propositional natural deduction · example - transitivity of implication (derived rule) we prove...
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![Page 1: Propositional natural deduction · Example - transitivity of implication (derived rule) We prove p!q q!r p!r 1 p!q 2 q!r 3 p 4 q!-E, 1, 3 5 r!-E, 2, 4 6 p!r!-I, 3–5 Lines 1 and](https://reader034.vdocuments.us/reader034/viewer/2022050302/5f6b9035cd10764f32676d3f/html5/thumbnails/1.jpg)
Propositional natural deductionCOMP2600 / COMP6260
Dirk PattinsonAustralian National University
Semester 2, 2016
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Major proof techniques
Three major styles of proof in logic and mathematicsModel based computation: truth tables for propositional logicAlgebraic proof: simplification rules e.g. De Morgan’sDeductive reasoning: rules of inferenceI Natural deduction is one exampleI Other examples: tableau calculi, resolution, Hilbert calculi
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Example proof using truth tables
Statement to be proved:
(p∧ (q∨ r))→ ((p∧q)∨ r))
For all 8 (= 23) possibilities of p,q, r , calculate truth value of the statement
p q r q∨ r p∧ (q∨ r) p∧q (p∧q)∨ r (p∧ (q∨ r))→ ((p∧q)∨ r))
T T T T T T T TT T F T T T T TT F T T T F T TT F F F F F F TF T T T F F T TF T F T F F F TF F T T F F T TF F F F F F F T
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Example proof using truth tables
Statement to be proved:
(p∧ (q∨ r))→ ((p∧q)∨ r))
For all 8 (= 23) possibilities of p,q, r , calculate truth value of the statement
p q r q∨ r p∧ (q∨ r) p∧q (p∧q)∨ r (p∧ (q∨ r))→ ((p∧q)∨ r))T T T T T T T T
T T F T T T T TT F T T T F T TT F F F F F F TF T T T F F T TF T F T F F F TF F T T F F T TF F F F F F F T
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Example proof using truth tables
Statement to be proved:
(p∧ (q∨ r))→ ((p∧q)∨ r))
For all 8 (= 23) possibilities of p,q, r , calculate truth value of the statement
p q r q∨ r p∧ (q∨ r) p∧q (p∧q)∨ r (p∧ (q∨ r))→ ((p∧q)∨ r))T T T T T T T TT T F T T T T T
T F T T T F T TT F F F F F F TF T T T F F T TF T F T F F F TF F T T F F T TF F F F F F F T
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Example proof using truth tables
Statement to be proved:
(p∧ (q∨ r))→ ((p∧q)∨ r))
For all 8 (= 23) possibilities of p,q, r , calculate truth value of the statement
p q r q∨ r p∧ (q∨ r) p∧q (p∧q)∨ r (p∧ (q∨ r))→ ((p∧q)∨ r))T T T T T T T TT T F T T T T TT F T T T F T T
T F F F F F F TF T T T F F T TF T F T F F F TF F T T F F T TF F F F F F F T
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Example proof using truth tables
Statement to be proved:
(p∧ (q∨ r))→ ((p∧q)∨ r))
For all 8 (= 23) possibilities of p,q, r , calculate truth value of the statement
p q r q∨ r p∧ (q∨ r) p∧q (p∧q)∨ r (p∧ (q∨ r))→ ((p∧q)∨ r))T T T T T T T TT T F T T T T TT F T T T F T TT F F F F F F T
F T T T F F T TF T F T F F F TF F T T F F T TF F F F F F F T
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Example proof using truth tables
Statement to be proved:
(p∧ (q∨ r))→ ((p∧q)∨ r))
For all 8 (= 23) possibilities of p,q, r , calculate truth value of the statement
p q r q∨ r p∧ (q∨ r) p∧q (p∧q)∨ r (p∧ (q∨ r))→ ((p∧q)∨ r))T T T T T T T TT T F T T T T TT F T T T F T TT F F F F F F TF T T T F F T T
F T F T F F F TF F T T F F T TF F F F F F F T
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Example proof using truth tables
Statement to be proved:
(p∧ (q∨ r))→ ((p∧q)∨ r))
For all 8 (= 23) possibilities of p,q, r , calculate truth value of the statement
p q r q∨ r p∧ (q∨ r) p∧q (p∧q)∨ r (p∧ (q∨ r))→ ((p∧q)∨ r))T T T T T T T TT T F T T T T TT F T T T F T TT F F F F F F TF T T T F F T TF T F T F F F T
F F T T F F T TF F F F F F F T
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Example proof using truth tables
Statement to be proved:
(p∧ (q∨ r))→ ((p∧q)∨ r))
For all 8 (= 23) possibilities of p,q, r , calculate truth value of the statement
p q r q∨ r p∧ (q∨ r) p∧q (p∧q)∨ r (p∧ (q∨ r))→ ((p∧q)∨ r))T T T T T T T TT T F T T T T TT F T T T F T TT F F F F F F TF T T T F F T TF T F T F F F TF F T T F F T T
F F F F F F F T
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Example proof using truth tables
Statement to be proved:
(p∧ (q∨ r))→ ((p∧q)∨ r))
For all 8 (= 23) possibilities of p,q, r , calculate truth value of the statement
p q r q∨ r p∧ (q∨ r) p∧q (p∧q)∨ r (p∧ (q∨ r))→ ((p∧q)∨ r))T T T T T T T TT T F T T T T TT F T T T F T TT F F F F F F TF T T T F F T TF T F T F F F TF F T T F F T TF F F F F F F T
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The notion of a deductive proof
A proof is a sequence of steps.Each step is either:I an axiom or an assumption; orI a statement which follows from previous steps via a valid rule of inference.
Natural deduction:I for each connective, there is an introduction and an elimination ruleI rules are formal, but resemble natural, i.e., human reasoning
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Example of a natural deduction proof
Statement to be proved: (p∧ (q∨ r))→ ((q→ s)∨p)
1 p∧ (q∨ r) Assumption
2 p ∧-E, 1
3 (q→ s)∨p ∨-I, 2
4 (p∧ (q∨ r))→ ((q→ s)∨p) →-I, 1–3
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Conjunction rules
∧-I (and introduction)
p q
p ∧ q
∧-E (and elimination)
p ∧ q
p
p ∧ q
q
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Conjunction rules
∧-I (and introduction)
p q
p ∧ q
∧-E (and elimination)
p ∧ q
p
p ∧ q
q
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Example
Commutativity of conjunction (derived rule)
p ∧ q
q ∧ p
1 p∧q
2 p ∧-E, 1
3 q ∧-E, 1
4 q∧p ∧-I, 2, 3
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Implication rules
→-I (implication introduction)[ p ]
...q
p→ q
This notation means that by assuming p, you can prove q.p is then discharged - no longer an assumption.
→-E (implication elimination)p p→ q
q
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Implication rules
→-I (implication introduction)[ p ]
...q
p→ q
This notation means that by assuming p, you can prove q.p is then discharged - no longer an assumption.
