lecture 6: propositional calculus: part 2 s. choimathsci.kaist.ac.kr/~schoi/logiclec6.pdflogic and...

Logic and the set theoryLecture 6: Propositional Calculus: part 2

S. Choi

Department of Mathematical ScienceKAIST, Daejeon, South Korea

Fall semester, 2012

S. Choi (KAIST) Logic and set theory September 18, 2012 1 / 18

Introduction

About this lecture

Notions of Inference

Inference RulesHypothetical RulesDerived RulesThe Propositional RulesEquivalencesThe soundness and the completeness of deductions.We go over the last three Hypothetical Rules in Lecture 6.Course homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://moodle.kaist.ac.kr

Grading and so on in the moodle. Ask questions in moodle.

Introduction

About this lecture

Notions of InferenceInference Rules

Hypothetical RulesDerived RulesThe Propositional RulesEquivalencesThe soundness and the completeness of deductions.We go over the last three Hypothetical Rules in Lecture 6.Course homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://moodle.kaist.ac.kr

Introduction

About this lecture

Notions of InferenceInference RulesHypothetical Rules

Derived RulesThe Propositional RulesEquivalencesThe soundness and the completeness of deductions.We go over the last three Hypothetical Rules in Lecture 6.Course homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://moodle.kaist.ac.kr

Introduction

About this lecture

Notions of InferenceInference RulesHypothetical RulesDerived Rules

The Propositional RulesEquivalencesThe soundness and the completeness of deductions.We go over the last three Hypothetical Rules in Lecture 6.Course homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://moodle.kaist.ac.kr

Introduction

About this lecture

Notions of InferenceInference RulesHypothetical RulesDerived RulesThe Propositional Rules

EquivalencesThe soundness and the completeness of deductions.We go over the last three Hypothetical Rules in Lecture 6.Course homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://moodle.kaist.ac.kr

Introduction

About this lecture

Notions of InferenceInference RulesHypothetical RulesDerived RulesThe Propositional RulesEquivalences

The soundness and the completeness of deductions.We go over the last three Hypothetical Rules in Lecture 6.Course homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://moodle.kaist.ac.kr

Introduction

About this lecture

Notions of InferenceInference RulesHypothetical RulesDerived RulesThe Propositional RulesEquivalencesThe soundness and the completeness of deductions.

We go over the last three Hypothetical Rules in Lecture 6.Course homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://moodle.kaist.ac.kr

Introduction

About this lecture

Notions of InferenceInference RulesHypothetical RulesDerived RulesThe Propositional RulesEquivalencesThe soundness and the completeness of deductions.We go over the last three Hypothetical Rules in Lecture 6.

Course homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://moodle.kaist.ac.kr

Introduction

About this lecture

Notions of InferenceInference RulesHypothetical RulesDerived RulesThe Propositional RulesEquivalencesThe soundness and the completeness of deductions.We go over the last three Hypothetical Rules in Lecture 6.Course homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://moodle.kaist.ac.kr

Introduction

About this lecture

Notions of InferenceInference RulesHypothetical RulesDerived RulesThe Propositional RulesEquivalencesThe soundness and the completeness of deductions.We go over the last three Hypothetical Rules in Lecture 6.Course homepages:http://mathsci.kaist.ac.kr/~schoi/logic.html andthe moodle page http://moodle.kaist.ac.kr

Introduction

Some helpful references

Sets, Logic and Categories, Peter J. Cameron, Springer

A mathematical introduction to logic, H. Enderton, AcademicPress.Whitehead, Russell, Principia Mathematica (our library). (Thiscould be a project idea. )http://plato.stanford.edu/contents.html has muchresource. See “classical logic”.http://ocw.mit.edu/OcwWeb/Linguistics-and-Philosophy/24-241Fall-2005/CourseHome/ See "Derivations inSentential Calculus". (or SC Derivations.) and “The completenessof the SC rules.”http://jvrosset.free.fr/Goedel-Proof-Truth.pdf“Does Godels incompleteness prove that truth transcends proof?”

Introduction

Sets, Logic and Categories, Peter J. Cameron, SpringerA mathematical introduction to logic, H. Enderton, AcademicPress.

Whitehead, Russell, Principia Mathematica (our library). (Thiscould be a project idea. )http://plato.stanford.edu/contents.html has muchresource. See “classical logic”.http://ocw.mit.edu/OcwWeb/Linguistics-and-Philosophy/24-241Fall-2005/CourseHome/ See "Derivations inSentential Calculus". (or SC Derivations.) and “The completenessof the SC rules.”http://jvrosset.free.fr/Goedel-Proof-Truth.pdf“Does Godels incompleteness prove that truth transcends proof?”

Introduction

Sets, Logic and Categories, Peter J. Cameron, SpringerA mathematical introduction to logic, H. Enderton, AcademicPress.Whitehead, Russell, Principia Mathematica (our library). (Thiscould be a project idea. )

http://plato.stanford.edu/contents.html has muchresource. See “classical logic”.http://ocw.mit.edu/OcwWeb/Linguistics-and-Philosophy/24-241Fall-2005/CourseHome/ See "Derivations inSentential Calculus". (or SC Derivations.) and “The completenessof the SC rules.”http://jvrosset.free.fr/Goedel-Proof-Truth.pdf“Does Godels incompleteness prove that truth transcends proof?”

Introduction

Sets, Logic and Categories, Peter J. Cameron, SpringerA mathematical introduction to logic, H. Enderton, AcademicPress.Whitehead, Russell, Principia Mathematica (our library). (Thiscould be a project idea. )http://plato.stanford.edu/contents.html has muchresource. See “classical logic”.

http://ocw.mit.edu/OcwWeb/Linguistics-and-Philosophy/24-241Fall-2005/CourseHome/ See "Derivations inSentential Calculus". (or SC Derivations.) and “The completenessof the SC rules.”http://jvrosset.free.fr/Goedel-Proof-Truth.pdf“Does Godels incompleteness prove that truth transcends proof?”

Introduction

Sets, Logic and Categories, Peter J. Cameron, SpringerA mathematical introduction to logic, H. Enderton, AcademicPress.Whitehead, Russell, Principia Mathematica (our library). (Thiscould be a project idea. )http://plato.stanford.edu/contents.html has muchresource. See “classical logic”.http://ocw.mit.edu/OcwWeb/Linguistics-and-Philosophy/24-241Fall-2005/CourseHome/ See "Derivations inSentential Calculus". (or SC Derivations.) and “The completenessof the SC rules.”

http://jvrosset.free.fr/Goedel-Proof-Truth.pdf“Does Godels incompleteness prove that truth transcends proof?”

