and set theory October 7, 2012 12 / 26

Inference rules for existential quantifiersInference rules for existential quantifiersExample: ¬∃xFx ⊢ ∀x¬Fx.1. ¬∃xFx. A.2.: Fa H. (You can do that ...)S. Choi (KAIST) Logic and set theory October 7, 2012 12 / 26

Inference rules for existential quantifiersInference rules for existential quantifiersExample: ¬∃xFx ⊢ ∀x¬Fx.1. ¬∃xFx. A.2.: Fa H. (You can do that ...)3.: ∃xFx. from 2.S. Choi (KAIST) Logic and set theory October 7, 2012 12 / 26

Inference rules for existential quantifiersInference rules for existential quantifiersExample: ¬∃xFx ⊢ ∀x¬Fx.1. ¬∃xFx. A.2.: Fa H. (You can do that ...)3.: ∃xFx. from 2.4.: ∃xFx ∧ ¬∃xFx.S. Choi (KAIST) Logic and set theory October 7, 2012 12 / 26

Inference rules for existential quantifiersInference rules for existential quantifiersExample: ¬∃xFx ⊢ ∀x¬Fx.1. ¬∃xFx. A.2.: Fa H. (You can do that ...)3.: ∃xFx. from 2.4.: ∃xFx ∧ ¬∃xFx.5. ¬Fa.S. Choi (KAIST) Logic and set theory October 7, 2012 12 / 26

Inference rules for existential quantifiersInference rules for existential quantifiersExample: ¬∃xFx ⊢ ∀x¬Fx.1. ¬∃xFx. A.2.: Fa H. (You can do that ...)3.: ∃xFx. from 2.4.: ∃xFx ∧ ¬∃xFx.5. ¬Fa.6. ∀x¬Fx.S. Choi (KAIST) Logic and set theory October 7, 2012 12 / 26

Inference rules for existential quantifiersInference rules for existential quantifiersExample: ¬∃xFx ⊢ ∀x¬Fx.1. ¬∃xFx. A.2.: Fa H. (You can do that ...)3.: ∃xFx. from 2.4.: ∃xFx ∧ ¬∃xFx.5. ¬Fa.6. ∀x¬Fx.The converse can be proven also.S. Choi (KAIST) Logic and set theory October 7, 2012 12 / 26

Inference rules for existential quantifiersExampleExample ¬∀xFx ⊢ ∃x¬Fx.S. Choi (KAIST) Logic and set theory October 7, 2012 13 / 26

Inference rules for existential quantifiersExampleExample ¬∀xFx ⊢ ∃x¬Fx.1. ¬∀xFx.S. Choi (KAIST) Logic and set theory October 7, 2012 13 / 26

Inference rules for existential quantifiersExampleExample ¬∀xFx ⊢ ∃x¬Fx.1. ¬∀xFx.2.: ¬∃x¬Fx. (H)S. Choi (KAIST) Logic and set theory October 7, 2012 13 / 26

Inference rules for existential quantifiersExampleExample ¬∀xFx ⊢ ∃x¬Fx.1. ¬∀xFx.2.: ¬∃x¬Fx. (H)3.:: ¬Fa (H) (note this also.)S. Choi (KAIST) Logic and set theory October 7, 2012 13 / 26

Inference rules for existential quantifiersExampleExample ¬∀xFx ⊢ ∃x¬Fx.1. ¬∀xFx.2.: ¬∃x¬Fx. (H)3.:: ¬Fa (H) (note this also.)4.:: ∃x¬Fx. 3 (∃I).S. Choi (KAIST) Logic and set theory October 7, 2012 13 / 26

Inference rules for existential quantifiersExampleExample ¬∀xFx ⊢ ∃x¬Fx.1. ¬∀xFx.2.: ¬∃x¬Fx. (H)3.:: ¬Fa (H) (note this also.)4.:: ∃x¬Fx. 3 (∃I).5.:: ∃x¬Fx ∧ ¬∃x¬Fx 2.4.S. Choi (KAIST) Logic and set theory October 7, 2012 13 / 26

Inference rules for existential quantifiersExampleExample ¬∀xFx ⊢ ∃x¬Fx.1. ¬∀xFx.2.: ¬∃x¬Fx. (H)3.:: ¬Fa (H) (note this also.)4.:: ∃x¬Fx. 3 (∃I).5.:: ∃x¬Fx ∧ ¬∃x¬Fx 2.4.6.: Fa. 3-5.S. Choi (KAIST) Logic and set theory October 7, 2012 13 / 26

Inference rules for existential quantifiersExampleExample ¬∀xFx ⊢ ∃x¬Fx.1. ¬∀xFx.2.: ¬∃x¬Fx. (H)3.:: ¬Fa (H) (note this also.)4.:: ∃x¬Fx. 3 (∃I).5.:: ∃x¬Fx ∧ ¬∃x¬Fx 2.4.6.: Fa. 3-5.7.: ∀xFx. 6. (∀I).S. Choi (KAIST) Logic and set theory October 7, 2012 13 / 26

Inference rules for existential quantifiersExampleExample ¬∀xFx ⊢ ∃x¬Fx.1. ¬∀xFx.2.: ¬∃x¬Fx. (H)3.:: ¬Fa (H) (note this also.)4.:: ∃x¬Fx. 3 (∃I).5.:: ∃x¬Fx ∧ ¬∃x¬Fx 2.4.6.: Fa. 3-5.7.: ∀xFx. 6. (∀I).8.: ∀xFx ∧ ¬∀xFx. 1. 7.S. Choi (KAIST) Logic and set theory October 7, 2012 13 / 26