→-E (implication elimination)p p→ q
q
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Example - transitivity of implication (derived rule)
We provep→ q q→ r
p→ r
1 p→ q
2 q→ r
3 p
4 q →-E, 1, 3
5 r →-E, 2, 4
6 p→ r →-I, 3–5
Lines 1 and 2 are the assumptions given, you may use these throughout theproofLine 3 is an assumption made within the proof, you may use it only within itsscope (lines 3 to 5)
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Example - transitivity of implication (derived rule)
We provep→ q q→ r
p→ r
1 p→ q
2 q→ r
3 p
4 q →-E, 1, 3
5 r →-E, 2, 4
6 p→ r →-I, 3–5
Lines 1 and 2 are the assumptions given, you may use these throughout theproofLine 3 is an assumption made within the proof, you may use it only within itsscope (lines 3 to 5)
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Example - transitivity of implication (derived rule)
We provep→ q q→ r
p→ r
1 p→ q
2 q→ r
3 p
4 q →-E, 1, 3
5 r →-E, 2, 4
6 p→ r →-I, 3–5
Lines 1 and 2 are the assumptions given, you may use these throughout theproofLine 3 is an assumption made within the proof, you may use it only within itsscope (lines 3 to 5)
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Example - transitivity of implication (derived rule)
We provep→ q q→ r
p→ r
1 p→ q
2 q→ r
3 p
4 q →-E, 1, 3
5 r →-E, 2, 4
6 p→ r →-I, 3–5
Lines 1 and 2 are the assumptions given, you may use these throughout theproofLine 3 is an assumption made within the proof, you may use it only within itsscope (lines 3 to 5)
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Example - transitivity of implication (derived rule)
We provep→ q q→ r
p→ r
1 p→ q
2 q→ r
3 p
4 q →-E, 1, 3
5 r →-E, 2, 4
6 p→ r →-I, 3–5
Lines 1 and 2 are the assumptions given, you may use these throughout theproofLine 3 is an assumption made within the proof, you may use it only within itsscope (lines 3 to 5)
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Example - transitivity of implication (derived rule)
We provep→ q q→ r
p→ r
1 p→ q
2 q→ r
3 p
4 q →-E, 1, 3
5 r →-E, 2, 4
6 p→ r →-I, 3–5
Lines 1 and 2 are the assumptions given, you may use these throughout theproofLine 3 is an assumption made within the proof, you may use it only within itsscope (lines 3 to 5)
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Example - transitivity of implication (derived rule)
We provep→ q q→ r
p→ r
1 p→ q
2 q→ r
3 p
4 q →-E, 1, 3
5 r →-E, 2, 4
6 p→ r →-I, 3–5
Lines 1 and 2 are the assumptions given, you may use these throughout theproofLine 3 is an assumption made within the proof, you may use it only within itsscope (lines 3 to 5)
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Example - transitivity of implication (derived rule)
We provep→ q q→ r
p→ r
1 p→ q
2 q→ r
3 p
4 q →-E, 1, 3
5 r →-E, 2, 4
6 p→ r →-I, 3–5
Lines 1 and 2 are the assumptions given, you may use these throughout theproofLine 3 is an assumption made within the proof, you may use it only within itsscope (lines 3 to 5)
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Notation: justification of a step
1 p→ q
2 p
3 q →-E, 1, 2
4 p→ q →-I, 2–3This is a rather silly proof, we succeed in proving what we started with.But it illustrates the meaning of the line number notation:→-E,1,2 means that rule→-E proves line 3 from lines 1 and 2→-I,2-3 means rule→-I proves line 4 from the fact that we could assume line2 and (using that assumption) prove line 3
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Notation: justification of a step
1 p→ q
2 p
3 q →-E, 1, 2
4 p→ q →-I, 2–3This is a rather silly proof, we succeed in proving what we started with.But it illustrates the meaning of the line number notation:→-E,1,2 means that rule→-E proves line 3 from lines 1 and 2→-I,2-3 means rule→-I proves line 4 from the fact that we could assume line2 and (using that assumption) prove line 3
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Notation: justification of a step
1 p→ q
2 p
3 q →-E, 1, 2
4 p→ q →-I, 2–3This is a rather silly proof, we succeed in proving what we started with.But it illustrates the meaning of the line number notation:→-E,1,2 means that rule→-E proves line 3 from lines 1 and 2→-I,2-3 means rule→-I proves line 4 from the fact that we could assume line2 and (using that assumption) prove line 3
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Notation: justification of a step
1 p→ q
2 p
3 q →-E, 1, 2
4 p→ q →-I, 2–3This is a rather silly proof, we succeed in proving what we started with.But it illustrates the meaning of the line number notation:→-E,1,2 means that rule→-E proves line 3 from lines 1 and 2→-I,2-3 means rule→-I proves line 4 from the fact that we could assume line2 and (using that assumption) prove line 3
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Notation: justification of a step
1 p→ q
2 p
3 q →-E, 1, 2
4 p→ q →-I, 2–3
This is a rather silly proof, we succeed in proving what we started with.But it illustrates the meaning of the line number notation:→-E,1,2 means that rule→-E proves line 3 from lines 1 and 2→-I,2-3 means rule→-I proves line 4 from the fact that we could assume line2 and (using that assumption) prove line 3
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Notation: justification of a step
1 p→ q
2 p
3 q →-E, 1, 2
4 p→ q →-I, 2–3This is a rather silly proof, we succeed in proving what we started with.But it illustrates the meaning of the line number notation:→-E,1,2 means that rule→-E proves line 3 from lines 1 and 2→-I,2-3 means rule→-I proves line 4 from the fact that we could assume line2 and (using that assumption) prove line 3
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Rules involving assumptions
1 p→ q
2 q→ r
3 p
4 q →-E, 1, 3
5 r →-E, 2, 4
6 q∧ r WRONG ∧-I, 4, 5
If a statement is inside the scope of an assumption, then it depends on thatassumption.Given p→ q and q→ r , we then assumed p and ”proved” q∧ r , but q∧ rdepends on p.Indentation and vertical lines indicate scopingSimilar to scoping in program code: eg lines 3 to 5 are a a method, and p is alocal variable to that method.
