Logic and the Set Theory Lecture 5: Propositional Calculus: Part 1

Logic and the set theory Lecture 5: Propositional Calculus: part 1

S. Choi

Department of Mathematical Science KAIST, Daejeon, South Korea

Fall semester, 2012

S. Choi (KAIST) Logic and set theory September 18, 2012 1 / 16 Inference Rules Hypothetical Rules Derived Rules The Propositional Rules Equivalences Course homepages: http://mathsci.kaist.ac.kr/~schoi/logic.html and the moodle page http://moodle.kaist.ac.kr Grading and so on in the moodle. Ask questions in moodle.

Introduction About this lecture

Notions of Inference

S. Choi (KAIST) Logic and set theory September 18, 2012 2 / 16 Hypothetical Rules Derived Rules The Propositional Rules Equivalences Course homepages: http://mathsci.kaist.ac.kr/~schoi/logic.html and the moodle page http://moodle.kaist.ac.kr Grading and so on in the moodle. Ask questions in moodle.

Introduction About this lecture

Notions of Inference Inference Rules

S. Choi (KAIST) Logic and set theory September 18, 2012 2 / 16 Derived Rules The Propositional Rules Equivalences Course homepages: http://mathsci.kaist.ac.kr/~schoi/logic.html and the moodle page http://moodle.kaist.ac.kr Grading and so on in the moodle. Ask questions in moodle.

Introduction About this lecture

Notions of Inference Inference Rules Hypothetical Rules

S. Choi (KAIST) Logic and set theory September 18, 2012 2 / 16 The Propositional Rules Equivalences Course homepages: http://mathsci.kaist.ac.kr/~schoi/logic.html and the moodle page http://moodle.kaist.ac.kr Grading and so on in the moodle. Ask questions in moodle.

Introduction About this lecture

Notions of Inference Inference Rules Hypothetical Rules Derived Rules

S. Choi (KAIST) Logic and set theory September 18, 2012 2 / 16 Equivalences Course homepages: http://mathsci.kaist.ac.kr/~schoi/logic.html and the moodle page http://moodle.kaist.ac.kr Grading and so on in the moodle. Ask questions in moodle.

Introduction About this lecture

Notions of Inference Inference Rules Hypothetical Rules Derived Rules The Propositional Rules

S. Choi (KAIST) Logic and set theory September 18, 2012 2 / 16 Course homepages: http://mathsci.kaist.ac.kr/~schoi/logic.html and the moodle page http://moodle.kaist.ac.kr Grading and so on in the moodle. Ask questions in moodle.

Introduction About this lecture

Notions of Inference Inference Rules Hypothetical Rules Derived Rules The Propositional Rules Equivalences

S. Choi (KAIST) Logic and set theory September 18, 2012 2 / 16 Grading and so on in the moodle. Ask questions in moodle.

Introduction About this lecture

Notions of Inference Inference Rules Hypothetical Rules Derived Rules The Propositional Rules Equivalences Course homepages: http://mathsci.kaist.ac.kr/~schoi/logic.html and the moodle page http://moodle.kaist.ac.kr

S. Choi (KAIST) Logic and set theory September 18, 2012 2 / 16 Introduction About this lecture

S. Choi (KAIST) Logic and set theory September 18, 2012 2 / 16 A mathematical introduction to logic, H. Enderton, Academic Press. Whitehead, Russell, Principia Mathematica (our library). (This could be a project idea. ) http://plato.stanford.edu/contents.html has much resource. See “Realism, Informal logic 2. Deductivism and beyond,” and “Nondeductive methods in mathematics.” http: //ocw.mit.edu/OcwWeb/Linguistics-and-Philosophy/ 24-241Fall-2005/CourseHome/ See "Derivations in Sentential Calculus". (or SC Derivations.) http://jvrosset.free.fr/Goedel-Proof-Truth.pdf “Does Godels incompleteness prove that truth transcends proof?”

Introduction Some helpful references

Richard Jeffrey, Formal logic: its scope and limits, Mc Graw Hill

S. Choi (KAIST) Logic and set theory September 18, 2012 3 / 16 Whitehead, Russell, Principia Mathematica (our library). (This could be a project idea. ) http://plato.stanford.edu/contents.html has much resource. See “Realism, Informal logic 2. Deductivism and beyond,” and “Nondeductive methods in mathematics.” http: //ocw.mit.edu/OcwWeb/Linguistics-and-Philosophy/ 24-241Fall-2005/CourseHome/ See "Derivations in Sentential Calculus". (or SC Derivations.) http://jvrosset.free.fr/Goedel-Proof-Truth.pdf “Does Godels incompleteness prove that truth transcends proof?”

Introduction Some helpful references

Richard Jeffrey, Formal logic: its scope and limits, Mc Graw Hill A mathematical introduction to logic, H. Enderton, Academic Press.

