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Introductory LogicPHI 120
Presentation: “Solving Proofs"
Bring the Rules Handout to lecture
Homework• Memorize the primitive rules, except ->I and
RAA
• Ex. 1.4.2 (according to these directions)
For Each Sequent, answer these two questions:1. What is the conclusion?2. How is the conclusion embedded in the
premises?
Homework I• Memorize the primitive rules
– Capable of writing the annotationm vI
– Cite how many premises make up each ruleone premise rule
– Cite what kind of premises make up each rulecan be any kind of wff (i.e., one of the disjuncts)
– Cite what sort of conclusion may be deriveda disjunction
See The Rules Handout
Except ->I and RAA
Homework I• Memorize the primitive rules
– Capable of writing the annotationm vI
– Cite how many premises make up each ruleone premise rule
– Cite what kind of premises make up each rulecan be any kind of wff (i.e., one of the disjuncts)
– Cite what sort of conclusion may be deriveda disjunction
See The Rules Handout
Except ->I and RAA
Content of Today’s Lesson
1. Proof Solving Strategy
2. The Rules
3. Doing Proofs
Expect a Learning Curve with this New Material
Homework is imperative
Study these presentations
SOLVING PROOFS“Natural Deduction”
Strategy
Key Lesson Today
(1) Read Conclusion
(2) Find Conclusion in Premises
P -> Q, Q -> R ⊢ P -> R
Valid Argument:True Premises Guarantee a True Conclusion
Ex. 1.4.2
My DirectionsConclusion
(1) What is the conclusion?
Conclusion in Premises
(2.a) Is the conclusion as a whole embedded in any premise?
If yes, where? Else…
(2.b) Where are the parts that make up the conclusion embedded in the premise(s)?
S1 – S10
2) How is the conclusion embedded in the premises?
Homework II
Conclusion in Premises• Example: S16
P -> Q, Q -> R ⊢ P -> R
Conclusion in Premises• Example: S16
P -> Q, Q -> R ⊢ P -> RC1.Conclusion:
a conditional statement
2.Conclusion in the premises: The conditional is not embedded in any premise Its antecedent “P” is the antecedent of the first premise. Its consequent “R” is the consequent of the second
premise.
Answers:
SOLVING PROOFS“Natural Deduction”
The Rules
Proofs
• Rule based system– 10 “primitive” rules
• Aim of Proofs– To derive conclusions on basis of given premises
using the primitive rules
See page 17 – “proof”
What is a Primitive Rule of Proof?
• Primitive Rules are Basic Argument Forms– simple valid argument forms
• Rule Structure– One conclusion
– Premises• Some rules employ one premise• Some rules employ two premises
Φ & Ψ ⊢ Φ
m &E Ampersand-EliminationGiven a sentence that is a conjunction, conclude either conjunct
m,n &I Ampersand-IntroductionGiven two sentences, conclude a conjunction of them.
Φ , Ψ ⊢ Φ & Ψ
Catch-22You have to memorize the rules!
1. To memorize the rules, you need to practice doing proofs.
2. To practice proofs, you need to have the rules memorized
A Solution of Sorts
"Rules to Memorize" on The Rules handout
&E ampersand elimination
vE wedge elimination
->E arrow elimination
<->E double-arrow elimination
&I ampersand introduction
vI wedge introduction
->I arrow introduction
<->I double-arrow introduction
Elimination Introduction
Make a conclusion
Break a premise
THE TEN “PRIMITIVE” RULESProofs
Elimination Rules (break a premise) Introduction Rules (make a conclusion)
* &E (ampersand Elimination) * &I (ampersand Introduction)
* vE (wedge Elimination) * vI (wedge Introduction)
* ->E (arrow Elimination) * ->I (arrow Introduction)
* <->E (double arrow Elimination) * <->I (double arrow Introduction)
A (Rule of Assumption)
RAA (Reductio ad absurdum)
The 10 Rules
Rules of Derivation
1 rule of "assumption": A
4 "elimination" rules: &E, vE, ->E, <->E
4 "introduction" rules: &I, vI, ->I, <->I
1 more rule: “RAA” (reductio ad absurdum)
= 10 rules
The 10 Rules
Rules of Derivation
1 rule of "assumption": A
4 "elimination" rules: &E, vE, ->E, <->E
4 "introduction" rules: &I, vI, ->I, <->I
1 more rule: “RAA” (reductio ad absurdum)
= 10 rules
Proofs: 1st Rule
• The most basic rule: <A> Rule of Assumption
a) Every proof begins with assumptions (i.e., basic premises)
b) You may assume any WFF at any point in a proof
Assumption Numberthe line number on which the “A” occurs.