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Rules involving assumptions
1 p→ q
2 q→ r
3 p
4 q →-E, 1, 3
5 r →-E, 2, 4
6 q∧ r WRONG ∧-I, 4, 5
If a statement is inside the scope of an assumption, then it depends on thatassumption.Given p→ q and q→ r , we then assumed p and ”proved” q∧ r , but q∧ rdepends on p.Indentation and vertical lines indicate scopingSimilar to scoping in program code: eg lines 3 to 5 are a a method, and p is alocal variable to that method.
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Rules involving assumptions
1 p→ q
2 q→ r
3 p
4 q →-E, 1, 3
5 r →-E, 2, 4
6 q∧ r WRONG ∧-I, 4, 5
If a statement is inside the scope of an assumption, then it depends on thatassumption.Given p→ q and q→ r , we then assumed p and ”proved” q∧ r , but q∧ rdepends on p.Indentation and vertical lines indicate scopingSimilar to scoping in program code: eg lines 3 to 5 are a a method, and p is alocal variable to that method.
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Rules involving assumptions
1 p→ q
2 q→ r
3 p
4 q →-E, 1, 3
5 r →-E, 2, 4
6 q∧ r WRONG ∧-I, 4, 5
If a statement is inside the scope of an assumption, then it depends on thatassumption.Given p→ q and q→ r , we then assumed p and ”proved” q∧ r , but q∧ rdepends on p.Indentation and vertical lines indicate scopingSimilar to scoping in program code: eg lines 3 to 5 are a a method, and p is alocal variable to that method.
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Rules involving assumptions
1 p→ q
2 q→ r
3 p
4 q →-E, 1, 3
5 r →-E, 2, 4
6 q∧ r WRONG ∧-I, 4, 5
If a statement is inside the scope of an assumption, then it depends on thatassumption.Given p→ q and q→ r , we then assumed p and ”proved” q∧ r , but q∧ rdepends on p.Indentation and vertical lines indicate scopingSimilar to scoping in program code: eg lines 3 to 5 are a a method, and p is alocal variable to that method.
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Rules involving assumptions
1 p→ q
2 q→ r
3 p
4 q →-E, 1, 3
5 r →-E, 2, 4
6 q∧ r WRONG ∧-I, 4, 5
If a statement is inside the scope of an assumption, then it depends on thatassumption.Given p→ q and q→ r , we then assumed p and ”proved” q∧ r , but q∧ rdepends on p.Indentation and vertical lines indicate scopingSimilar to scoping in program code: eg lines 3 to 5 are a a method, and p is alocal variable to that method.
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Rules involving assumptions
1 p→ q
2 q→ r
3 p
4 q →-E, 1, 3
5 r →-E, 2, 4
6 q∧ r WRONG ∧-I, 4, 5
If a statement is inside the scope of an assumption, then it depends on thatassumption.Given p→ q and q→ r , we then assumed p and ”proved” q∧ r , but q∧ rdepends on p.Indentation and vertical lines indicate scopingSimilar to scoping in program code: eg lines 3 to 5 are a a method, and p is alocal variable to that method.