Introduction

Sets, Logic and Categories, Peter J. Cameron, SpringerA mathematical introduction to logic, H. Enderton, AcademicPress.Whitehead, Russell, Principia Mathematica (our library). (Thiscould be a project idea. )http://plato.stanford.edu/contents.html has muchresource. See “classical logic”.http://ocw.mit.edu/OcwWeb/Linguistics-and-Philosophy/24-241Fall-2005/CourseHome/ See "Derivations inSentential Calculus". (or SC Derivations.) and “The completenessof the SC rules.”http://jvrosset.free.fr/Goedel-Proof-Truth.pdf“Does Godels incompleteness prove that truth transcends proof?”

Introduction

http://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.

Introduction

http://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.

Introduction

http://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.

Derived Rules

Suppose that one proved a logical formula, which are not in theten elementary rules. Then we can substitute the symbols withwffs and still obtain valid logical formula.

Example: P → Q,¬Q ` ¬P. (Modus Tollens (MT)).Substitution instance: P to (R ∨ S) and Q to ¬C. Then obtain(R ∨ S)→ ¬C, ¬¬C. ` ¬(R ∨ S).

Derived Rules

Suppose that one proved a logical formula, which are not in theten elementary rules. Then we can substitute the symbols withwffs and still obtain valid logical formula.Example: P → Q,¬Q ` ¬P. (Modus Tollens (MT)).

Substitution instance: P to (R ∨ S) and Q to ¬C. Then obtain(R ∨ S)→ ¬C, ¬¬C. ` ¬(R ∨ S).

Derived Rules

Suppose that one proved a logical formula, which are not in theten elementary rules. Then we can substitute the symbols withwffs and still obtain valid logical formula.Example: P → Q,¬Q ` ¬P. (Modus Tollens (MT)).Substitution instance: P to (R ∨ S) and Q to ¬C. Then obtain(R ∨ S)→ ¬C, ¬¬C. ` ¬(R ∨ S).

Derived Rules

Examples

Prove MT:

1. P → Q A2. ¬Q A.3.: ¬¬P. for ¬I.4.: P. ¬E .5.: Q 1,4← E .6.: Q ∧ ¬Q. 2.5. ∧I.7. ¬P.

Derived Rules

Examples

Prove MT:1. P → Q A

2. ¬Q A.3.: ¬¬P. for ¬I.4.: P. ¬E .5.: Q 1,4← E .6.: Q ∧ ¬Q. 2.5. ∧I.7. ¬P.

Derived Rules

Examples

Prove MT:1. P → Q A2. ¬Q A.

3.: ¬¬P. for ¬I.4.: P. ¬E .5.: Q 1,4← E .6.: Q ∧ ¬Q. 2.5. ∧I.7. ¬P.

Derived Rules

Examples

Prove MT:1. P → Q A2. ¬Q A.3.: ¬¬P. for ¬I.

4.: P. ¬E .5.: Q 1,4← E .6.: Q ∧ ¬Q. 2.5. ∧I.7. ¬P.

Derived Rules

Examples

Prove MT:1. P → Q A2. ¬Q A.3.: ¬¬P. for ¬I.4.: P. ¬E .

5.: Q 1,4← E .6.: Q ∧ ¬Q. 2.5. ∧I.7. ¬P.

Derived Rules

Examples

Prove MT:1. P → Q A2. ¬Q A.3.: ¬¬P. for ¬I.4.: P. ¬E .5.: Q 1,4← E .

6.: Q ∧ ¬Q. 2.5. ∧I.7. ¬P.

Derived Rules

Examples

Prove MT:1. P → Q A2. ¬Q A.3.: ¬¬P. for ¬I.4.: P. ¬E .5.: Q 1,4← E .6.: Q ∧ ¬Q. 2.5. ∧I.

7. ¬P.

Derived Rules

Examples

Prove MT:1. P → Q A2. ¬Q A.3.: ¬¬P. for ¬I.4.: P. ¬E .5.: Q 1,4← E .6.: Q ∧ ¬Q. 2.5. ∧I.7. ¬P.

Derived Rules

Modus Tollens (MT): P → Q,¬Q ` ¬P.

Hypothetical syllogism (HS): P → Q,Q → R ` P → R.Absorption (ABS): P → Q ` P → (P ∧Q).Constructive Dilemma (CD): P ∨Q, P → R, Q → S ` R ∨ S.Repeat or Reiteration (RE): P ` Q → P.Contradiction (CON): P,¬P ` Q.Disjunctive syllogism (DS): P ∨Q,¬P ` Q.

Derived Rules

Modus Tollens (MT): P → Q,¬Q ` ¬P.Hypothetical syllogism (HS): P → Q,Q → R ` P → R.

Absorption (ABS): P → Q ` P → (P ∧Q).Constructive Dilemma (CD): P ∨Q, P → R, Q → S ` R ∨ S.Repeat or Reiteration (RE): P ` Q → P.Contradiction (CON): P,¬P ` Q.Disjunctive syllogism (DS): P ∨Q,¬P ` Q.

Derived Rules

Modus Tollens (MT): P → Q,¬Q ` ¬P.Hypothetical syllogism (HS): P → Q,Q → R ` P → R.Absorption (ABS): P → Q ` P → (P ∧Q).

Constructive Dilemma (CD): P ∨Q, P → R, Q → S ` R ∨ S.Repeat or Reiteration (RE): P ` Q → P.Contradiction (CON): P,¬P ` Q.Disjunctive syllogism (DS): P ∨Q,¬P ` Q.

Derived Rules

Modus Tollens (MT): P → Q,¬Q ` ¬P.Hypothetical syllogism (HS): P → Q,Q → R ` P → R.Absorption (ABS): P → Q ` P → (P ∧Q).Constructive Dilemma (CD): P ∨Q, P → R, Q → S ` R ∨ S.

Repeat or Reiteration (RE): P ` Q → P.Contradiction (CON): P,¬P ` Q.Disjunctive syllogism (DS): P ∨Q,¬P ` Q.

Derived Rules

Modus Tollens (MT): P → Q,¬Q ` ¬P.Hypothetical syllogism (HS): P → Q,Q → R ` P → R.Absorption (ABS): P → Q ` P → (P ∧Q).Constructive Dilemma (CD): P ∨Q, P → R, Q → S ` R ∨ S.Repeat or Reiteration (RE): P ` Q → P.

Contradiction (CON): P,¬P ` Q.Disjunctive syllogism (DS): P ∨Q,¬P ` Q.

Derived Rules

Modus Tollens (MT): P → Q,¬Q ` ¬P.Hypothetical syllogism (HS): P → Q,Q → R ` P → R.Absorption (ABS): P → Q ` P → (P ∧Q).Constructive Dilemma (CD): P ∨Q, P → R, Q → S ` R ∨ S.Repeat or Reiteration (RE): P ` Q → P.Contradiction (CON): P,¬P ` Q.

Disjunctive syllogism (DS): P ∨Q,¬P ` Q.