Inference rules for existential quantifiersExampleExample ¬∀xFx ⊢ ∃x¬Fx.1. ¬∀xFx.2.: ¬∃x¬Fx. (H)3.:: ¬Fa (H) (note this also.)4.:: ∃x¬Fx. 3 (∃I).5.:: ∃x¬Fx ∧ ¬∃x¬Fx 2.4.6.: Fa. 3-5.7.: ∀xFx. 6. (∀I).8.: ∀xFx ∧ ¬∀xFx. 1. 7.9. ∃x¬Fx. 2-8.S. Choi (KAIST) Logic and set theory October 7, 2012 13 / 26

Inference rules for existential quantifiersExistential EliminationExistential Elimination ∃E:S. Choi (KAIST) Logic and set theory October 7, 2012 14 / 26

Inference rules for existential quantifiersExistential EliminationExistential Elimination ∃E:∃βφ. We derive φ α/β → ψ. Discharge φ and assert ψ.S. Choi (KAIST) Logic and set theory October 7, 2012 14 / 26

Inference rules for existential quantifiersExistential EliminationExistential Elimination ∃E:∃βφ. We derive φ α/β → ψ. Discharge φ and assert ψ.◮ The name letter α may not have occurred earlier.S. Choi (KAIST) Logic and set theory October 7, 2012 14 / 26

Inference rules for existential quantifiersExistential EliminationExistential Elimination ∃E:∃βφ. We derive φ α/β → ψ. Discharge φ and assert ψ.◮ The name letter α may not have occurred earlier.◮ α may not occur at ψ.S. Choi (KAIST) Logic and set theory October 7, 2012 14 / 26

Inference rules for existential quantifiersExistential EliminationExistential Elimination ∃E:∃βφ. We derive φ α/β → ψ. Discharge φ and assert ψ.◮ The name letter α may not have occurred earlier.◮ α may not occur at ψ.◮ α may not occur in assumptions.S. Choi (KAIST) Logic and set theory October 7, 2012 14 / 26

Inference rules for existential quantifiersExistential EliminationExistential Elimination ∃E:∃βφ. We derive φ α/β → ψ. Discharge φ and assert ψ.◮ The name letter α may not have occurred earlier.◮ α may not occur at ψ.◮ α may not occur in assumptions.◮ α may not occur in hypothesis in effect.S. Choi (KAIST) Logic and set theory October 7, 2012 14 / 26

Inference rules for existential quantifiersExampleExample: ∀x(Fx → Gx), ∃xFx ⊢ ∃xGx.S. Choi (KAIST) Logic and set theory October 7, 2012 15 / 26

Inference rules for existential quantifiersExampleExample: ∀x(Fx → Gx), ∃xFx ⊢ ∃xGx.1. ∀x(Fx → Gx) AS. Choi (KAIST) Logic and set theory October 7, 2012 15 / 26

Inference rules for existential quantifiersExampleExample: ∀x(Fx → Gx), ∃xFx ⊢ ∃xGx.1. ∀x(Fx → Gx) A2. ∃xFx. A.S. Choi (KAIST) Logic and set theory October 7, 2012 15 / 26

Inference rules for existential quantifiersExampleExample: ∀x(Fx → Gx), ∃xFx ⊢ ∃xGx.1. ∀x(Fx → Gx) A2. ∃xFx. A.3.: Fa. for ∃E.S. Choi (KAIST) Logic and set theory October 7, 2012 15 / 26

Inference rules for existential quantifiersExampleExample: ∀x(Fx → Gx), ∃xFx ⊢ ∃xGx.1. ∀x(Fx → Gx) A2. ∃xFx. A.3.: Fa. for ∃E.4.: Fa → Ga. 1,3.S. Choi (KAIST) Logic and set theory October 7, 2012 15 / 26

Inference rules for existential quantifiersExampleExample: ∀x(Fx → Gx), ∃xFx ⊢ ∃xGx.1. ∀x(Fx → Gx) A2. ∃xFx. A.3.: Fa. for ∃E.4.: Fa → Ga. 1,3.5.: Ga.S. Choi (KAIST) Logic and set theory October 7, 2012 15 / 26

Inference rules for existential quantifiersExampleExample: ∀x(Fx → Gx), ∃xFx ⊢ ∃xGx.1. ∀x(Fx → Gx) A2. ∃xFx. A.3.: Fa. for ∃E.4.: Fa → Ga. 1,3.5.: Ga.6.: ∃xGx.S. Choi (KAIST) Logic and set theory October 7, 2012 15 / 26

Inference rules for existential quantifiersExampleExample: ∀x(Fx → Gx), ∃xFx ⊢ ∃xGx.1. ∀x(Fx → Gx) A2. ∃xFx. A.3.: Fa. for ∃E.4.: Fa → Ga. 1,3.5.: Ga.6.: ∃xGx.7. ∃xGx.S. Choi (KAIST) Logic and set theory October 7, 2012 15 / 26

Inference rules for existential quantifiersExampleExample: ∀x¬Fx ⊢ ¬∃xFx.S. Choi (KAIST) Logic and set theory October 7, 2012 16 / 26

Inference rules for existential quantifiersExampleExample: ∀x¬Fx ⊢ ¬∃xFx.1. ∀x¬Fx. A.S. Choi (KAIST) Logic and set theory October 7, 2012 16 / 26

Inference rules for existential quantifiersExampleExample: ∀x¬Fx ⊢ ¬∃xFx.1. ∀x¬Fx. A.2.: ∃xFx. H for ¬I.S. Choi (KAIST) Logic and set theory October 7, 2012 16 / 26

Inference rules for existential quantifiersExampleExample: ∀x¬Fx ⊢ ¬∃xFx.1. ∀x¬Fx. A.2.: ∃xFx. H for ¬I.3.:: Fa. by ∃E.S. Choi (KAIST) Logic and set theory October 7, 2012 16 / 26