S. Choi (KAIST) Logic and set theory September 18, 2012 3 / 16 http://plato.stanford.edu/contents.html has much resource. See “Realism, Informal logic 2. Deductivism and beyond,” and “Nondeductive methods in mathematics.” http: //ocw.mit.edu/OcwWeb/Linguistics-and-Philosophy/ 24-241Fall-2005/CourseHome/ See "Derivations in Sentential Calculus". (or SC Derivations.) http://jvrosset.free.fr/Goedel-Proof-Truth.pdf “Does Godels incompleteness prove that truth transcends proof?”

Introduction Some helpful references

S. Choi (KAIST) Logic and set theory September 18, 2012 3 / 16 http: //ocw.mit.edu/OcwWeb/Linguistics-and-Philosophy/ 24-241Fall-2005/CourseHome/ See "Derivations in Sentential Calculus". (or SC Derivations.) http://jvrosset.free.fr/Goedel-Proof-Truth.pdf “Does Godels incompleteness prove that truth transcends proof?”

Introduction Some helpful references

S. Choi (KAIST) Logic and set theory September 18, 2012 3 / 16 http://jvrosset.free.fr/Goedel-Proof-Truth.pdf “Does Godels incompleteness prove that truth transcends proof?”

Introduction Some helpful references

Richard Jeffrey, Formal logic: its scope and limits, Mc Graw Hill A mathematical introduction to logic, H. Enderton, Academic Press. Whitehead, Russell, Principia Mathematica (our library). (This could be a project idea. ) http://plato.stanford.edu/contents.html has much resource. See “Realism, Informal logic 2. Deductivism and beyond,” and “Nondeductive methods in mathematics.” http: //ocw.mit.edu/OcwWeb/Linguistics-and-Philosophy/ 24-241Fall-2005/CourseHome/ See "Derivations in Sentential Calculus". (or SC Derivations.)

S. Choi (KAIST) Logic and set theory September 18, 2012 3 / 16 Introduction Some helpful references

S. Choi (KAIST) Logic and set theory September 18, 2012 3 / 16 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 Some helpful references

http://en.wikipedia.org/wiki/Truth_table,

S. Choi (KAIST) Logic and set theory September 18, 2012 4 / 16 http: //svn.oriontransfer.org/TruthTable/index.rhtml, has xor, complete.

Introduction Some helpful references

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)

S. Choi (KAIST) Logic and set theory September 18, 2012 4 / 16 Introduction Some helpful references

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.

S. Choi (KAIST) Logic and set theory September 18, 2012 4 / 16 Realism believes in the existence and the independence of certain objects and so on. This is very close to the logical atomism. Antirealism: One has to test to ﬁnd out before it can considered to exists and so on. Since we do not know everything, which should we take as our position?

The notion of inference The realism and antirealism

If the tree fall in a forest, and no one was there to heard it, did it make a sound? (Berkeley)

S. Choi (KAIST) Logic and set theory September 18, 2012 5 / 16 Antirealism: One has to test to ﬁnd out before it can considered to exists and so on. Since we do not know everything, which should we take as our position?

The notion of inference The realism and antirealism

S. Choi (KAIST) Logic and set theory September 18, 2012 5 / 16 Since we do not know everything, which should we take as our position?

The notion of inference The realism and antirealism

If the tree fall in a forest, and no one was there to heard it, did it make a sound? (Berkeley) Realism believes in the existence and the independence of certain objects and so on. This is very close to the logical atomism. Antirealism: One has to test to ﬁnd out before it can considered to exists and so on.

S. Choi (KAIST) Logic and set theory September 18, 2012 5 / 16 The notion of inference The realism and antirealism

S. Choi (KAIST) Logic and set theory September 18, 2012 5 / 16 We give a collection of ten rules of inference that gives you true statements from assumptions. (The collection of rules depend on books but essentially equivalent.) (Some Postmordernist will call these just Rhetorics.) Inference = Deduction = Proof. This is actually weaker than TF table or truth tree method. (in higher-order logic) If you take the antirealist’s position, the deductions are only valid method. But we could also take the realist’s position. The reason for doing it is that for Predicate calculus, TF methods cannot work since we have to check inﬁnitely many cases. (incompleteness)

The notion of inference The notion of inference

From a valid set of "assumptions" or "theorems" we wish to deduce more true statements.

S. Choi (KAIST) Logic and set theory September 18, 2012 6 / 16 Inference = Deduction = Proof. This is actually weaker than TF table or truth tree method. (in higher-order logic) If you take the antirealist’s position, the deductions are only valid method. But we could also take the realist’s position. The reason for doing it is that for Predicate calculus, TF methods cannot work since we have to check inﬁnitely many cases. (incompleteness)

The notion of inference The notion of inference

S. Choi (KAIST) Logic and set theory September 18, 2012 6 / 16 This is actually weaker than TF table or truth tree method. (in higher-order logic) If you take the antirealist’s position, the deductions are only valid method. But we could also take the realist’s position. The reason for doing it is that for Predicate calculus, TF methods cannot work since we have to check inﬁnitely many cases. (incompleteness)

The notion of inference The notion of inference

S. Choi (KAIST) Logic and set theory September 18, 2012 6 / 16 If you take the antirealist’s position, the deductions are only valid method. But we could also take the realist’s position. The reason for doing it is that for Predicate calculus, TF methods cannot work since we have to check inﬁnitely many cases. (incompleteness)