The 10 Rules
Rules of Derivation
1 rule of "assumption": A
4 "elimination" rules: &E, vE, ->E, <->E
4 "introduction" rules: &I, vI, ->I, <->I
1 more rule: “RAA” (reductio ad absurdum)
= 10 rules
The 10 Rules
Rules of Derivation
1 rule of "assumption": A
4 "elimination" rules: &E, vE, ->E, <->E
4 "introduction" rules: &I, vI, ->I, <->I
1 more rule: “RAA” (reductio ad absurdum)
= 10 rules
Proofs: 2nd – 9th Rules
–Elimination Rules – break premises
–Introduction Rules – make conclusions
The Guts of the System
&I, vI, ->I, <->I
&E, vE, ->E, <->E
The 10 Rules
Rules of Derivation
1 rule of "assumption": A
4 "elimination" rules: &E, vE, ->E, <->E
4 "introduction" rules: &I, vI, ->I, <->I
1 more rule: “RAA” (reductio ad absurdum)
= 10 rules
The 10 Rules
Rules of Derivation
1 rule of "assumption": A
4 "elimination" rules: &E, vE, ->E, <->E
4 "introduction" rules: &I, vI, ->I, <->I
1 more rule: “RAA” (reductio ad absurdum)
= 10 rules
(later)
SOLVING PROOFS“Natural Deduction”
Doing Proofs
m &EDoing Proofs
The “annotation”
page 18
P & Q P⊢
A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)
P & Q P⊢(1)
A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)
P & Q P⊢(1) A
A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)
P & Q P⊢(1) P & Q A
A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)
P & Q P⊢1 (1) P & Q A
A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)
P & Q P⊢1 (1) P & Q A
(2)
P & Q P ⊢1 (1) P & Q A
(2) P ???
Read the sequent!
"P" is embedded in the premise.
We will have to break it out of the conjunction. Hence &E.
P & Q P⊢1 (1) P & Q A
(2) P ???
P & Q P⊢1 (1) P & Q A
(2) P 1 &E
P & Q P⊢1 (1) P & Q A
(2) P 1 &E
P & Q P⊢1 (1) P & Q A
(2) P 1 &E
P & Q P⊢1 (1) P & Q A
(2) P 1 &E
P & Q P⊢1 (1) P & Q A1 (2) P 1 &E
m,n &IDoing Proofs
The “annotation”
P, Q Q & P⊢
A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)
P, Q Q & P⊢(1)
A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)
P, Q Q & P⊢(1) A
A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)
P, Q Q & P⊢(1) P A
A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)
P, Q Q & P⊢1 (1) P A
A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)
P, Q Q & P⊢1 (1) P A
(2)A line of a proof contains four elements: (i) line number (number within parentheses)
P, Q Q & P⊢1 (1) P A
(2) AA line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right)
P, Q Q & P⊢1 (1) P A
(2) Q AA line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number)
P, Q Q & P⊢1 (1) P A2 (2) Q A
A line of a proof contains four elements: (i) line number (number within parentheses) (ii) annotation (at the very right) (iii) sentence derived (next to line number) (iv) assumption set (number to very left)
P, Q Q & P⊢1 (1) P A2 (2) Q A
(3)
P, Q Q & P⊢1 (1) P A2 (2) Q A
(3) ???
Read the sequent!
"P & Q" is not embedded in any premise.
We will have to make the conjunction. Hence &I
P, Q Q & P⊢1 (1) P A2 (2) Q A
(3) ?, ? &I
P, Q Q & P⊢1 (1) P A2 (2) Q A
(3) Q & P ?, ? &I
P, Q Q & P⊢1 (1) P A2 (2) Q A
(3) Q & P 1, 2 &I
P, Q Q & P⊢1 (1) P A2 (2) Q A
(3) Q & P 1, 2 &I
P, Q Q & P⊢1 (1) P A2 (2) Q A1,2 (3) Q & P 1, 2 &I
Don't forget to define the assumption set!
P, Q Q & P⊢1 (1) P A2 (2) Q A1, 2 (3) Q & P 1, 2 &I
Homework• Memorize the primitive rules, except ->I and
RAA
• Ex. 1.4.2 (according to these directions)
For Each Sequent, answer these two questions:1. What is the conclusion?2. How is the conclusion embedded in the
premises?