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Rules involving assumptions
1 p→ q
2 q→ r
3 p
4 q →-E, 1, 3
5 r →-E, 2, 4
6 q∧ r WRONG ∧-I, 4, 5
If a statement is inside the scope of an assumption, then it depends on thatassumption.Given p→ q and q→ r , we then assumed p and ”proved” q∧ r , but q∧ rdepends on p.Indentation and vertical lines indicate scopingSimilar to scoping in program code: eg lines 3 to 5 are a a method, and p is alocal variable to that method.
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Useless assumptions
You can assume anything, but it might not be useful.
1 p∧q
2 q ∧-E, 1
3 (p∧q)→ q →-I, 1–2
1 p ∧ You are a giraffe
2 You are a giraffe ∧-E, 1
3 p ∧ You are a giraffe → You are a giraffe →-I, 1–2
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Useless assumptions
You can assume anything, but it might not be useful.
1 p∧q
2 q ∧-E, 1
3 (p∧q)→ q →-I, 1–2
1 p ∧ You are a giraffe
2 You are a giraffe ∧-E, 1
3 p ∧ You are a giraffe → You are a giraffe →-I, 1–2
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Useless assumptions
You can assume anything, but it might not be useful.
1 p∧q
2 q ∧-E, 1
3 (p∧q)→ q →-I, 1–2
1 p ∧ You are a giraffe
2 You are a giraffe ∧-E, 1
3 p ∧ You are a giraffe → You are a giraffe →-I, 1–2
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Useless assumptions
You can assume anything, but it might not be useful.
1 p∧q
2 q ∧-E, 1
3 (p∧q)→ q →-I, 1–2
1 p ∧ You are a giraffe
2 You are a giraffe ∧-E, 1
3 p ∧ You are a giraffe → You are a giraffe →-I, 1–2
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Useless assumptions
You can assume anything, but it might not be useful.
1 p∧q
2 q ∧-E, 1
3 (p∧q)→ q →-I, 1–2
1 p ∧ You are a giraffe
2 You are a giraffe ∧-E, 1
3 p ∧ You are a giraffe → You are a giraffe →-I, 1–2
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Disjunction rules
∨-I (or introduction)
p
p ∨ q
p
q ∨ p
∨-E (or elimination)
[p] [q]... ...
p ∨ q r r
r
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Disjunction rules
∨-I (or introduction)
p
p ∨ q
p
q ∨ p
∨-E (or elimination)
[p] [q]... ...
p ∨ q r r
r
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∨-E template
1 p∨q
2 p
... ...
a r
b q
... ...
c r
d r ∨-E, 1, 2–a, b–c
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∨-E template
1 p∨q
2 p
... ...
a r
b q
... ...
c r
d r ∨-E, 1, 2–a, b–c
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∨-E template
1 p∨q
2 p
... ...
a r
b q
... ...
c r
d r ∨-E, 1, 2–a, b–c
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∨-E template
1 p∨q
2 p
... ...
a r
b q
... ...
c r
d r ∨-E, 1, 2–a, b–c
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∨-E template
1 p∨q
2 p
... ...
a r
b q
... ...
c r
d r ∨-E, 1, 2–a, b–c
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∨-E template
1 p∨q
2 p
... ...
a r
b q
... ...
c r
d r ∨-E, 1, 2–a, b–c
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∨-E template
1 p∨q
2 p
... ...
a r
b q
... ...
c r
d r ∨-E, 1, 2–a, b–c
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∨-E template
1 p∨q
2 p
... ...
a r
b q
... ...
c r
d r ∨-E, 1, 2–a, b–c
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∨-E template
1 p∨q
2 p
... ...
a r
b q
... ...