The notion of inference The notion of inference

S. Choi (KAIST) Logic and set theory September 18, 2012 6 / 16 The reason for doing it is that for Predicate calculus, TF methods cannot work since we have to check inﬁnitely many cases. (incompleteness)

The notion of inference The notion of inference

From a valid set of "assumptions" or "theorems" we wish to deduce more true statements. We give a collection of ten rules of inference that gives you true statements from assumptions. (The collection of rules depend on books but essentially equivalent.) (Some Postmordernist will call these just Rhetorics.) Inference = Deduction = Proof. This is actually weaker than TF table or truth tree method. (in higher-order logic) If you take the antirealist’s position, the deductions are only valid method. But we could also take the realist’s position.

S. Choi (KAIST) Logic and set theory September 18, 2012 6 / 16 The notion of inference The notion of inference

S. Choi (KAIST) Logic and set theory September 18, 2012 6 / 16 P, P → Q, ` Q. We can check in truth table. Example: P, Q → R, P → Q, ` R. We need two (→ E).

Nonhypothetical Inference Rules Nonhypothetical Inference Rules

Modus Ponens or condition eliminations (→ E). From a conditional and its antecendent, we can infer the consequent.

S. Choi (KAIST) Logic and set theory September 18, 2012 7 / 16 We can check in truth table. Example: P, Q → R, P → Q, ` R. We need two (→ E).

Nonhypothetical Inference Rules Nonhypothetical Inference Rules

Modus Ponens or condition eliminations (→ E). From a conditional and its antecendent, we can infer the consequent. P, P → Q, ` Q.

S. Choi (KAIST) Logic and set theory September 18, 2012 7 / 16 Example: P, Q → R, P → Q, ` R. We need two (→ E).

Nonhypothetical Inference Rules Nonhypothetical Inference Rules

Modus Ponens or condition eliminations (→ E). From a conditional and its antecendent, we can infer the consequent. P, P → Q, ` Q. We can check in truth table.

S. Choi (KAIST) Logic and set theory September 18, 2012 7 / 16 P, Q → R, P → Q, ` R. We need two (→ E).

Nonhypothetical Inference Rules Nonhypothetical Inference Rules

Modus Ponens or condition eliminations (→ E). From a conditional and its antecendent, we can infer the consequent. P, P → Q, ` Q. We can check in truth table. Example:

S. Choi (KAIST) Logic and set theory September 18, 2012 7 / 16 We need two (→ E).

Nonhypothetical Inference Rules Nonhypothetical Inference Rules

Modus Ponens or condition eliminations (→ E). From a conditional and its antecendent, we can infer the consequent. P, P → Q, ` Q. We can check in truth table. Example: P, Q → R, P → Q, ` R.

S. Choi (KAIST) Logic and set theory September 18, 2012 7 / 16 Nonhypothetical Inference Rules Nonhypothetical Inference Rules

Modus Ponens or condition eliminations (→ E). From a conditional and its antecendent, we can infer the consequent. P, P → Q, ` Q. We can check in truth table. Example: P, Q → R, P → Q, ` R. We need two (→ E).

S. Choi (KAIST) Logic and set theory September 18, 2012 7 / 16 Conjunction introduction (∧I): φ, ψ → φ ∧ ψ. Conjunction elimination (∧E): φ ∧ ψ → φ, ψ. Disjunction introduction (∨I): φ → φ ∨ ψ for any wff ψ. Disjuction elimination (∨E): φ ∨ ψ, φ → χ, ψ → χ. Then infer χ. Biconditional introduction.(↔ I): φ → ψ, ψ → φ. Then φ ↔ ψ. Biconditional elimination. (↔ E): φ ↔ ψ. Then φ → ψ, ψ → φ.

Nonhypothetical Inference Rules More rules

Negation elimination (¬E): ¬¬φ → φ.

S. Choi (KAIST) Logic and set theory September 18, 2012 8 / 16 Conjunction elimination (∧E): φ ∧ ψ → φ, ψ. Disjunction introduction (∨I): φ → φ ∨ ψ for any wff ψ. Disjuction elimination (∨E): φ ∨ ψ, φ → χ, ψ → χ. Then infer χ. Biconditional introduction.(↔ I): φ → ψ, ψ → φ. Then φ ↔ ψ. Biconditional elimination. (↔ E): φ ↔ ψ. Then φ → ψ, ψ → φ.

Nonhypothetical Inference Rules More rules

Negation elimination (¬E): ¬¬φ → φ. Conjunction introduction (∧I): φ, ψ → φ ∧ ψ.

S. Choi (KAIST) Logic and set theory September 18, 2012 8 / 16 Disjunction introduction (∨I): φ → φ ∨ ψ for any wff ψ. Disjuction elimination (∨E): φ ∨ ψ, φ → χ, ψ → χ. Then infer χ. Biconditional introduction.(↔ I): φ → ψ, ψ → φ. Then φ ↔ ψ. Biconditional elimination. (↔ E): φ ↔ ψ. Then φ → ψ, ψ → φ.