Derived Rules

Modus Tollens (MT): P → Q,¬Q ` ¬P.Hypothetical syllogism (HS): P → Q,Q → R ` P → R.Absorption (ABS): P → Q ` P → (P ∧Q).Constructive Dilemma (CD): P ∨Q, P → R, Q → S ` R ∨ S.Repeat or Reiteration (RE): P ` Q → P.Contradiction (CON): P,¬P ` Q.Disjunctive syllogism (DS): P ∨Q,¬P ` Q.

Derived Rules

ExamplesProve CD: P ∨Q, P → R, Q → S ` R ∨ S.

1. P ∨Q A2. P → R A.3. Q → S. A.4.: P for→ I.5.: R from→ E .6.: R ∨ S ∨I.7. P → (R ∨ S).8.: Q for→ I.9.: S.10.: R ∨ S.11. Q → (R ∨ S).12. R ∨ S.

Derived Rules

ExamplesProve CD: P ∨Q, P → R, Q → S ` R ∨ S.1. P ∨Q A

2. P → R A.3. Q → S. A.4.: P for→ I.5.: R from→ E .6.: R ∨ S ∨I.7. P → (R ∨ S).8.: Q for→ I.9.: S.10.: R ∨ S.11. Q → (R ∨ S).12. R ∨ S.

Derived Rules

ExamplesProve CD: P ∨Q, P → R, Q → S ` R ∨ S.1. P ∨Q A2. P → R A.

3. Q → S. A.4.: P for→ I.5.: R from→ E .6.: R ∨ S ∨I.7. P → (R ∨ S).8.: Q for→ I.9.: S.10.: R ∨ S.11. Q → (R ∨ S).12. R ∨ S.

Derived Rules

ExamplesProve CD: P ∨Q, P → R, Q → S ` R ∨ S.1. P ∨Q A2. P → R A.3. Q → S. A.

4.: P for→ I.5.: R from→ E .6.: R ∨ S ∨I.7. P → (R ∨ S).8.: Q for→ I.9.: S.10.: R ∨ S.11. Q → (R ∨ S).12. R ∨ S.

Derived Rules

ExamplesProve CD: P ∨Q, P → R, Q → S ` R ∨ S.1. P ∨Q A2. P → R A.3. Q → S. A.4.: P for→ I.

5.: R from→ E .6.: R ∨ S ∨I.7. P → (R ∨ S).8.: Q for→ I.9.: S.10.: R ∨ S.11. Q → (R ∨ S).12. R ∨ S.

Derived Rules

ExamplesProve CD: P ∨Q, P → R, Q → S ` R ∨ S.1. P ∨Q A2. P → R A.3. Q → S. A.4.: P for→ I.5.: R from→ E .

6.: R ∨ S ∨I.7. P → (R ∨ S).8.: Q for→ I.9.: S.10.: R ∨ S.11. Q → (R ∨ S).12. R ∨ S.

Derived Rules

ExamplesProve CD: P ∨Q, P → R, Q → S ` R ∨ S.1. P ∨Q A2. P → R A.3. Q → S. A.4.: P for→ I.5.: R from→ E .6.: R ∨ S ∨I.

7. P → (R ∨ S).8.: Q for→ I.9.: S.10.: R ∨ S.11. Q → (R ∨ S).12. R ∨ S.

Derived Rules

ExamplesProve CD: P ∨Q, P → R, Q → S ` R ∨ S.1. P ∨Q A2. P → R A.3. Q → S. A.4.: P for→ I.5.: R from→ E .6.: R ∨ S ∨I.7. P → (R ∨ S).

8.: Q for→ I.9.: S.10.: R ∨ S.11. Q → (R ∨ S).12. R ∨ S.

Derived Rules

ExamplesProve CD: P ∨Q, P → R, Q → S ` R ∨ S.1. P ∨Q A2. P → R A.3. Q → S. A.4.: P for→ I.5.: R from→ E .6.: R ∨ S ∨I.7. P → (R ∨ S).8.: Q for→ I.

9.: S.10.: R ∨ S.11. Q → (R ∨ S).12. R ∨ S.

Derived Rules

ExamplesProve CD: P ∨Q, P → R, Q → S ` R ∨ S.1. P ∨Q A2. P → R A.3. Q → S. A.4.: P for→ I.5.: R from→ E .6.: R ∨ S ∨I.7. P → (R ∨ S).8.: Q for→ I.9.: S.

10.: R ∨ S.11. Q → (R ∨ S).12. R ∨ S.

Derived Rules

ExamplesProve CD: P ∨Q, P → R, Q → S ` R ∨ S.1. P ∨Q A2. P → R A.3. Q → S. A.4.: P for→ I.5.: R from→ E .6.: R ∨ S ∨I.7. P → (R ∨ S).8.: Q for→ I.9.: S.10.: R ∨ S.

11. Q → (R ∨ S).12. R ∨ S.

Derived Rules

ExamplesProve CD: P ∨Q, P → R, Q → S ` R ∨ S.1. P ∨Q A2. P → R A.3. Q → S. A.4.: P for→ I.5.: R from→ E .6.: R ∨ S ∨I.7. P → (R ∨ S).8.: Q for→ I.9.: S.10.: R ∨ S.11. Q → (R ∨ S).

12. R ∨ S.

Derived Rules

ExamplesProve CD: P ∨Q, P → R, Q → S ` R ∨ S.1. P ∨Q A2. P → R A.3. Q → S. A.4.: P for→ I.5.: R from→ E .6.: R ∨ S ∨I.7. P → (R ∨ S).8.: Q for→ I.9.: S.10.: R ∨ S.11. Q → (R ∨ S).12. R ∨ S.

Derived Rules

Examples

Prove DS: P ∨Q,¬P ` Q.

1. P ∨Q A2. ¬P A.3.: P for→ I.4.: Q. 2.3. (CON)5. P → Q.6.: Q for→ I.7:: Q.8. Q → Q.9. Q by ∨E .

Derived Rules

Examples

Prove DS: P ∨Q,¬P ` Q.1. P ∨Q A

2. ¬P A.3.: P for→ I.4.: Q. 2.3. (CON)5. P → Q.6.: Q for→ I.7:: Q.8. Q → Q.9. Q by ∨E .

Derived Rules

Examples

Prove DS: P ∨Q,¬P ` Q.1. P ∨Q A2. ¬P A.

3.: P for→ I.4.: Q. 2.3. (CON)5. P → Q.6.: Q for→ I.7:: Q.8. Q → Q.9. Q by ∨E .

Derived Rules

Examples

Prove DS: P ∨Q,¬P ` Q.1. P ∨Q A2. ¬P A.3.: P for→ I.

4.: Q. 2.3. (CON)5. P → Q.6.: Q for→ I.7:: Q.8. Q → Q.9. Q by ∨E .