Inference rules for existential quantifiersExampleExample: ∀x¬Fx ⊢ ¬∃xFx.1. ∀x¬Fx. A.2.: ∃xFx. H for ¬I.3.:: Fa. by ∃E.4.:: ¬Fa. 1,3.S. Choi (KAIST) Logic and set theory October 7, 2012 16 / 26

Inference rules for existential quantifiersExampleExample: ∀x¬Fx ⊢ ¬∃xFx.1. ∀x¬Fx. A.2.: ∃xFx. H for ¬I.3.:: Fa. by ∃E.4.:: ¬Fa. 1,3.5.:: Fa ∧ ¬Fa.S. Choi (KAIST) Logic and set theory October 7, 2012 16 / 26

Inference rules for existential quantifiersExampleExample: ∀x¬Fx ⊢ ¬∃xFx.1. ∀x¬Fx. A.2.: ∃xFx. H for ¬I.3.:: Fa. by ∃E.4.:: ¬Fa. 1,3.5.:: Fa ∧ ¬Fa.6.: ¬∃xFx.S. Choi (KAIST) Logic and set theory October 7, 2012 16 / 26

Inference rules for existential quantifiersExampleExample: ∀x¬Fx ⊢ ¬∃xFx.1. ∀x¬Fx. A.2.: ∃xFx. H for ¬I.3.:: Fa. by ∃E.4.:: ¬Fa. 1,3.5.:: Fa ∧ ¬Fa.6.: ¬∃xFx.7. ∃xFx → ¬∃xFx. 2-7.S. Choi (KAIST) Logic and set theory October 7, 2012 16 / 26

Inference rules for existential quantifiersExampleExample: ∀x¬Fx ⊢ ¬∃xFx.1. ∀x¬Fx. A.2.: ∃xFx. H for ¬I.3.:: Fa. by ∃E.4.:: ¬Fa. 1,3.5.:: Fa ∧ ¬Fa.6.: ¬∃xFx.7. ∃xFx → ¬∃xFx. 2-7.8. ¬∃xFx. (Use. ¬P → P, ⊢ P or P → ¬P, ⊢ ¬P)S. Choi (KAIST) Logic and set theory October 7, 2012 16 / 26

Inference rules for existential quantifiersExample∃x¬Fx ⊢ ¬∀xFx.S. Choi (KAIST) Logic and set theory October 7, 2012 17 / 26

Inference rules for existential quantifiersExample∃x¬Fx ⊢ ¬∀xFx.1. ∃x¬Fx.S. Choi (KAIST) Logic and set theory October 7, 2012 17 / 26

Inference rules for existential quantifiersExample∃x¬Fx ⊢ ¬∀xFx.1. ∃x¬Fx.2.: ∀xFx. (H)S. Choi (KAIST) Logic and set theory October 7, 2012 17 / 26

Inference rules for existential quantifiersExample∃x¬Fx ⊢ ¬∀xFx.1. ∃x¬Fx.2.: ∀xFx. (H)3.:: ¬Fa. 1. (for ∃E).S. Choi (KAIST) Logic and set theory October 7, 2012 17 / 26

Inference rules for existential quantifiersExample∃x¬Fx ⊢ ¬∀xFx.1. ∃x¬Fx.2.: ∀xFx. (H)3.:: ¬Fa. 1. (for ∃E).4.:: Fa 2.S. Choi (KAIST) Logic and set theory October 7, 2012 17 / 26

Inference rules for existential quantifiersExample∃x¬Fx ⊢ ¬∀xFx.1. ∃x¬Fx.2.: ∀xFx. (H)3.:: ¬Fa. 1. (for ∃E).4.:: Fa 2.5.:: Fa ∧ ¬Fa. 3.4.S. Choi (KAIST) Logic and set theory October 7, 2012 17 / 26

Inference rules for existential quantifiersExample∃x¬Fx ⊢ ¬∀xFx.1. ∃x¬Fx.2.: ∀xFx. (H)3.:: ¬Fa. 1. (for ∃E).4.:: Fa 2.5.:: Fa ∧ ¬Fa. 3.4.6.:: ¬∀xFx.S. Choi (KAIST) Logic and set theory October 7, 2012 17 / 26

Inference rules for existential quantifiersExample∃x¬Fx ⊢ ¬∀xFx.1. ∃x¬Fx.2.: ∀xFx. (H)3.:: ¬Fa. 1. (for ∃E).4.:: Fa 2.5.:: Fa ∧ ¬Fa. 3.4.6.:: ¬∀xFx.7.: ¬∀xFx. 3-6 (∃E).S. Choi (KAIST) Logic and set theory October 7, 2012 17 / 26

Inference rules for existential quantifiersExample∃x¬Fx ⊢ ¬∀xFx.1. ∃x¬Fx.2.: ∀xFx. (H)3.:: ¬Fa. 1. (for ∃E).4.:: Fa 2.5.:: Fa ∧ ¬Fa. 3.4.6.:: ¬∀xFx.7.: ¬∀xFx. 3-6 (∃E).8. ∀xFx → ¬∀xFx. 2-7S. Choi (KAIST) Logic and set theory October 7, 2012 17 / 26

Inference rules for existential quantifiersExample∃x¬Fx ⊢ ¬∀xFx.1. ∃x¬Fx.2.: ∀xFx. (H)3.:: ¬Fa. 1. (for ∃E).4.:: Fa 2.5.:: Fa ∧ ¬Fa. 3.4.6.:: ¬∀xFx.7.: ¬∀xFx. 3-6 (∃E).8. ∀xFx → ¬∀xFx. 2-79. ¬∀xFx.S. Choi (KAIST) Logic and set theory October 7, 2012 17 / 26

Inference rules for existential quantifiersSome strategies1. Be careful about where the quantifiers apply. Noticeparanthesis well.S. Choi (KAIST) Logic and set theory October 7, 2012 18 / 26