Nonhypothetical Inference Rules More rules

Negation elimination (¬E): ¬¬φ → φ. Conjunction introduction (∧I): φ, ψ → φ ∧ ψ. Conjunction elimination (∧E): φ ∧ ψ → φ, ψ.

S. Choi (KAIST) Logic and set theory September 18, 2012 8 / 16 Disjuction elimination (∨E): φ ∨ ψ, φ → χ, ψ → χ. Then infer χ. Biconditional introduction.(↔ I): φ → ψ, ψ → φ. Then φ ↔ ψ. Biconditional elimination. (↔ E): φ ↔ ψ. Then φ → ψ, ψ → φ.

Nonhypothetical Inference Rules More rules

Negation elimination (¬E): ¬¬φ → φ. Conjunction introduction (∧I): φ, ψ → φ ∧ ψ. Conjunction elimination (∧E): φ ∧ ψ → φ, ψ. Disjunction introduction (∨I): φ → φ ∨ ψ for any wff ψ.

S. Choi (KAIST) Logic and set theory September 18, 2012 8 / 16 Biconditional introduction.(↔ I): φ → ψ, ψ → φ. Then φ ↔ ψ. Biconditional elimination. (↔ E): φ ↔ ψ. Then φ → ψ, ψ → φ.

Nonhypothetical Inference Rules More rules

S. Choi (KAIST) Logic and set theory September 18, 2012 8 / 16 Biconditional elimination. (↔ E): φ ↔ ψ. Then φ → ψ, ψ → φ.

Nonhypothetical Inference Rules More rules

Negation elimination (¬E): ¬¬φ → φ. Conjunction introduction (∧I): φ, ψ → φ ∧ ψ. Conjunction elimination (∧E): φ ∧ ψ → φ, ψ. Disjunction introduction (∨I): φ → φ ∨ ψ for any wff ψ. Disjuction elimination (∨E): φ ∨ ψ, φ → χ, ψ → χ. Then infer χ. Biconditional introduction.(↔ I): φ → ψ, ψ → φ. Then φ ↔ ψ.

S. Choi (KAIST) Logic and set theory September 18, 2012 8 / 16 Nonhypothetical Inference Rules More rules

S. Choi (KAIST) Logic and set theory September 18, 2012 8 / 16 1. P. Assumption 2. P ∨ Q. 1. ∨I. 3. P ∨ R. 1. ∨I. 4. (P ∨ Q) ∧ (P ∨ R). 2.3. ∧I.

Nonhypothetical Inference Rules Example 1

P ` (P ∨ Q) ∧ (P ∨ R).

S. Choi (KAIST) Logic and set theory September 18, 2012 9 / 16 2. P ∨ Q. 1. ∨I. 3. P ∨ R. 1. ∨I. 4. (P ∨ Q) ∧ (P ∨ R). 2.3. ∧I.

Nonhypothetical Inference Rules Example 1

P ` (P ∨ Q) ∧ (P ∨ R). 1. P. Assumption

S. Choi (KAIST) Logic and set theory September 18, 2012 9 / 16 3. P ∨ R. 1. ∨I. 4. (P ∨ Q) ∧ (P ∨ R). 2.3. ∧I.

Nonhypothetical Inference Rules Example 1

P ` (P ∨ Q) ∧ (P ∨ R). 1. P. Assumption 2. P ∨ Q. 1. ∨I.

S. Choi (KAIST) Logic and set theory September 18, 2012 9 / 16 4. (P ∨ Q) ∧ (P ∨ R). 2.3. ∧I.

Nonhypothetical Inference Rules Example 1

P ` (P ∨ Q) ∧ (P ∨ R). 1. P. Assumption 2. P ∨ Q. 1. ∨I. 3. P ∨ R. 1. ∨I.

S. Choi (KAIST) Logic and set theory September 18, 2012 9 / 16 Nonhypothetical Inference Rules Example 1

P ` (P ∨ Q) ∧ (P ∨ R). 1. P. Assumption 2. P ∨ Q. 1. ∨I. 3. P ∨ R. 1. ∨I. 4. (P ∨ Q) ∧ (P ∨ R). 2.3. ∧I.

S. Choi (KAIST) Logic and set theory September 18, 2012 9 / 16 1. P. Assumption 2. ¬¬(P → Q). A. 3. P → Q. 2. ¬E. 4. Q. 1.3. → E. 5. (R ∧ S) ∨ Q 4. ∨I.

Nonhypothetical Inference Rules Example 2

P, ¬¬(P → Q) ` (R ∧ S) ∨ Q.

S. Choi (KAIST) Logic and set theory September 18, 2012 10 / 16 2. ¬¬(P → Q). A. 3. P → Q. 2. ¬E. 4. Q. 1.3. → E. 5. (R ∧ S) ∨ Q 4. ∨I.