Derived Rules

Examples

Prove DS: P ∨Q,¬P ` Q.1. P ∨Q A2. ¬P A.3.: P for→ I.4.: Q. 2.3. (CON)

5. P → Q.6.: Q for→ I.7:: Q.8. Q → Q.9. Q by ∨E .

Derived Rules

Examples

Prove DS: P ∨Q,¬P ` Q.1. P ∨Q A2. ¬P A.3.: P for→ I.4.: Q. 2.3. (CON)5. P → Q.

6.: Q for→ I.7:: Q.8. Q → Q.9. Q by ∨E .

Derived Rules

Examples

Prove DS: P ∨Q,¬P ` Q.1. P ∨Q A2. ¬P A.3.: P for→ I.4.: Q. 2.3. (CON)5. P → Q.6.: Q for→ I.

7:: Q.8. Q → Q.9. Q by ∨E .

Derived Rules

Examples

Prove DS: P ∨Q,¬P ` Q.1. P ∨Q A2. ¬P A.3.: P for→ I.4.: Q. 2.3. (CON)5. P → Q.6.: Q for→ I.7:: Q.

8. Q → Q.9. Q by ∨E .

Derived Rules

Examples

Prove DS: P ∨Q,¬P ` Q.1. P ∨Q A2. ¬P A.3.: P for→ I.4.: Q. 2.3. (CON)5. P → Q.6.: Q for→ I.7:: Q.8. Q → Q.

9. Q by ∨E .

Derived Rules

Examples

Prove DS: P ∨Q,¬P ` Q.1. P ∨Q A2. ¬P A.3.: P for→ I.4.: Q. 2.3. (CON)5. P → Q.6.: Q for→ I.7:: Q.8. Q → Q.9. Q by ∨E .

Theorems

Theorems are wffs deduced from no assumptions. They are justtautologies. (At least in this book)

¬(P ∧ ¬P), or ¬P ∨ P.P → ((P → Q)→ Q).P → (Q → P).(P → (Q → R))→ ((P → Q)→ (P → R)).(((¬P → ¬Q)→ (Q → P)).

Theorems

Theorems are wffs deduced from no assumptions. They are justtautologies. (At least in this book)¬(P ∧ ¬P), or ¬P ∨ P.

P → ((P → Q)→ Q).P → (Q → P).(P → (Q → R))→ ((P → Q)→ (P → R)).(((¬P → ¬Q)→ (Q → P)).

Theorems

Theorems are wffs deduced from no assumptions. They are justtautologies. (At least in this book)¬(P ∧ ¬P), or ¬P ∨ P.P → ((P → Q)→ Q).

P → (Q → P).(P → (Q → R))→ ((P → Q)→ (P → R)).(((¬P → ¬Q)→ (Q → P)).

Theorems

Theorems are wffs deduced from no assumptions. They are justtautologies. (At least in this book)¬(P ∧ ¬P), or ¬P ∨ P.P → ((P → Q)→ Q).P → (Q → P).

(P → (Q → R))→ ((P → Q)→ (P → R)).(((¬P → ¬Q)→ (Q → P)).

Theorems

Theorems are wffs deduced from no assumptions. They are justtautologies. (At least in this book)¬(P ∧ ¬P), or ¬P ∨ P.P → ((P → Q)→ Q).P → (Q → P).(P → (Q → R))→ ((P → Q)→ (P → R)).

(((¬P → ¬Q)→ (Q → P)).

Theorems

Theorems are wffs deduced from no assumptions. They are justtautologies. (At least in this book)¬(P ∧ ¬P), or ¬P ∨ P.P → ((P → Q)→ Q).P → (Q → P).(P → (Q → R))→ ((P → Q)→ (P → R)).(((¬P → ¬Q)→ (Q → P)).

Theorems

Example

Deduce (Prove) ` ((¬P → ¬Q)→ (Q → P)).

1. :¬P → ¬Q H. for→ I.2. :: Q. H for→ I.3. ::: ¬P4. ::: ¬Q. 1.3.5. ::: Q ∧ ¬Q.6. :: P7. : Q → P. 2-58. ¬P → ¬Q → (Q → P).

Theorems

Example

Deduce (Prove) ` ((¬P → ¬Q)→ (Q → P)).1. :¬P → ¬Q H. for→ I.

2. :: Q. H for→ I.3. ::: ¬P4. ::: ¬Q. 1.3.5. ::: Q ∧ ¬Q.6. :: P7. : Q → P. 2-58. ¬P → ¬Q → (Q → P).

Theorems

Example

Deduce (Prove) ` ((¬P → ¬Q)→ (Q → P)).1. :¬P → ¬Q H. for→ I.2. :: Q. H for→ I.

3. ::: ¬P4. ::: ¬Q. 1.3.5. ::: Q ∧ ¬Q.6. :: P7. : Q → P. 2-58. ¬P → ¬Q → (Q → P).

Theorems

Example

Deduce (Prove) ` ((¬P → ¬Q)→ (Q → P)).1. :¬P → ¬Q H. for→ I.2. :: Q. H for→ I.3. ::: ¬P

4. ::: ¬Q. 1.3.5. ::: Q ∧ ¬Q.6. :: P7. : Q → P. 2-58. ¬P → ¬Q → (Q → P).

Theorems

Example

Deduce (Prove) ` ((¬P → ¬Q)→ (Q → P)).1. :¬P → ¬Q H. for→ I.2. :: Q. H for→ I.3. ::: ¬P4. ::: ¬Q. 1.3.

5. ::: Q ∧ ¬Q.6. :: P7. : Q → P. 2-58. ¬P → ¬Q → (Q → P).

Theorems

Example

Deduce (Prove) ` ((¬P → ¬Q)→ (Q → P)).1. :¬P → ¬Q H. for→ I.2. :: Q. H for→ I.3. ::: ¬P4. ::: ¬Q. 1.3.5. ::: Q ∧ ¬Q.

6. :: P7. : Q → P. 2-58. ¬P → ¬Q → (Q → P).

Theorems

Example

Deduce (Prove) ` ((¬P → ¬Q)→ (Q → P)).1. :¬P → ¬Q H. for→ I.2. :: Q. H for→ I.3. ::: ¬P4. ::: ¬Q. 1.3.5. ::: Q ∧ ¬Q.6. :: P

7. : Q → P. 2-58. ¬P → ¬Q → (Q → P).

Theorems

Example

Deduce (Prove) ` ((¬P → ¬Q)→ (Q → P)).1. :¬P → ¬Q H. for→ I.2. :: Q. H for→ I.3. ::: ¬P4. ::: ¬Q. 1.3.5. ::: Q ∧ ¬Q.6. :: P7. : Q → P. 2-5

8. ¬P → ¬Q → (Q → P).