c r
d r ∨-E, 1, 2–a, b–c
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Example: commutativity of disjunction (derived rule)
p∨q
q∨p
1 p∨q
2 p
3 q∨p ∨-I, 2
4 q
5 q∨p ∨-I, 4
6 q∨p ∨-E, 1, 2–3, 4–5
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Example: commutativity of disjunction (derived rule)
p∨q
q∨p
1 p∨q
2 p
3 q∨p ∨-I, 2
4 q
5 q∨p ∨-I, 4
6 q∨p ∨-E, 1, 2–3, 4–5
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Example: commutativity of disjunction (derived rule)
p∨q
q∨p
1 p∨q
2 p
3 q∨p ∨-I, 2
4 q
5 q∨p ∨-I, 4
6 q∨p ∨-E, 1, 2–3, 4–5
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Example: commutativity of disjunction (derived rule)
p∨q
q∨p
1 p∨q
2 p
3 q∨p ∨-I, 2
4 q
5 q∨p ∨-I, 4
6 q∨p ∨-E, 1, 2–3, 4–5
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Example: commutativity of disjunction (derived rule)
p∨q
q∨p
1 p∨q
2 p
3 q∨p ∨-I, 2
4 q
5 q∨p ∨-I, 4
6 q∨p ∨-E, 1, 2–3, 4–5
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Example: commutativity of disjunction (derived rule)
p∨q
q∨p
1 p∨q
2 p
3 q∨p ∨-I, 2
4 q
5 q∨p ∨-I, 4
6 q∨p ∨-E, 1, 2–3, 4–5
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Example: commutativity of disjunction (derived rule)
p∨q
q∨p
1 p∨q
2 p
3 q∨p ∨-I, 2
4 q
5 q∨p ∨-I, 4
6 q∨p ∨-E, 1, 2–3, 4–5
14 / 25
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Negation rules
Idea: assume the opposite of what you want to prove and find a contradiction —so your assumption must have been wrong
¬-I (not introduction)[p]...
q ∧ ¬q
¬p
¬-E (not elimination)(eliminates ¬ froman assumption)
[¬p]...
q ∧ ¬q
p
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Negation rules
Idea: assume the opposite of what you want to prove and find a contradiction —so your assumption must have been wrong
¬-I (not introduction)[p]...
q ∧ ¬q
¬p
¬-E (not elimination)(eliminates ¬ froman assumption)
[¬p]...
q ∧ ¬q
p
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Example: double negation introduction (derived rule)
p
¬¬pIt is raining
It is not the case that is is not raining
1 p
2 ¬p
3 p∧¬p ∧-I, 1, 2
4 ¬¬p ¬I, 2–3
16 / 25
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Example: double negation introduction (derived rule)
p
¬¬pIt is raining
It is not the case that is is not raining
1 p
2 ¬p
3 p∧¬p ∧-I, 1, 2
4 ¬¬p ¬I, 2–3
16 / 25
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Example: double negation introduction (derived rule)
p
¬¬pIt is raining
It is not the case that is is not raining
1 p
2 ¬p
3 p∧¬p ∧-I, 1, 2
4 ¬¬p ¬I, 2–3
16 / 25
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Example: double negation introduction (derived rule)
p
¬¬pIt is raining
It is not the case that is is not raining
1 p
2 ¬p
3 p∧¬p ∧-I, 1, 2
4 ¬¬p ¬I, 2–3
16 / 25
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Example: double negation introduction (derived rule)
p
¬¬pIt is raining
It is not the case that is is not raining
1 p
2 ¬p
3 p∧¬p ∧-I, 1, 2
4 ¬¬p ¬I, 2–3
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Example: contradiction elimination (derived rule)
p∧¬p
q
1 p∧¬p
2 ¬q
3 p∧¬p R, 1
4 q ¬E, 2–3
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Example: contradiction elimination (derived rule)
p∧¬p
q
1 p∧¬p
2 ¬q
3 p∧¬p R, 1
4 q ¬E, 2–3
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Example: contradiction elimination (derived rule)
p∧¬p
q
1 p∧¬p