Nonhypothetical Inference Rules Example 2

P, ¬¬(P → Q) ` (R ∧ S) ∨ Q. 1. P. Assumption

S. Choi (KAIST) Logic and set theory September 18, 2012 10 / 16 3. P → Q. 2. ¬E. 4. Q. 1.3. → E. 5. (R ∧ S) ∨ Q 4. ∨I.

Nonhypothetical Inference Rules Example 2

P, ¬¬(P → Q) ` (R ∧ S) ∨ Q. 1. P. Assumption 2. ¬¬(P → Q). A.

S. Choi (KAIST) Logic and set theory September 18, 2012 10 / 16 4. Q. 1.3. → E. 5. (R ∧ S) ∨ Q 4. ∨I.

Nonhypothetical Inference Rules Example 2

P, ¬¬(P → Q) ` (R ∧ S) ∨ Q. 1. P. Assumption 2. ¬¬(P → Q). A. 3. P → Q. 2. ¬E.

S. Choi (KAIST) Logic and set theory September 18, 2012 10 / 16 5. (R ∧ S) ∨ Q 4. ∨I.

Nonhypothetical Inference Rules Example 2

P, ¬¬(P → Q) ` (R ∧ S) ∨ Q. 1. P. Assumption 2. ¬¬(P → Q). A. 3. P → Q. 2. ¬E. 4. Q. 1.3. → E.

S. Choi (KAIST) Logic and set theory September 18, 2012 10 / 16 Nonhypothetical Inference Rules Example 2

P, ¬¬(P → Q) ` (R ∧ S) ∨ Q. 1. P. Assumption 2. ¬¬(P → Q). A. 3. P → Q. 2. ¬E. 4. Q. 1.3. → E. 5. (R ∧ S) ∨ Q 4. ∨I.

S. Choi (KAIST) Logic and set theory September 18, 2012 10 / 16 1. P ∨ P. A. 2. P → (Q ∧ R). A. 3. Q ∧ R. 1, 2. ∨E. 4. R. 3. ∧E.

Nonhypothetical Inference Rules Example 3

P ∨ P, P → (Q ∧ R) ` R.

S. Choi (KAIST) Logic and set theory September 18, 2012 11 / 16 2. P → (Q ∧ R). A. 3. Q ∧ R. 1, 2. ∨E. 4. R. 3. ∧E.

Nonhypothetical Inference Rules Example 3

P ∨ P, P → (Q ∧ R) ` R. 1. P ∨ P. A.

S. Choi (KAIST) Logic and set theory September 18, 2012 11 / 16 3. Q ∧ R. 1, 2. ∨E. 4. R. 3. ∧E.

Nonhypothetical Inference Rules Example 3

P ∨ P, P → (Q ∧ R) ` R. 1. P ∨ P. A. 2. P → (Q ∧ R). A.

S. Choi (KAIST) Logic and set theory September 18, 2012 11 / 16 4. R. 3. ∧E.

Nonhypothetical Inference Rules Example 3

P ∨ P, P → (Q ∧ R) ` R. 1. P ∨ P. A. 2. P → (Q ∧ R). A. 3. Q ∧ R. 1, 2. ∨E.

S. Choi (KAIST) Logic and set theory September 18, 2012 11 / 16 Nonhypothetical Inference Rules Example 3

P ∨ P, P → (Q ∧ R) ` R. 1. P ∨ P. A. 2. P → (Q ∧ R). A. 3. Q ∧ R. 1, 2. ∨E. 4. R. 3. ∧E.

S. Choi (KAIST) Logic and set theory September 18, 2012 11 / 16 Example: P → Q, Q → R ` P → R. (Socrates is human, Humans are mortal, Thus, Socrates is mortal.) P → Q. A. Q → R. A. : P. H. : Q. 1,3, → E. : R. 2.4, → E. P → R. 3-5. → I.

Hypothetical Rules Hypothetical Rules

Conditional introduction (→ I): Given a derivation of φ with help of ψ, we infer ψ → φ.

S. Choi (KAIST) Logic and set theory September 18, 2012 12 / 16 P → Q, Q → R ` P → R. (Socrates is human, Humans are mortal, Thus, Socrates is mortal.) P → Q. A. Q → R. A. : P. H. : Q. 1,3, → E. : R. 2.4, → E. P → R. 3-5. → I.

Hypothetical Rules Hypothetical Rules

Conditional introduction (→ I): Given a derivation of φ with help of ψ, we infer ψ → φ. Example:

S. Choi (KAIST) Logic and set theory September 18, 2012 12 / 16 P → Q. A. Q → R. A. : P. H. : Q. 1,3, → E. : R. 2.4, → E. P → R. 3-5. → I.

Hypothetical Rules Hypothetical Rules

Conditional introduction (→ I): Given a derivation of φ with help of ψ, we infer ψ → φ. Example: P → Q, Q → R ` P → R. (Socrates is human, Humans are mortal, Thus, Socrates is mortal.)

S. Choi (KAIST) Logic and set theory September 18, 2012 12 / 16 Q → R. A. : P. H. : Q. 1,3, → E. : R. 2.4, → E. P → R. 3-5. → I.