Theorems

Example

Deduce (Prove) ` ((¬P → ¬Q)→ (Q → P)).1. :¬P → ¬Q H. for→ I.2. :: Q. H for→ I.3. ::: ¬P4. ::: ¬Q. 1.3.5. ::: Q ∧ ¬Q.6. :: P7. : Q → P. 2-58. ¬P → ¬Q → (Q → P).

Theorems

Example

Deduce ` (P → (Q → R))→ ((P → Q)→ (P → R)).

1. : (P → (Q → R)) for→ I.2. :: (P → Q) for→ I.3. ::: P for→ I.4. ::: (Q → R) 1.3.5. ::: P → R. 2.4.6. ::: R.7. :: P → R. 3-68. : (P → Q)→ (P → R). 2-79. (P → (Q → R))→ ((P → Q)→ (P → R)) 1-8.

Theorems

Example

Deduce ` (P → (Q → R))→ ((P → Q)→ (P → R)).1. : (P → (Q → R)) for→ I.

2. :: (P → Q) for→ I.3. ::: P for→ I.4. ::: (Q → R) 1.3.5. ::: P → R. 2.4.6. ::: R.7. :: P → R. 3-68. : (P → Q)→ (P → R). 2-79. (P → (Q → R))→ ((P → Q)→ (P → R)) 1-8.

Theorems

Example

Deduce ` (P → (Q → R))→ ((P → Q)→ (P → R)).1. : (P → (Q → R)) for→ I.2. :: (P → Q) for→ I.

3. ::: P for→ I.4. ::: (Q → R) 1.3.5. ::: P → R. 2.4.6. ::: R.7. :: P → R. 3-68. : (P → Q)→ (P → R). 2-79. (P → (Q → R))→ ((P → Q)→ (P → R)) 1-8.

Theorems

Example

Deduce ` (P → (Q → R))→ ((P → Q)→ (P → R)).1. : (P → (Q → R)) for→ I.2. :: (P → Q) for→ I.3. ::: P for→ I.

4. ::: (Q → R) 1.3.5. ::: P → R. 2.4.6. ::: R.7. :: P → R. 3-68. : (P → Q)→ (P → R). 2-79. (P → (Q → R))→ ((P → Q)→ (P → R)) 1-8.

Theorems

Example

Deduce ` (P → (Q → R))→ ((P → Q)→ (P → R)).1. : (P → (Q → R)) for→ I.2. :: (P → Q) for→ I.3. ::: P for→ I.4. ::: (Q → R) 1.3.

5. ::: P → R. 2.4.6. ::: R.7. :: P → R. 3-68. : (P → Q)→ (P → R). 2-79. (P → (Q → R))→ ((P → Q)→ (P → R)) 1-8.

Theorems

Example

Deduce ` (P → (Q → R))→ ((P → Q)→ (P → R)).1. : (P → (Q → R)) for→ I.2. :: (P → Q) for→ I.3. ::: P for→ I.4. ::: (Q → R) 1.3.5. ::: P → R. 2.4.

6. ::: R.7. :: P → R. 3-68. : (P → Q)→ (P → R). 2-79. (P → (Q → R))→ ((P → Q)→ (P → R)) 1-8.

Theorems

Example

Deduce ` (P → (Q → R))→ ((P → Q)→ (P → R)).1. : (P → (Q → R)) for→ I.2. :: (P → Q) for→ I.3. ::: P for→ I.4. ::: (Q → R) 1.3.5. ::: P → R. 2.4.6. ::: R.

7. :: P → R. 3-68. : (P → Q)→ (P → R). 2-79. (P → (Q → R))→ ((P → Q)→ (P → R)) 1-8.

Theorems

Example

Deduce ` (P → (Q → R))→ ((P → Q)→ (P → R)).1. : (P → (Q → R)) for→ I.2. :: (P → Q) for→ I.3. ::: P for→ I.4. ::: (Q → R) 1.3.5. ::: P → R. 2.4.6. ::: R.7. :: P → R. 3-6

8. : (P → Q)→ (P → R). 2-79. (P → (Q → R))→ ((P → Q)→ (P → R)) 1-8.

Theorems

Example

Deduce ` (P → (Q → R))→ ((P → Q)→ (P → R)).1. : (P → (Q → R)) for→ I.2. :: (P → Q) for→ I.3. ::: P for→ I.4. ::: (Q → R) 1.3.5. ::: P → R. 2.4.6. ::: R.7. :: P → R. 3-68. : (P → Q)→ (P → R). 2-7

9. (P → (Q → R))→ ((P → Q)→ (P → R)) 1-8.

Theorems

Example

Deduce ` (P → (Q → R))→ ((P → Q)→ (P → R)).1. : (P → (Q → R)) for→ I.2. :: (P → Q) for→ I.3. ::: P for→ I.4. ::: (Q → R) 1.3.5. ::: P → R. 2.4.6. ::: R.7. :: P → R. 3-68. : (P → Q)→ (P → R). 2-79. (P → (Q → R))→ ((P → Q)→ (P → R)) 1-8.

Equivalences

Equivalences φ↔ ψ for two wff φ and ψ. We prove by φ→ ψ andψ → φ.

Clearly, equivalence is exactly a tautology for the form φ↔ ψ.The equivalences can be used to replace some subwffs withequivalent subwffs.¬(P ∧Q)↔ (¬P ∨ ¬Q). DeMorgan’s law. (DM)¬(P ∨Q)↔ ¬P ∧ ¬Q. (DM)P ∨Q ↔ Q ∨ P. Commutation (COM)P ∧Q ↔ Q ∧ P. (COM)P ∨ (Q ∨ R)↔ (P ∨Q) ∨ R. Association (ASSOC).P ∧ (Q ∧ R)↔ (P ∧Q) ∧ R. Association (ASSOC).

Equivalences

Equivalences φ↔ ψ for two wff φ and ψ. We prove by φ→ ψ andψ → φ.Clearly, equivalence is exactly a tautology for the form φ↔ ψ.

The equivalences can be used to replace some subwffs withequivalent subwffs.¬(P ∧Q)↔ (¬P ∨ ¬Q). DeMorgan’s law. (DM)¬(P ∨Q)↔ ¬P ∧ ¬Q. (DM)P ∨Q ↔ Q ∨ P. Commutation (COM)P ∧Q ↔ Q ∧ P. (COM)P ∨ (Q ∨ R)↔ (P ∨Q) ∨ R. Association (ASSOC).P ∧ (Q ∧ R)↔ (P ∧Q) ∧ R. Association (ASSOC).

Equivalences

Equivalences φ↔ ψ for two wff φ and ψ. We prove by φ→ ψ andψ → φ.Clearly, equivalence is exactly a tautology for the form φ↔ ψ.The equivalences can be used to replace some subwffs withequivalent subwffs.