Inference rules for existential quantifiersSome strategies1. Be careful about where the quantifiers apply. Noticeparanthesis well.2. To proveS. Choi (KAIST) Logic and set theory October 7, 2012 18 / 26

Inference rules for existential quantifiersSome strategies1. Be careful about where the quantifiers apply. Noticeparanthesis well.2. To prove◮ ∃xFx: We prove Fa.S. Choi (KAIST) Logic and set theory October 7, 2012 18 / 26

Inference rules for existential quantifiersSome strategies1. Be careful about where the quantifiers apply. Noticeparanthesis well.2. To prove◮ ∃xFx: We prove Fa.◮ ∀x(Fx → Gx): We prove Fa → Ga.S. Choi (KAIST) Logic and set theory October 7, 2012 18 / 26

Inference rules for existential quantifiersSome strategies1. Be careful about where the quantifiers apply. Noticeparanthesis well.2. To prove◮ ∃xFx: We prove Fa.◮ ∀x(Fx → Gx): We prove Fa → Ga.◮ ∀x¬Fx: We prove ¬Fa.S. Choi (KAIST) Logic and set theory October 7, 2012 18 / 26

Inference rules for existential quantifiersSome strategies1. Be careful about where the quantifiers apply. Noticeparanthesis well.2. To prove◮ ∃xFx: We prove Fa.◮ ∀x(Fx → Gx): We prove Fa → Ga.◮ ∀x¬Fx: We prove ¬Fa.◮ ∀x∃yFxy: We prove ∃yFay.S. Choi (KAIST) Logic and set theory October 7, 2012 18 / 26

Inference rules for existential quantifiersSome strategies1. Be careful about where the quantifiers apply. Noticeparanthesis well.2. To prove◮ ∃xFx: We prove Fa.◮ ∀x(Fx → Gx): We prove Fa → Ga.◮ ∀x¬Fx: We prove ¬Fa.◮ ∀x∃yFxy: We prove ∃yFay.◮ ∃yFay: We prove Fab.S. Choi (KAIST) Logic and set theory October 7, 2012 18 / 26

Inference rules for existential quantifiersSome strategies1. Be careful about where the quantifiers apply. Noticeparanthesis well.2. To prove◮ ∃xFx: We prove Fa.◮ ∀x(Fx → Gx): We prove Fa → Ga.◮ ∀x¬Fx: We prove ¬Fa.◮ ∀x∃yFxy: We prove ∃yFay.◮ ∃yFay: We prove Fab.◮ ∃xFxx: We prove Faa.S. Choi (KAIST) Logic and set theory October 7, 2012 18 / 26