Hypothetical Rules Hypothetical Rules

Conditional introduction (→ I): Given a derivation of φ with help of ψ, we infer ψ → φ. Example: P → Q, Q → R ` P → R. (Socrates is human, Humans are mortal, Thus, Socrates is mortal.) P → Q. A.

S. Choi (KAIST) Logic and set theory September 18, 2012 12 / 16 : P. H. : Q. 1,3, → E. : R. 2.4, → E. P → R. 3-5. → I.

Hypothetical Rules Hypothetical Rules

S. Choi (KAIST) Logic and set theory September 18, 2012 12 / 16 : Q. 1,3, → E. : R. 2.4, → E. P → R. 3-5. → I.

Hypothetical Rules Hypothetical Rules

S. Choi (KAIST) Logic and set theory September 18, 2012 12 / 16 : R. 2.4, → E. P → R. 3-5. → I.

Hypothetical Rules Hypothetical Rules

S. Choi (KAIST) Logic and set theory September 18, 2012 12 / 16 P → R. 3-5. → I.

Hypothetical Rules Hypothetical Rules

S. Choi (KAIST) Logic and set theory September 18, 2012 12 / 16 Hypothetical Rules Hypothetical Rules

S. Choi (KAIST) Logic and set theory September 18, 2012 12 / 16 1. (P ∧ Q) ∨ (P ∧ R).A 2. : P ∧ Q.H 3. : P 2. ∧E. 4. : Q 2. ∧E. 5. : Q ∨ R. 4. ∨I. 6 : P ∧ (Q ∨ R). 3.5. ∧I. 7. (P ∧ Q) → (P ∧ (Q ∨ R)). 2-6 → I. 8. : P ∧ R.H 9. : P 8. ∧E. 10. : R 8. ∧E. 11. : Q ∨ R. 10. ∨I. 12 : P ∧ (Q ∨ R). 9.11. ∧I. 13. (P ∧ R) → (P ∧ (Q ∨ R)). 2-6 → I. 14. P ∧ (Q ∨ R). 1.7.13 ∨E.

Hypothetical Rules Example (P ∧ Q) ∨ (P ∧ R) ` P ∧ (Q ∨ R).

S. Choi (KAIST) Logic and set theory September 18, 2012 13 / 16 2. : P ∧ Q.H 3. : P 2. ∧E. 4. : Q 2. ∧E. 5. : Q ∨ R. 4. ∨I. 6 : P ∧ (Q ∨ R). 3.5. ∧I. 7. (P ∧ Q) → (P ∧ (Q ∨ R)). 2-6 → I. 8. : P ∧ R.H 9. : P 8. ∧E. 10. : R 8. ∧E. 11. : Q ∨ R. 10. ∨I. 12 : P ∧ (Q ∨ R). 9.11. ∧I. 13. (P ∧ R) → (P ∧ (Q ∨ R)). 2-6 → I. 14. P ∧ (Q ∨ R). 1.7.13 ∨E.

Hypothetical Rules Example (P ∧ Q) ∨ (P ∧ R) ` P ∧ (Q ∨ R). 1. (P ∧ Q) ∨ (P ∧ R).A

S. Choi (KAIST) Logic and set theory September 18, 2012 13 / 16 3. : P 2. ∧E. 4. : Q 2. ∧E. 5. : Q ∨ R. 4. ∨I. 6 : P ∧ (Q ∨ R). 3.5. ∧I. 7. (P ∧ Q) → (P ∧ (Q ∨ R)). 2-6 → I. 8. : P ∧ R.H 9. : P 8. ∧E. 10. : R 8. ∧E. 11. : Q ∨ R. 10. ∨I. 12 : P ∧ (Q ∨ R). 9.11. ∧I. 13. (P ∧ R) → (P ∧ (Q ∨ R)). 2-6 → I. 14. P ∧ (Q ∨ R). 1.7.13 ∨E.

Hypothetical Rules Example (P ∧ Q) ∨ (P ∧ R) ` P ∧ (Q ∨ R). 1. (P ∧ Q) ∨ (P ∧ R).A 2. : P ∧ Q.H

S. Choi (KAIST) Logic and set theory September 18, 2012 13 / 16 4. : Q 2. ∧E. 5. : Q ∨ R. 4. ∨I. 6 : P ∧ (Q ∨ R). 3.5. ∧I. 7. (P ∧ Q) → (P ∧ (Q ∨ R)). 2-6 → I. 8. : P ∧ R.H 9. : P 8. ∧E. 10. : R 8. ∧E. 11. : Q ∨ R. 10. ∨I. 12 : P ∧ (Q ∨ R). 9.11. ∧I. 13. (P ∧ R) → (P ∧ (Q ∨ R)). 2-6 → I. 14. P ∧ (Q ∨ R). 1.7.13 ∨E.

Hypothetical Rules Example (P ∧ Q) ∨ (P ∧ R) ` P ∧ (Q ∨ R). 1. (P ∧ Q) ∨ (P ∧ R).A 2. : P ∧ Q.H 3. : P 2. ∧E.