¬(P ∧Q)↔ (¬P ∨ ¬Q). DeMorgan’s law. (DM)¬(P ∨Q)↔ ¬P ∧ ¬Q. (DM)P ∨Q ↔ Q ∨ P. Commutation (COM)P ∧Q ↔ Q ∧ P. (COM)P ∨ (Q ∨ R)↔ (P ∨Q) ∨ R. Association (ASSOC).P ∧ (Q ∧ R)↔ (P ∧Q) ∧ R. Association (ASSOC).

Equivalences

Equivalences φ↔ ψ for two wff φ and ψ. We prove by φ→ ψ andψ → φ.Clearly, equivalence is exactly a tautology for the form φ↔ ψ.The equivalences can be used to replace some subwffs withequivalent subwffs.¬(P ∧Q)↔ (¬P ∨ ¬Q). DeMorgan’s law. (DM)

¬(P ∨Q)↔ ¬P ∧ ¬Q. (DM)P ∨Q ↔ Q ∨ P. Commutation (COM)P ∧Q ↔ Q ∧ P. (COM)P ∨ (Q ∨ R)↔ (P ∨Q) ∨ R. Association (ASSOC).P ∧ (Q ∧ R)↔ (P ∧Q) ∧ R. Association (ASSOC).

Equivalences

Equivalences φ↔ ψ for two wff φ and ψ. We prove by φ→ ψ andψ → φ.Clearly, equivalence is exactly a tautology for the form φ↔ ψ.The equivalences can be used to replace some subwffs withequivalent subwffs.¬(P ∧Q)↔ (¬P ∨ ¬Q). DeMorgan’s law. (DM)¬(P ∨Q)↔ ¬P ∧ ¬Q. (DM)

P ∨Q ↔ Q ∨ P. Commutation (COM)P ∧Q ↔ Q ∧ P. (COM)P ∨ (Q ∨ R)↔ (P ∨Q) ∨ R. Association (ASSOC).P ∧ (Q ∧ R)↔ (P ∧Q) ∧ R. Association (ASSOC).

Equivalences

Equivalences φ↔ ψ for two wff φ and ψ. We prove by φ→ ψ andψ → φ.Clearly, equivalence is exactly a tautology for the form φ↔ ψ.The equivalences can be used to replace some subwffs withequivalent subwffs.¬(P ∧Q)↔ (¬P ∨ ¬Q). DeMorgan’s law. (DM)¬(P ∨Q)↔ ¬P ∧ ¬Q. (DM)P ∨Q ↔ Q ∨ P. Commutation (COM)

P ∧Q ↔ Q ∧ P. (COM)P ∨ (Q ∨ R)↔ (P ∨Q) ∨ R. Association (ASSOC).P ∧ (Q ∧ R)↔ (P ∧Q) ∧ R. Association (ASSOC).

Equivalences

Equivalences φ↔ ψ for two wff φ and ψ. We prove by φ→ ψ andψ → φ.Clearly, equivalence is exactly a tautology for the form φ↔ ψ.The equivalences can be used to replace some subwffs withequivalent subwffs.¬(P ∧Q)↔ (¬P ∨ ¬Q). DeMorgan’s law. (DM)¬(P ∨Q)↔ ¬P ∧ ¬Q. (DM)P ∨Q ↔ Q ∨ P. Commutation (COM)P ∧Q ↔ Q ∧ P. (COM)

P ∨ (Q ∨ R)↔ (P ∨Q) ∨ R. Association (ASSOC).P ∧ (Q ∧ R)↔ (P ∧Q) ∧ R. Association (ASSOC).

Equivalences

Equivalences φ↔ ψ for two wff φ and ψ. We prove by φ→ ψ andψ → φ.Clearly, equivalence is exactly a tautology for the form φ↔ ψ.The equivalences can be used to replace some subwffs withequivalent subwffs.¬(P ∧Q)↔ (¬P ∨ ¬Q). DeMorgan’s law. (DM)¬(P ∨Q)↔ ¬P ∧ ¬Q. (DM)P ∨Q ↔ Q ∨ P. Commutation (COM)P ∧Q ↔ Q ∧ P. (COM)P ∨ (Q ∨ R)↔ (P ∨Q) ∨ R. Association (ASSOC).

P ∧ (Q ∧ R)↔ (P ∧Q) ∧ R. Association (ASSOC).

Equivalences

Equivalences φ↔ ψ for two wff φ and ψ. We prove by φ→ ψ andψ → φ.Clearly, equivalence is exactly a tautology for the form φ↔ ψ.The equivalences can be used to replace some subwffs withequivalent subwffs.¬(P ∧Q)↔ (¬P ∨ ¬Q). DeMorgan’s law. (DM)¬(P ∨Q)↔ ¬P ∧ ¬Q. (DM)P ∨Q ↔ Q ∨ P. Commutation (COM)P ∧Q ↔ Q ∧ P. (COM)P ∨ (Q ∨ R)↔ (P ∨Q) ∨ R. Association (ASSOC).P ∧ (Q ∧ R)↔ (P ∧Q) ∧ R. Association (ASSOC).

Equivalences

More equivalences

P ∧ (Q ∨ R)↔ (P ∧Q) ∨ (P ∧ R). Distribution (DIST)

P ∨ (Q ∧ R)↔ (P ∨Q) ∧ (P ∨ R). (DIST)P ↔ ¬¬P. Double negation (DN)(P → Q)↔ (¬Q → ¬P). Transposition (TRANS)(P → Q)↔ (¬P ∨Q). Material Implication (MI)(P ∧Q)→ R ↔ (P → (Q → R)). Exportation (EXP)P ↔ (P ∧ P). Tautology (TAUT)P ↔ (P ∨ P). (TAUT)The equivalences can be verified by the truth table method or bydeduction.

Equivalences

More equivalences

P ∧ (Q ∨ R)↔ (P ∧Q) ∨ (P ∧ R). Distribution (DIST)P ∨ (Q ∧ R)↔ (P ∨Q) ∧ (P ∨ R). (DIST)

P ↔ ¬¬P. Double negation (DN)(P → Q)↔ (¬Q → ¬P). Transposition (TRANS)(P → Q)↔ (¬P ∨Q). Material Implication (MI)(P ∧Q)→ R ↔ (P → (Q → R)). Exportation (EXP)P ↔ (P ∧ P). Tautology (TAUT)P ↔ (P ∨ P). (TAUT)The equivalences can be verified by the truth table method or bydeduction.

Equivalences

More equivalences

P ∧ (Q ∨ R)↔ (P ∧Q) ∨ (P ∧ R). Distribution (DIST)P ∨ (Q ∧ R)↔ (P ∨Q) ∧ (P ∨ R). (DIST)P ↔ ¬¬P. Double negation (DN)

(P → Q)↔ (¬Q → ¬P). Transposition (TRANS)(P → Q)↔ (¬P ∨Q). Material Implication (MI)(P ∧Q)→ R ↔ (P → (Q → R)). Exportation (EXP)P ↔ (P ∧ P). Tautology (TAUT)P ↔ (P ∨ P). (TAUT)The equivalences can be verified by the truth table method or bydeduction.