Inference rules for existential quantifiersSome strategies1. Be careful about where the quantifiers apply. Noticeparanthesis well.2. To prove◮ ∃xFx: We prove Fa.◮ ∀x(Fx → Gx): We prove Fa → Ga.◮ ∀x¬Fx: We prove ¬Fa.◮ ∀x∃yFxy: We prove ∃yFay.◮ ∃yFay: We prove Fab.◮ ∃xFxx: We prove Faa.3. To prove the forms in negation, conjunction, disjunction,conditional, or biconditional, then use propositional calculusmethods.... (Similar to (∀xP) → (∀xQ)).S. Choi (KAIST) Logic and set theory October 7, 2012 18 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsTheoremsThe truth that follows from no assumptions.S. Choi (KAIST) Logic and set theory October 7, 2012 19 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsTheoremsThe truth that follows from no assumptions.These hold in every model.S. Choi (KAIST) Logic and set theory October 7, 2012 19 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsTheoremsThe truth that follows from no assumptions.These hold in every model.These are called theorems.S. Choi (KAIST) Logic and set theory October 7, 2012 19 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsTheoremsThe truth that follows from no assumptions.These hold in every model.These are called theorems.Example: ⊢ ¬(∀xFx ∧ ∃x¬Fx)S. Choi (KAIST) Logic and set theory October 7, 2012 19 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsTheoremsThe truth that follows from no assumptions.These hold in every model.These are called theorems.Example: ⊢ ¬(∀xFx ∧ ∃x¬Fx)1.: (∀xFx ∧ ∃x¬Fx). H.S. Choi (KAIST) Logic and set theory October 7, 2012 19 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsTheoremsThe truth that follows from no assumptions.These hold in every model.These are called theorems.Example: ⊢ ¬(∀xFx ∧ ∃x¬Fx)1.: (∀xFx ∧ ∃x¬Fx). H.2.: ∀xFx.S. Choi (KAIST) Logic and set theory October 7, 2012 19 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsTheoremsThe truth that follows from no assumptions.These hold in every model.These are called theorems.Example: ⊢ ¬(∀xFx ∧ ∃x¬Fx)1.: (∀xFx ∧ ∃x¬Fx). H.2.: ∀xFx.3.: ∃x¬Fx.S. Choi (KAIST) Logic and set theory October 7, 2012 19 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsTheoremsThe truth that follows from no assumptions.These hold in every model.These are called theorems.Example: ⊢ ¬(∀xFx ∧ ∃x¬Fx)1.: (∀xFx ∧ ∃x¬Fx). H.2.: ∀xFx.3.: ∃x¬Fx.4.:: ¬Fa. (for ∃E).S. Choi (KAIST) Logic and set theory October 7, 2012 19 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsTheoremsThe truth that follows from no assumptions.These hold in every model.These are called theorems.Example: ⊢ ¬(∀xFx ∧ ∃x¬Fx)1.: (∀xFx ∧ ∃x¬Fx). H.2.: ∀xFx.3.: ∃x¬Fx.4.:: ¬Fa. (for ∃E).5.:: Fa 2.S. Choi (KAIST) Logic and set theory October 7, 2012 19 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsTheoremsThe truth that follows from no assumptions.These hold in every model.These are called theorems.Example: ⊢ ¬(∀xFx ∧ ∃x¬Fx)1.: (∀xFx ∧ ∃x¬Fx). H.2.: ∀xFx.3.: ∃x¬Fx.4.:: ¬Fa. (for ∃E).5.:: Fa 2.6.:: ¬Fa ∧ Fa.S. Choi (KAIST) Logic and set theory October 7, 2012 19 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsTheoremsThe truth that follows from no assumptions.These hold in every model.These are called theorems.Example: ⊢ ¬(∀xFx ∧ ∃x¬Fx)1.: (∀xFx ∧ ∃x¬Fx). H.2.: ∀xFx.3.: ∃x¬Fx.4.:: ¬Fa. (for ∃E).5.:: Fa 2.6.:: ¬Fa ∧ Fa.7.: ¬Fa ∧ Fa.S. Choi (KAIST) Logic and set theory October 7, 2012 19 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsTheoremsThe truth that follows from no assumptions.These hold in every model.These are called theorems.Example: ⊢ ¬(∀xFx ∧ ∃x¬Fx)1.: (∀xFx ∧ ∃x¬Fx). H.2.: ∀xFx.3.: ∃x¬Fx.4.:: ¬Fa. (for ∃E).5.:: Fa 2.6.:: ¬Fa ∧ Fa.7.: ¬Fa ∧ Fa.8. ¬(∀xFx ∧ ∃x¬Fx)S. Choi (KAIST) Logic and set theory October 7, 2012 19 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamples⊢ ∀xFx ∨ ∃x¬Fx.S. Choi (KAIST) Logic and set theory October 7, 2012 20 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamples⊢ ∀xFx ∨ ∃x¬Fx.1.: ¬∀xFx. H for → I.S. Choi (KAIST) Logic and set theory October 7, 2012 20 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamples⊢ ∀xFx ∨ ∃x¬Fx.1.: ¬∀xFx. H for → I.2.:: ¬∃x¬Fx. H for ¬I.S. Choi (KAIST) Logic and set theory October 7, 2012 20 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamples⊢ ∀xFx ∨ ∃x¬Fx.1.: ¬∀xFx. H for → I.2.:: ¬∃x¬Fx. H for ¬I.3.::: ¬Fa H. for ¬I.S. Choi (KAIST) Logic and set theory October 7, 2012 20 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamples⊢ ∀xFx ∨ ∃x¬Fx.1.: ¬∀xFx. H for → I.2.:: ¬∃x¬Fx. H for ¬I.3.::: ¬Fa H. for ¬I.4.::: ∃x¬FxS. Choi (KAIST) Logic and set theory October 7, 2012 20 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamples⊢ ∀xFx ∨ ∃x¬Fx.1.: ¬∀xFx. H for → I.2.:: ¬∃x¬Fx. H for ¬I.3.::: ¬Fa H. for ¬I.4.::: ∃x¬Fx5.::: ∃x¬Fx ∧ ¬∃x¬Fx.S. Choi (KAIST) Logic