S. Choi (KAIST) Logic and set theory September 18, 2012 13 / 16 5. : Q ∨ R. 4. ∨I. 6 : P ∧ (Q ∨ R). 3.5. ∧I. 7. (P ∧ Q) → (P ∧ (Q ∨ R)). 2-6 → I. 8. : P ∧ R.H 9. : P 8. ∧E. 10. : R 8. ∧E. 11. : Q ∨ R. 10. ∨I. 12 : P ∧ (Q ∨ R). 9.11. ∧I. 13. (P ∧ R) → (P ∧ (Q ∨ R)). 2-6 → I. 14. P ∧ (Q ∨ R). 1.7.13 ∨E.

Hypothetical Rules Example (P ∧ Q) ∨ (P ∧ R) ` P ∧ (Q ∨ R). 1. (P ∧ Q) ∨ (P ∧ R).A 2. : P ∧ Q.H 3. : P 2. ∧E. 4. : Q 2. ∧E.

S. Choi (KAIST) Logic and set theory September 18, 2012 13 / 16 6 : P ∧ (Q ∨ R). 3.5. ∧I. 7. (P ∧ Q) → (P ∧ (Q ∨ R)). 2-6 → I. 8. : P ∧ R.H 9. : P 8. ∧E. 10. : R 8. ∧E. 11. : Q ∨ R. 10. ∨I. 12 : P ∧ (Q ∨ R). 9.11. ∧I. 13. (P ∧ R) → (P ∧ (Q ∨ R)). 2-6 → I. 14. P ∧ (Q ∨ R). 1.7.13 ∨E.

Hypothetical Rules Example (P ∧ Q) ∨ (P ∧ R) ` P ∧ (Q ∨ R). 1. (P ∧ Q) ∨ (P ∧ R).A 2. : P ∧ Q.H 3. : P 2. ∧E. 4. : Q 2. ∧E. 5. : Q ∨ R. 4. ∨I.

S. Choi (KAIST) Logic and set theory September 18, 2012 13 / 16 7. (P ∧ Q) → (P ∧ (Q ∨ R)). 2-6 → I. 8. : P ∧ R.H 9. : P 8. ∧E. 10. : R 8. ∧E. 11. : Q ∨ R. 10. ∨I. 12 : P ∧ (Q ∨ R). 9.11. ∧I. 13. (P ∧ R) → (P ∧ (Q ∨ R)). 2-6 → I. 14. P ∧ (Q ∨ R). 1.7.13 ∨E.

Hypothetical Rules Example (P ∧ Q) ∨ (P ∧ R) ` P ∧ (Q ∨ R). 1. (P ∧ Q) ∨ (P ∧ R).A 2. : P ∧ Q.H 3. : P 2. ∧E. 4. : Q 2. ∧E. 5. : Q ∨ R. 4. ∨I. 6 : P ∧ (Q ∨ R). 3.5. ∧I.

S. Choi (KAIST) Logic and set theory September 18, 2012 13 / 16 8. : P ∧ R.H 9. : P 8. ∧E. 10. : R 8. ∧E. 11. : Q ∨ R. 10. ∨I. 12 : P ∧ (Q ∨ R). 9.11. ∧I. 13. (P ∧ R) → (P ∧ (Q ∨ R)). 2-6 → I. 14. P ∧ (Q ∨ R). 1.7.13 ∨E.

Hypothetical Rules Example (P ∧ Q) ∨ (P ∧ R) ` P ∧ (Q ∨ R). 1. (P ∧ Q) ∨ (P ∧ R).A 2. : P ∧ Q.H 3. : P 2. ∧E. 4. : Q 2. ∧E. 5. : Q ∨ R. 4. ∨I. 6 : P ∧ (Q ∨ R). 3.5. ∧I. 7. (P ∧ Q) → (P ∧ (Q ∨ R)). 2-6 → I.

S. Choi (KAIST) Logic and set theory September 18, 2012 13 / 16 9. : P 8. ∧E. 10. : R 8. ∧E. 11. : Q ∨ R. 10. ∨I. 12 : P ∧ (Q ∨ R). 9.11. ∧I. 13. (P ∧ R) → (P ∧ (Q ∨ R)). 2-6 → I. 14. P ∧ (Q ∨ R). 1.7.13 ∨E.

Hypothetical Rules Example (P ∧ Q) ∨ (P ∧ R) ` P ∧ (Q ∨ R). 1. (P ∧ Q) ∨ (P ∧ R).A 2. : P ∧ Q.H 3. : P 2. ∧E. 4. : Q 2. ∧E. 5. : Q ∨ R. 4. ∨I. 6 : P ∧ (Q ∨ R). 3.5. ∧I. 7. (P ∧ Q) → (P ∧ (Q ∨ R)). 2-6 → I. 8. : P ∧ R.H

S. Choi (KAIST) Logic and set theory September 18, 2012 13 / 16 10. : R 8. ∧E. 11. : Q ∨ R. 10. ∨I. 12 : P ∧ (Q ∨ R). 9.11. ∧I. 13. (P ∧ R) → (P ∧ (Q ∨ R)). 2-6 → I. 14. P ∧ (Q ∨ R). 1.7.13 ∨E.