Equivalences

More equivalences

P ∧ (Q ∨ R)↔ (P ∧Q) ∨ (P ∧ R). Distribution (DIST)P ∨ (Q ∧ R)↔ (P ∨Q) ∧ (P ∨ R). (DIST)P ↔ ¬¬P. Double negation (DN)(P → Q)↔ (¬Q → ¬P). Transposition (TRANS)

(P → Q)↔ (¬P ∨Q). Material Implication (MI)(P ∧Q)→ R ↔ (P → (Q → R)). Exportation (EXP)P ↔ (P ∧ P). Tautology (TAUT)P ↔ (P ∨ P). (TAUT)The equivalences can be verified by the truth table method or bydeduction.

Equivalences

More equivalences

P ∧ (Q ∨ R)↔ (P ∧Q) ∨ (P ∧ R). Distribution (DIST)P ∨ (Q ∧ R)↔ (P ∨Q) ∧ (P ∨ R). (DIST)P ↔ ¬¬P. Double negation (DN)(P → Q)↔ (¬Q → ¬P). Transposition (TRANS)(P → Q)↔ (¬P ∨Q). Material Implication (MI)

(P ∧Q)→ R ↔ (P → (Q → R)). Exportation (EXP)P ↔ (P ∧ P). Tautology (TAUT)P ↔ (P ∨ P). (TAUT)The equivalences can be verified by the truth table method or bydeduction.

Equivalences

More equivalences

P ∧ (Q ∨ R)↔ (P ∧Q) ∨ (P ∧ R). Distribution (DIST)P ∨ (Q ∧ R)↔ (P ∨Q) ∧ (P ∨ R). (DIST)P ↔ ¬¬P. Double negation (DN)(P → Q)↔ (¬Q → ¬P). Transposition (TRANS)(P → Q)↔ (¬P ∨Q). Material Implication (MI)(P ∧Q)→ R ↔ (P → (Q → R)). Exportation (EXP)

P ↔ (P ∧ P). Tautology (TAUT)P ↔ (P ∨ P). (TAUT)The equivalences can be verified by the truth table method or bydeduction.

Equivalences

More equivalences

P ∧ (Q ∨ R)↔ (P ∧Q) ∨ (P ∧ R). Distribution (DIST)P ∨ (Q ∧ R)↔ (P ∨Q) ∧ (P ∨ R). (DIST)P ↔ ¬¬P. Double negation (DN)(P → Q)↔ (¬Q → ¬P). Transposition (TRANS)(P → Q)↔ (¬P ∨Q). Material Implication (MI)(P ∧Q)→ R ↔ (P → (Q → R)). Exportation (EXP)P ↔ (P ∧ P). Tautology (TAUT)

P ↔ (P ∨ P). (TAUT)The equivalences can be verified by the truth table method or bydeduction.

Equivalences

More equivalences

P ∧ (Q ∨ R)↔ (P ∧Q) ∨ (P ∧ R). Distribution (DIST)P ∨ (Q ∧ R)↔ (P ∨Q) ∧ (P ∨ R). (DIST)P ↔ ¬¬P. Double negation (DN)(P → Q)↔ (¬Q → ¬P). Transposition (TRANS)(P → Q)↔ (¬P ∨Q). Material Implication (MI)(P ∧Q)→ R ↔ (P → (Q → R)). Exportation (EXP)P ↔ (P ∧ P). Tautology (TAUT)P ↔ (P ∨ P). (TAUT)

The equivalences can be verified by the truth table method or bydeduction.

Equivalences

More equivalences

P ∧ (Q ∨ R)↔ (P ∧Q) ∨ (P ∧ R). Distribution (DIST)P ∨ (Q ∧ R)↔ (P ∨Q) ∧ (P ∨ R). (DIST)P ↔ ¬¬P. Double negation (DN)(P → Q)↔ (¬Q → ¬P). Transposition (TRANS)(P → Q)↔ (¬P ∨Q). Material Implication (MI)(P ∧Q)→ R ↔ (P → (Q → R)). Exportation (EXP)P ↔ (P ∧ P). Tautology (TAUT)P ↔ (P ∨ P). (TAUT)The equivalences can be verified by the truth table method or bydeduction.

Equivalences

More derived rules

Theorem introduction (TI): Any substituted version of a theoremmay be introduced with at any line of the proof.

Equivalence introduction (using above notations): Given χ withsubwff φ and an equivalence φ↔ ψ, we deduce χ′ with somesubwffs of form φ replaced with subwffs of form ψ.

Equivalences

More derived rules

Theorem introduction (TI): Any substituted version of a theoremmay be introduced with at any line of the proof.Equivalence introduction (using above notations): Given χ withsubwff φ and an equivalence φ↔ ψ, we deduce χ′ with somesubwffs of form φ replaced with subwffs of form ψ.

Equivalences

Example

We use the equivalence ¬P ∨Q ↔ ¬(P ∧ ¬Q). DM.

We shall prove that ¬P ∨Q, ` P → Q1. ¬P ∨Q. A2. : P H. for→ I.3. ::¬Q H for ¬I.4. :: P ∧ ¬Q.5. :: ¬(P ∧ ¬Q). 1. (DM)6. :: (P ∧ ¬Q) ∧ ¬(P ∧ ¬Q).7. : Q. 3-68. P → Q. 2-7.

Equivalences

Example

We use the equivalence ¬P ∨Q ↔ ¬(P ∧ ¬Q). DM.We shall prove that ¬P ∨Q, ` P → Q

1. ¬P ∨Q. A2. : P H. for→ I.3. ::¬Q H for ¬I.4. :: P ∧ ¬Q.5. :: ¬(P ∧ ¬Q). 1. (DM)6. :: (P ∧ ¬Q) ∧ ¬(P ∧ ¬Q).7. : Q. 3-68. P → Q. 2-7.

Equivalences

Example

We use the equivalence ¬P ∨Q ↔ ¬(P ∧ ¬Q). DM.We shall prove that ¬P ∨Q, ` P → Q1. ¬P ∨Q. A

2. : P H. for→ I.3. ::¬Q H for ¬I.4. :: P ∧ ¬Q.5. :: ¬(P ∧ ¬Q). 1. (DM)6. :: (P ∧ ¬Q) ∧ ¬(P ∧ ¬Q).7. : Q. 3-68. P → Q. 2-7.

Equivalences

Example

We use the equivalence ¬P ∨Q ↔ ¬(P ∧ ¬Q). DM.We shall prove that ¬P ∨Q, ` P → Q1. ¬P ∨Q. A2. : P H. for→ I.