2 ¬q
3 p∧¬p R, 1
4 q ¬E, 2–3
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Example: contradiction elimination (derived rule)
p∧¬p
q
1 p∧¬p
2 ¬q
3 p∧¬p R, 1
4 q ¬E, 2–3
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Example: contradiction elimination (derived rule)
p∧¬p
q
1 p∧¬p
2 ¬q
3 p∧¬p R, 1
4 q ¬E, 2–3
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Example: contradiction elimination (derived rule)
p∧¬p
q
1 p∧¬p
2 ¬q
3 p∧¬p R, 1
4 q ¬E, 2–3
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Example: double negation elimination (derived rule)
¬¬p
p
1 ¬¬p
2 ¬p
3 ¬p∧¬¬p ∧-I, 1, 2
4 p ¬E, 2–3
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Example: double negation elimination (derived rule)
¬¬p
p
1 ¬¬p
2 ¬p
3 ¬p∧¬¬p ∧-I, 1, 2
4 p ¬E, 2–3
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Example: double negation elimination (derived rule)
¬¬p
p
1 ¬¬p
2 ¬p
3 ¬p∧¬¬p ∧-I, 1, 2
4 p ¬E, 2–3
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Example: double negation elimination (derived rule)
¬¬p
p
1 ¬¬p
2 ¬p
3 ¬p∧¬¬p ∧-I, 1, 2
4 p ¬E, 2–3
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Example: double negation elimination (derived rule)
¬¬p
p
1 ¬¬p
2 ¬p
3 ¬p∧¬¬p ∧-I, 1, 2
4 p ¬E, 2–3
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Equivalence
p↔ q means p is true if and only if q is trueWe can make the definition
p↔ q ≡ (p→ q)∧ (q→ p)
which would naturally give us these rulesintroduction rule:
p→ q q→ p
p↔ qelimination rules:
p↔ q
p→ q
p↔ q
q→ p
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Equivalence — Rules
Alternatively we can get rules which don’t involve the→ symbol↔-I (↔ introduction) [p] [q]
... ...q p
p↔ q↔-E (↔ elimination)
p↔ q p
q
p↔ q q
pNote the similarities to the→-I and→-E rules
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Which rule to use next?
Guided by the “form” of your goal, and what you already have proved“form” — ie, look at the connective: ∧,∨,→,¬always can consider using ¬-E (not elimination) ruleto prove p∨q, ∨-I (or introduction) may not work
p
p ∨ q
q
p ∨ q
p may not be necessarily true, q may not be necessarily true
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To prove p∨q, sometimes you need to do this:
1 Using ¬-E, assume ¬(p∨q) (hoping to prove some contradiction)2 When is ¬(p∨q) true ? When both p and q false!3 From ¬(p∨q) how to prove ¬p ? (next slide)4 Having proved both ¬p and ¬q, prove some further contradiction
Exercise: prove¬p→ q
p∨q
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Not-or elimination (derived rule)
¬(p∨q)
¬p
1 ¬(p∨q)
2 p
3 p∨q ∨-I, 2
4 (p∨q)∧¬(p∨q) ∧-I, 1, 3
5 ¬p ¬I, 2–4
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Not-or elimination (derived rule)
¬(p∨q)
¬p
1 ¬(p∨q)
2 p
3 p∨q ∨-I, 2
4 (p∨q)∧¬(p∨q) ∧-I, 1, 3
5 ¬p ¬I, 2–4
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Not-or elimination (derived rule)
¬(p∨q)
¬p
1 ¬(p∨q)
2 p
3 p∨q ∨-I, 2
4 (p∨q)∧¬(p∨q) ∧-I, 1, 3
5 ¬p ¬I, 2–4
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Not-or elimination (derived rule)
¬(p∨q)
¬p
1 ¬(p∨q)
2 p
3 p∨q ∨-I, 2
4 (p∨q)∧¬(p∨q) ∧-I, 1, 3
5 ¬p ¬I, 2–4
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Not-or elimination (derived rule)
¬(p∨q)
¬p
1 ¬(p∨q)
2 p
3 p∨q ∨-I, 2
4 (p∨q)∧¬(p∨q) ∧-I, 1, 3
5 ¬p ¬I, 2–4
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Not-or elimination (derived rule)