S. Choi (KAIST) Logic and set theory September 18, 2012 13 / 16 11. : Q ∨ R. 10. ∨I. 12 : P ∧ (Q ∨ R). 9.11. ∧I. 13. (P ∧ R) → (P ∧ (Q ∨ R)). 2-6 → I. 14. P ∧ (Q ∨ R). 1.7.13 ∨E.

Hypothetical Rules Example (P ∧ Q) ∨ (P ∧ R) ` P ∧ (Q ∨ R). 1. (P ∧ Q) ∨ (P ∧ R).A 2. : P ∧ Q.H 3. : P 2. ∧E. 4. : Q 2. ∧E. 5. : Q ∨ R. 4. ∨I. 6 : P ∧ (Q ∨ R). 3.5. ∧I. 7. (P ∧ Q) → (P ∧ (Q ∨ R)). 2-6 → I. 8. : P ∧ R.H 9. : P 8. ∧E. 10. : R 8. ∧E.

S. Choi (KAIST) Logic and set theory September 18, 2012 13 / 16 12 : P ∧ (Q ∨ R). 9.11. ∧I. 13. (P ∧ R) → (P ∧ (Q ∨ R)). 2-6 → I. 14. P ∧ (Q ∨ R). 1.7.13 ∨E.

S. Choi (KAIST) Logic and set theory September 18, 2012 13 / 16 13. (P ∧ R) → (P ∧ (Q ∨ R)). 2-6 → I. 14. P ∧ (Q ∨ R). 1.7.13 ∨E.

S. Choi (KAIST) Logic and set theory September 18, 2012 13 / 16 14. P ∧ (Q ∨ R). 1.7.13 ∨E.

S. Choi (KAIST) Logic and set theory September 18, 2012 13 / 16 Hypothetical Rules Example (P ∧ Q) ∨ (P ∧ R) ` P ∧ (Q ∨ R). 1. (P ∧ Q) ∨ (P ∧ R).A 2. : P ∧ Q.H 3. : P 2. ∧E. 4. : Q 2. ∧E. 5. : Q ∨ R. 4. ∨I. 6 : P ∧ (Q ∨ R). 3.5. ∧I. 7. (P ∧ Q) → (P ∧ (Q ∨ R)). 2-6 → I. 8. : P ∧ R.H 9. : P 8. ∧E. 10. : R 8. ∧E. 11. : Q ∨ R. 10. ∨I. 12 : P ∧ (Q ∨ R). 9.11. ∧I. 13. (P ∧ R) → (P ∧ (Q ∨ R)). 2-6 → I. 14. P ∧ (Q ∨ R). 1.7.13 ∨E.

S. Choi (KAIST) Logic and set theory September 18, 2012 13 / 16 No occurance of a formula to the right of a vertical line may be cited after the line ended. (There may be multiple lines. See 4.20) If two or more hypotheses are ineffect, then the order that they are discharged is reversed. A proof is not valid until all the hypotheses are discharged.

Hypothetical Rules Note

Every hypothesis introduced begins at a new line.

S. Choi (KAIST) Logic and set theory September 18, 2012 14 / 16 If two or more hypotheses are ineffect, then the order that they are discharged is reversed. A proof is not valid until all the hypotheses are discharged.

Hypothetical Rules Note

Every hypothesis introduced begins at a new line. No occurance of a formula to the right of a vertical line may be cited after the line ended. (There may be multiple lines. See 4.20)

S. Choi (KAIST) Logic and set theory September 18, 2012 14 / 16 A proof is not valid until all the hypotheses are discharged.

Hypothetical Rules Note

Every hypothesis introduced begins at a new line. No occurance of a formula to the right of a vertical line may be cited after the line ended. (There may be multiple lines. See 4.20) If two or more hypotheses are ineffect, then the order that they are discharged is reversed.

S. Choi (KAIST) Logic and set theory September 18, 2012 14 / 16 Hypothetical Rules Note

S. Choi (KAIST) Logic and set theory September 18, 2012 14 / 16 I ¬P → P, ` P. I 1. ¬P → P. A. I 2. : ¬P. H (for ¬I) I 3. : P, 1.2, (→ E) I 4. : P ∧ ¬P. 2.3 (∧I). I 5. ¬¬P. 2-4 ¬I. I 6. P. ¬E.

Given a derivation of an absurdity from a hypothesis ¬φ, we infer φ.

Hypothetical Rules Negation introduction

Negation introduction (¬I). Reductio ad absurdum, indirect proof.

S. Choi (KAIST) Logic and set theory September 18, 2012 15 / 16 I ¬P → P, ` P. I 1. ¬P → P. A. I 2. : ¬P. H (for ¬I) I 3. : P, 1.2, (→ E) I 4. : P ∧ ¬P. 2.3 (∧I). I 5. ¬¬P. 2-4 ¬I. I 6. P. ¬E.

Hypothetical Rules Negation introduction