3. ::¬Q H for ¬I.4. :: P ∧ ¬Q.5. :: ¬(P ∧ ¬Q). 1. (DM)6. :: (P ∧ ¬Q) ∧ ¬(P ∧ ¬Q).7. : Q. 3-68. P → Q. 2-7.

Equivalences

Example

We use the equivalence ¬P ∨Q ↔ ¬(P ∧ ¬Q). DM.We shall prove that ¬P ∨Q, ` P → Q1. ¬P ∨Q. A2. : P H. for→ I.3. ::¬Q H for ¬I.

4. :: P ∧ ¬Q.5. :: ¬(P ∧ ¬Q). 1. (DM)6. :: (P ∧ ¬Q) ∧ ¬(P ∧ ¬Q).7. : Q. 3-68. P → Q. 2-7.

Equivalences

Example

We use the equivalence ¬P ∨Q ↔ ¬(P ∧ ¬Q). DM.We shall prove that ¬P ∨Q, ` P → Q1. ¬P ∨Q. A2. : P H. for→ I.3. ::¬Q H for ¬I.4. :: P ∧ ¬Q.

5. :: ¬(P ∧ ¬Q). 1. (DM)6. :: (P ∧ ¬Q) ∧ ¬(P ∧ ¬Q).7. : Q. 3-68. P → Q. 2-7.

Equivalences

Example

We use the equivalence ¬P ∨Q ↔ ¬(P ∧ ¬Q). DM.We shall prove that ¬P ∨Q, ` P → Q1. ¬P ∨Q. A2. : P H. for→ I.3. ::¬Q H for ¬I.4. :: P ∧ ¬Q.5. :: ¬(P ∧ ¬Q). 1. (DM)

6. :: (P ∧ ¬Q) ∧ ¬(P ∧ ¬Q).7. : Q. 3-68. P → Q. 2-7.

Equivalences

Example

We use the equivalence ¬P ∨Q ↔ ¬(P ∧ ¬Q). DM.We shall prove that ¬P ∨Q, ` P → Q1. ¬P ∨Q. A2. : P H. for→ I.3. ::¬Q H for ¬I.4. :: P ∧ ¬Q.5. :: ¬(P ∧ ¬Q). 1. (DM)6. :: (P ∧ ¬Q) ∧ ¬(P ∧ ¬Q).

7. : Q. 3-68. P → Q. 2-7.

Equivalences

Example

We use the equivalence ¬P ∨Q ↔ ¬(P ∧ ¬Q). DM.We shall prove that ¬P ∨Q, ` P → Q1. ¬P ∨Q. A2. : P H. for→ I.3. ::¬Q H for ¬I.4. :: P ∧ ¬Q.5. :: ¬(P ∧ ¬Q). 1. (DM)6. :: (P ∧ ¬Q) ∧ ¬(P ∧ ¬Q).7. : Q. 3-6

8. P → Q. 2-7.

Equivalences

Example

We use the equivalence ¬P ∨Q ↔ ¬(P ∧ ¬Q). DM.We shall prove that ¬P ∨Q, ` P → Q1. ¬P ∨Q. A2. : P H. for→ I.3. ::¬Q H for ¬I.4. :: P ∧ ¬Q.5. :: ¬(P ∧ ¬Q). 1. (DM)6. :: (P ∧ ¬Q) ∧ ¬(P ∧ ¬Q).7. : Q. 3-68. P → Q. 2-7.

Equivalences Soundness and completeness of deductions

Soundness

A logical system is a formal system with

I An alphabet, a set of statement symbols with logical connectives.I well-formed formulasI A set of axioms.I Rules of inference.

See also http://plato.stanford.edu/entries/logic-classical/

A logical system is consistent if not all wff can be deduced.(Equivalently, exactly one of φ and ¬φ can be deduced.)Given any truth-false assigment to atomic formula so that theaxioms are all true, the soundness means that by applying rules ofinference you obtain true statements only.That is, we cannot deduce a falsehood.

Soundness

A logical system is a formal system withI An alphabet, a set of statement symbols with logical connectives.

I well-formed formulasI A set of axioms.I Rules of inference.

Soundness

A logical system is a formal system withI An alphabet, a set of statement symbols with logical connectives.I well-formed formulas

I A set of axioms.I Rules of inference.

Soundness

A logical system is a formal system withI An alphabet, a set of statement symbols with logical connectives.I well-formed formulasI A set of axioms.

I Rules of inference.

Soundness

A logical system is a formal system withI An alphabet, a set of statement symbols with logical connectives.I well-formed formulasI A set of axioms.I Rules of inference.

Soundness

A logical system is consistent if not all wff can be deduced.(Equivalently, exactly one of φ and ¬φ can be deduced.)

Given any truth-false assigment to atomic formula so that theaxioms are all true, the soundness means that by applying rules ofinference you obtain true statements only.That is, we cannot deduce a falsehood.

Soundness

A logical system is consistent if not all wff can be deduced.(Equivalently, exactly one of φ and ¬φ can be deduced.)Given any truth-false assigment to atomic formula so that theaxioms are all true, the soundness means that by applying rules ofinference you obtain true statements only.

That is, we cannot deduce a falsehood.

Soundness

Completeness

The completeness means that if a formula is true from logical truthassignment from a set of assumptions Σ, then the formula can bededuced from Σ.

This is true for the first order theories but not true for higher-ordertheories. Also, true if there are finitely or countably manystatement symbols.See Theorem 4.3 of Cameron.If we go to a higher-order theory, this fails. (Gödel’sincompleteness theorems: Theorem 5.8 in Cameron)

Completeness

The completeness means that if a formula is true from logical truthassignment from a set of assumptions Σ, then the formula can bededuced from Σ.This is true for the first order theories but not true for higher-ordertheories. Also, true if there are finitely or countably manystatement symbols.

See Theorem 4.3 of Cameron.If we go to a higher-order theory, this fails. (Gödel’sincompleteness theorems: Theorem 5.8 in Cameron)

Completeness

The completeness means that if a formula is true from logical truthassignment from a set of assumptions Σ, then the formula can bededuced from Σ.This is true for the first order theories but not true for higher-ordertheories. Also, true if there are finitely or countably manystatement symbols.See Theorem 4.3 of Cameron.

If we go to a higher-order theory, this fails. (Gödel’sincompleteness theorems: Theorem 5.8 in Cameron)

Completeness

The completeness means that if a formula is true from logical truthassignment from a set of assumptions Σ, then the formula can bededuced from Σ.This is true for the first order theories but not true for higher-ordertheories. Also, true if there are finitely or countably manystatement symbols.See Theorem 4.3 of Cameron.If we go to a higher-order theory, this fails. (Gödel’sincompleteness theorems: Theorem 5.8 in Cameron)

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