and set theory October 7, 2012 20 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamples⊢ ∀xFx ∨ ∃x¬Fx.1.: ¬∀xFx. H for → I.2.:: ¬∃x¬Fx. H for ¬I.3.::: ¬Fa H. for ¬I.4.::: ∃x¬Fx5.::: ∃x¬Fx ∧ ¬∃x¬Fx.6.:: ¬¬Fa.S. Choi (KAIST) Logic and set theory October 7, 2012 20 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamples⊢ ∀xFx ∨ ∃x¬Fx.1.: ¬∀xFx. H for → I.2.:: ¬∃x¬Fx. H for ¬I.3.::: ¬Fa H. for ¬I.4.::: ∃x¬Fx5.::: ∃x¬Fx ∧ ¬∃x¬Fx.6.:: ¬¬Fa.7.:: Fa.S. Choi (KAIST) Logic and set theory October 7, 2012 20 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamples⊢ ∀xFx ∨ ∃x¬Fx.1.: ¬∀xFx. H for → I.2.:: ¬∃x¬Fx. H for ¬I.3.::: ¬Fa H. for ¬I.4.::: ∃x¬Fx5.::: ∃x¬Fx ∧ ¬∃x¬Fx.6.:: ¬¬Fa.7.:: Fa.8.:: ∀xFx.S. Choi (KAIST) Logic and set theory October 7, 2012 20 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamples⊢ ∀xFx ∨ ∃x¬Fx.1.: ¬∀xFx. H for → I.2.:: ¬∃x¬Fx. H for ¬I.3.::: ¬Fa H. for ¬I.4.::: ∃x¬Fx5.::: ∃x¬Fx ∧ ¬∃x¬Fx.6.:: ¬¬Fa.7.:: Fa.8.:: ∀xFx.9.:: ∀xFx ∧ ¬∀xFx.S. Choi (KAIST) Logic and set theory October 7, 2012 20 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamples⊢ ∀xFx ∨ ∃x¬Fx.1.: ¬∀xFx. H for → I.2.:: ¬∃x¬Fx. H for ¬I.3.::: ¬Fa H. for ¬I.4.::: ∃x¬Fx5.::: ∃x¬Fx ∧ ¬∃x¬Fx.6.:: ¬¬Fa.7.:: Fa.8.:: ∀xFx.9.:: ∀xFx ∧ ¬∀xFx.10.: ¬¬∃x¬Fx.S. Choi (KAIST) Logic and set theory October 7, 2012 20 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamples⊢ ∀xFx ∨ ∃x¬Fx.1.: ¬∀xFx. H for → I.2.:: ¬∃x¬Fx. H for ¬I.3.::: ¬Fa H. for ¬I.4.::: ∃x¬Fx5.::: ∃x¬Fx ∧ ¬∃x¬Fx.6.:: ¬¬Fa.7.:: Fa.8.:: ∀xFx.9.:: ∀xFx ∧ ¬∀xFx.10.: ¬¬∃x¬Fx.11.: ∃x¬Fx.S. Choi (KAIST) Logic and set theory October 7, 2012 20 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamples⊢ ∀xFx ∨ ∃x¬Fx.1.: ¬∀xFx. H for → I.2.:: ¬∃x¬Fx. H for ¬I.3.::: ¬Fa H. for ¬I.4.::: ∃x¬Fx5.::: ∃x¬Fx ∧ ¬∃x¬Fx.6.:: ¬¬Fa.7.:: Fa.8.:: ∀xFx.9.:: ∀xFx ∧ ¬∀xFx.10.: ¬¬∃x¬Fx.11.: ∃x¬Fx.12. ¬∀xFx → ∃x¬Fx.S. Choi (KAIST) Logic and set theory October 7, 2012 20 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamples⊢ ∀xFx ∨ ∃x¬Fx.1.: ¬∀xFx. H for → I.2.:: ¬∃x¬Fx. H for ¬I.3.::: ¬Fa H. for ¬I.4.::: ∃x¬Fx5.::: ∃x¬Fx ∧ ¬∃x¬Fx.6.:: ¬¬Fa.7.:: Fa.8.:: ∀xFx.9.:: ∀xFx ∧ ¬∀xFx.10.: ¬¬∃x¬Fx.11.: ∃x¬Fx.12. ¬∀xFx → ∃x¬Fx.13. ¬¬∀xFx ∨ ∃x¬Fx. MIS. Choi (KAIST) Logic and set theory October 7, 2012 20 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamples⊢ ∀xFx ∨ ∃x¬Fx.1.: ¬∀xFx. H for → I.2.:: ¬∃x¬Fx. H for ¬I.3.::: ¬Fa H. for ¬I.4.::: ∃x¬Fx5.::: ∃x¬Fx ∧ ¬∃x¬Fx.6.:: ¬¬Fa.7.:: Fa.8.:: ∀xFx.9.:: ∀xFx ∧ ¬∀xFx.10.: ¬¬∃x¬Fx.11.: ∃x¬Fx.12. ¬∀xFx → ∃x¬Fx.13. ¬¬∀xFx ∨ ∃x¬Fx. MI14. ∀xFx ∨ ∃x¬Fx. DN.S. Choi (KAIST) Logic and set theory October 7, 2012 20 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamples⊢ ∀xFx ∨ ∃x¬Fx.1.: ¬∀xFx. H for → I.2.:: ¬∃x¬Fx. H for ¬I.3.::: ¬Fa H. for ¬I.4.::: ∃x¬Fx5.::: ∃x¬Fx ∧ ¬∃x¬Fx.6.:: ¬¬Fa.7.:: Fa.8.:: ∀xFx.9.:: ∀xFx ∧ ¬∀xFx.10.: ¬¬∃x¬Fx.11.: ∃x¬Fx.12. ¬∀xFx → ∃x¬Fx.13. ¬¬∀xFx ∨ ∃x¬Fx. MI14. ∀xFx ∨ ∃x¬Fx. DN.There is a way using equivalences.S. Choi (KAIST) Logic and set theory October 7, 2012 20 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamplesThe rule ⊢ ∀xP → ∃xP.S. Choi (KAIST) Logic and set theory October 7, 2012 21 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamplesThe rule ⊢ ∀xP → ∃xP.We apply this rule to obtain:S. Choi (KAIST) Logic and set theory October 7, 2012 21 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamplesThe rule ⊢ ∀xP → ∃xP.We apply this rule to obtain:⊢ ∀x∀yP → ∀x∃yP.S. Choi (KAIST) Logic and set theory October 7, 2012 21 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamplesThe rule ⊢ ∀xP → ∃xP.We apply this rule to obtain:⊢ ∀x∀yP → ∀x∃yP.⊢ ∀x∀yP → ∃x∀yP.S. Choi (KAIST) Logic and set theory October 7, 2012 21 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamplesThe rule ⊢ ∀xP → ∃xP.We apply this rule to obtain:⊢ ∀x∀yP → ∀x∃yP.⊢ ∀x∀yP → ∃x∀yP.⊢ ∀x∀yP → ∃x∃yP.S. Choi (KAIST) Logic and set theory October 7, 2012 21 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamplesThe rule ⊢ ∀xP → ∃xP.We apply this rule to obtain:⊢ ∀x∀yP → ∀x∃yP.⊢ ∀x∀yP → ∃x∀yP.⊢ ∀x∀yP → ∃x∃yP.⊢ ∀x∃yP → ∃x∃yP.S. Choi (KAIST) Logic and set theory October 7, 2012 21 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsExamplesThe rule ⊢ ∀xP → ∃xP.We apply this rule to obtain:⊢ ∀x∀yP → ∀x∃yP.⊢ ∀x∀yP → ∃x∀yP.⊢ ∀x∀yP → ∃x∃yP.⊢ ∀x∃yP → ∃x∃yP.In fact, we can use a theorem to generate many more theorems....S. Choi (KAIST) Logic and set theory October 7, 2012 21 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsEquivalencesWe studied equivalences called interchanges: ∃x∃y ↔ ∃y∃x and∀x∀y ↔ ∀y∀x.S. Choi (KAIST) Logic and set theory October 7, 2012 22 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsEquivalencesWe studied equivalences called interchanges: ∃x∃y ↔ ∃y∃x and∀x∀y ↔ ∀y∀x.