¬(p∨q)
¬p
1 ¬(p∨q)
2 p
3 p∨q ∨-I, 2
4 (p∨q)∧¬(p∨q) ∧-I, 1, 3
5 ¬p ¬I, 2–4
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Not-or elimination (derived rule)
¬(p∨q)
¬p
1 ¬(p∨q)
2 p
3 p∨q ∨-I, 2
4 (p∨q)∧¬(p∨q) ∧-I, 1, 3
5 ¬p ¬I, 2–4
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Proving a contrapositive rule
In the same way, whenever you can prove anyp
q
then you can prove¬q¬p
1 ¬q
2 p
3 q your proof of q from p
4 q∧¬q ∧-I, 1, 3
5 ¬p ¬I, 2–4
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Proving a contrapositive rule
In the same way, whenever you can prove anyp
q
then you can prove¬q¬p
1 ¬q
2 p
3 q your proof of q from p
4 q∧¬q ∧-I, 1, 3
5 ¬p ¬I, 2–4
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Proving a contrapositive rule
In the same way, whenever you can prove anyp
q
then you can prove¬q¬p
1 ¬q
2 p
3 q your proof of q from p
4 q∧¬q ∧-I, 1, 3
5 ¬p ¬I, 2–4
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Proving a contrapositive rule
In the same way, whenever you can prove anyp
q
then you can prove¬q¬p
1 ¬q
2 p
3 q your proof of q from p
4 q∧¬q ∧-I, 1, 3
5 ¬p ¬I, 2–4
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Proving a contrapositive rule
In the same way, whenever you can prove anyp
q
then you can prove¬q¬p
1 ¬q
2 p
3 q your proof of q from p
4 q∧¬q ∧-I, 1, 3
5 ¬p ¬I, 2–4
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Proving a contrapositive rule
In the same way, whenever you can prove anyp
q
then you can prove¬q¬p
1 ¬q
2 p
3 q your proof of q from p
4 q∧¬q ∧-I, 1, 3
5 ¬p ¬I, 2–4
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Proving a contrapositive rule
In the same way, whenever you can prove anyp
q
then you can prove¬q¬p
1 ¬q
2 p
3 q your proof of q from p
4 q∧¬q ∧-I, 1, 3
5 ¬p ¬I, 2–4
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Law of the excluded middle (derived)
p ∨¬p“Everything must either be or not be.” – Russell
1 ¬(p∨¬p)
2 ¬p ¬∨-E (previous slide), 1
3 ¬¬p ¬∨-E (previous slide), 1
4 ¬p∧¬¬p ∧-I, 2, 3
5 p∨¬p ¬E, 1–4
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Law of the excluded middle (derived)
p ∨¬p“Everything must either be or not be.” – Russell
1 ¬(p∨¬p)
2 ¬p ¬∨-E (previous slide), 1
3 ¬¬p ¬∨-E (previous slide), 1
4 ¬p∧¬¬p ∧-I, 2, 3
5 p∨¬p ¬E, 1–4
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Law of the excluded middle (derived)
p ∨¬p“Everything must either be or not be.” – Russell
1 ¬(p∨¬p)
2 ¬p ¬∨-E (previous slide), 1
3 ¬¬p ¬∨-E (previous slide), 1
4 ¬p∧¬¬p ∧-I, 2, 3
5 p∨¬p ¬E, 1–4
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Law of the excluded middle (derived)
p ∨¬p“Everything must either be or not be.” – Russell
1 ¬(p∨¬p)
2 ¬p ¬∨-E (previous slide), 1
3 ¬¬p ¬∨-E (previous slide), 1
4 ¬p∧¬¬p ∧-I, 2, 3
5 p∨¬p ¬E, 1–4
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Law of the excluded middle (derived)
p ∨¬p“Everything must either be or not be.” – Russell
1 ¬(p∨¬p)
2 ¬p ¬∨-E (previous slide), 1
3 ¬¬p ¬∨-E (previous slide), 1
4 ¬p∧¬¬p ∧-I, 2, 3
5 p∨¬p ¬E, 1–4
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Law of the excluded middle (derived)
p ∨¬p“Everything must either be or not be.” – Russell
1 ¬(p∨¬p)
2 ¬p ¬∨-E (previous slide), 1
3 ¬¬p ¬∨-E (previous slide), 1
4 ¬p∧¬¬p ∧-I, 2, 3
5 p∨¬p ¬E, 1–4
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