⊢ ¬∀x¬Fx ↔ ∃xFx.S. Choi (KAIST) Logic and set theory October 7, 2012 22 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsEquivalencesWe studied equivalences called interchanges: ∃x∃y ↔ ∃y∃x and∀x∀y ↔ ∀y∀x.⊢ ¬∀x¬Fx ↔ ∃xFx.⊢ ¬∀xFx ↔ ∃x¬Fx. This was proved above.S. Choi (KAIST) Logic and set theory October 7, 2012 22 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsEquivalencesWe studied equivalences called interchanges: ∃x∃y ↔ ∃y∃x and∀x∀y ↔ ∀y∀x.⊢ ¬∀x¬Fx ↔ ∃xFx.⊢ ¬∀xFx ↔ ∃x¬Fx. This was proved above.⊢ ∀x¬Fx ↔ ¬∃xFx. This was proved above.S. Choi (KAIST) Logic and set theory October 7, 2012 22 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsEquivalencesWe studied equivalences called interchanges: ∃x∃y ↔ ∃y∃x and∀x∀y ↔ ∀y∀x.⊢ ¬∀x¬Fx ↔ ∃xFx.⊢ ¬∀xFx ↔ ∃x¬Fx. This was proved above.⊢ ∀x¬Fx ↔ ¬∃xFx. This was proved above.⊢ ∀xFx ↔ ¬∃x¬Fx.S. Choi (KAIST) Logic and set theory October 7, 2012 22 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsEquivalencesWe studied equivalences called interchanges: ∃x∃y ↔ ∃y∃x and∀x∀y ↔ ∀y∀x.⊢ ¬∀x¬Fx ↔ ∃xFx.⊢ ¬∀xFx ↔ ∃x¬Fx. This was proved above.⊢ ∀x¬Fx ↔ ¬∃xFx. This was proved above.⊢ ∀xFx ↔ ¬∃x¬Fx.The first and the fourth items are consequence of items two andthree.S. Choi (KAIST) Logic and set theory October 7, 2012 22 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsQuantifier exchangesUsing the above equivalences, we obtain the quantifier exchangerules.S. Choi (KAIST) Logic and set theory October 7, 2012 23 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsQuantifier exchangesUsing the above equivalences, we obtain the quantifier exchangerules.¬∀β¬φ, ∃βφ.S. Choi (KAIST) Logic and set theory October 7, 2012 23 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsQuantifier exchangesUsing the above equivalences, we obtain the quantifier exchangerules.¬∀β¬φ, ∃βφ.¬∀βφ, ∃β¬φ.S. Choi (KAIST) Logic and set theory October 7, 2012 23 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsQuantifier exchangesUsing the above equivalences, we obtain the quantifier exchangerules.¬∀β¬φ, ∃βφ.¬∀βφ, ∃β¬φ.∀β¬φ, ¬∃βφ.S. Choi (KAIST) Logic and set theory October 7, 2012 23 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsQuantifier exchangesUsing the above equivalences, we obtain the quantifier exchangerules.¬∀β¬φ, ∃βφ.¬∀βφ, ∃β¬φ.∀β¬φ, ¬∃βφ.∀βφ, ¬∃β¬φ.S. Choi (KAIST) Logic and set theory October 7, 2012 23 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsQuantifier exchangesUsing the above equivalences, we obtain the quantifier exchangerules.¬∀β¬φ, ∃βφ.¬∀βφ, ∃β¬φ.∀β¬φ, ¬∃βφ.∀βφ, ¬∃β¬φ.Using this many predicate calculus results are simply theconsequences of propositional caculus results.S. Choi (KAIST) Logic and set theory October 7, 2012 23 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsSome other equivalences (Repeated)How would one prove? :S. Choi (KAIST) Logic and set theory October 7, 2012 24 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsSome other equivalences (Repeated)How would one prove? :∃xf ↔ f if x is not a free variable of f .S. Choi (KAIST) Logic and set theory October 7, 2012 24 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsSome other equivalences (Repeated)How would one prove? :∃xf ↔ f if x is not a free variable of f .∀xf ↔ f if x is not a free variable of f .S. Choi (KAIST) Logic and set theory October 7, 2012 24 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsSome other equivalences (Repeated)How would one prove? :∃xf ↔ f if x is not a free variable of f .∀xf ↔ f if x is not a free variable of f .∃x(f ∨ g) ↔ ∃xf ∨ ∃xg.S. Choi (KAIST) Logic and set theory October 7, 2012 24 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsSome other equivalences (Repeated)How would one prove? :∃xf ↔ f if x is not a free variable of f .∀xf ↔ f if x is not a free variable of f .∃x(f ∨ g) ↔ ∃xf ∨ ∃xg.∀x(f ∧ g) ↔ ∀xf ∧ ∀yg.S. Choi (KAIST) Logic and set theory October 7, 2012 24 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsSome other equivalences (Repeated)How would one prove? :∃xf ↔ f if x is not a free variable of f .∀xf ↔ f if x is not a free variable of f .∃x(f ∨ g) ↔ ∃xf ∨ ∃xg.∀x(f ∧ g) ↔ ∀xf ∧ ∀yg.∃x(f ∧ g) ↔ (∃xf ) ∧ g if x does not occur as a free variable of g.And also ∃x(f ∨ g) ↔ (∃xf ) ∨ gS. Choi (KAIST) Logic and set theory October 7, 2012 24 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsSome other equivalences (Repeated)How would one prove? :∃xf ↔ f if x is not a free variable of f .∀xf ↔ f if x is not a free variable of f .∃x(f ∨ g) ↔ ∃xf ∨ ∃xg.∀x(f ∧ g) ↔ ∀xf ∧ ∀yg.∃x(f ∧ g) ↔ (∃xf ) ∧ g if x does not occur as a free variable of g.And also ∃x(f ∨ g) ↔ (∃xf ) ∨ g∀x(f ∨ g) ↔ (∀xf ) ∨ g if x does not occur as a free variable of g.And also ∀x(f ∧ g) ↔ (∀xf ) ∧ gS. Choi (KAIST) Logic and set theory October 7, 2012 24 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsSome other equivalences (Repeated)How would one prove? :∃xf ↔ f if x is not a free variable of f .∀xf ↔ f if x is not a free variable of f .∃x(f ∨ g) ↔ ∃xf ∨ ∃xg.∀x(f ∧ g) ↔ ∀xf ∧ ∀yg.∃x(f ∧ g) ↔ (∃xf ) ∧ g if x does not occur as a free variable of g.And also ∃x(f ∨ g) ↔ (∃xf ) ∨ g∀x(f ∨ g) ↔ (∀xf ) ∨ g if x does not occur as a free variable of g.And also ∀x(f ∧ g) ↔ (∀xf ) ∧ g∃yf (x 1 , ..., x n , y) ↔ ∃zf (x 1 , .., x n , z) if neither y, z are part ofx 1 , ..., x n .S. Choi (KAIST) Logic and set theory October 7, 2012 24 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsSome other equivalences (Repeated)How would one prove? :∃xf ↔ f if x is not a free variable of f .∀xf ↔ f if x is not a free variable of f .∃x(f ∨ g) ↔ ∃xf ∨ ∃xg.∀x(f ∧ g) ↔ ∀xf ∧ ∀yg.∃x(f ∧ g) ↔ (∃xf ) ∧ g if x does not occur as a free variable of g.And also ∃x(f ∨ g) ↔ (∃xf ) ∨ g∀x(f ∨ g) ↔ (∀xf ) ∨ g if x does not occur as a free variable of g.And also ∀x(f ∧ g) ↔ (∀xf ) ∧ g∃yf (x 1 , ..., x n , y) ↔ ∃zf (x 1 , .., x n , z) if neither y, z are part ofx 1 , ..., x n .∀yf (x 1 , ..., x n , y) ↔ ∀zf (x 1 , .., x n , z) if neither y, z are part ofx 1 , ..., x n .S. Choi (KAIST) Logic and set theory October 7, 2012 24 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsInference rules of =Identity introduction (= I):S. Choi (KAIST) Logic and set theory October 7, 2012 25 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsInference rules of =Identity introduction (= I):For any name letter α, assert α = α at any line.S. Choi (KAIST) Logic and set theory October 7, 2012 25 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsInference rules of =Identity introduction (= I):For any name letter α, assert α = α at any line.Example ⊢ ∃x, a = x.S. Choi (KAIST) Logic and set theory October 7, 2012 25 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsInference rules of =Identity introduction (= I):For any name letter α, assert α = α at any line.Example ⊢ ∃x, a = x.1. a = a (= I).S. Choi (KAIST) Logic and set theory October 7, 2012 25 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsInference rules of =Identity introduction (= I):For any name letter α, assert α = α at any line.Example ⊢ ∃x, a = x.1. a = a (= I).2. ∃x, a = x 1. (∃I).S. Choi (KAIST) Logic and set theory October 7, 2012 25 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsIdentity elimination (= E):S. Choi (KAIST) Logic and set theory October 7, 2012 26 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsIdentity elimination (= E):A wff φ containing α. We have α = β or β = α. Then infer φ β/α .S. Choi (KAIST) Logic and set theory October 7, 2012 26 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsIdentity elimination (= E):A wff φ containing α. We have α = β or β = α. Then infer φ β/α .Example: ⊢ ∀x∀y(x = y → y = x).S. Choi (KAIST) Logic and set theory October 7, 2012 26 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsIdentity elimination (= E):A wff φ containing α. We have α = β or β = α. Then infer φ β/α .Example: ⊢ ∀x∀y(x = y → y = x).1.: a = b H for → I.S. Choi (KAIST) Logic and set theory October 7, 2012 26 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsIdentity elimination (= E):A wff φ containing α. We have α = β or β = α. Then infer φ β/α .Example: ⊢ ∀x∀y(x = y → y = x).1.: a = b H for → I.2.: a = a.S. Choi (KAIST) Logic and set theory October 7, 2012 26 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsIdentity elimination (= E):A wff φ containing α. We have α = β or β = α. Then infer φ β/α .Example: ⊢ ∀x∀y(x = y → y = x).1.: a = b H for → I.2.: a = a.3.: b = a. (= E).S. Choi (KAIST) Logic and set theory October 7, 2012 26 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsIdentity elimination (= E):A wff φ containing α. We have α = β or β = α. Then infer φ β/α .Example: ⊢ ∀x∀y(x = y → y = x).1.: a = b H for → I.2.: a = a.3.: b = a. (= E).4. a = b → b = a. 1-3S. Choi (KAIST) Logic and set theory October 7, 2012 26 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsIdentity elimination (= E):A wff φ containing α. We have α = β or β = α. Then infer φ β/α .Example: ⊢ ∀x∀y(x = y → y = x).1.: a = b H for → I.2.: a = a.3.: b = a. (= E).4. a = b → b = a. 1-3∀y(a = y → y = a). (∀I).S. Choi (KAIST) Logic and set theory October 7, 2012 26 / 26

Inference rules for existential quantifiersTheorems and quantifier equivalence relationsIdentity elimination (= E):A wff φ containing α. We have α = β or β = α. Then infer φ β/α .Example: ⊢ ∀x∀y(x = y → y = x).1.: a = b H for → I.2.: a = a.3.: b = a. (= E).4. a = b → b = a. 1-3∀y(a = y → y = a). (∀I).∀x∀y(x = y → y = x). (∀I).S. Choi (KAIST) Logic and set theory October 7, 2012 26 / 26