Presentation: “Solving Proofs"
Introductory Logic
PHI 120
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
Except ->I and RAA
– Capable of writing the annotation
m vI
– Cite how many premises make up each rule
one premise rule
– Cite what kind of premises make up each rule
can be any kind of wff (i.e., one of the disjuncts)
– Cite what sort of conclusion may be derived
a disjunction
See The Rules Handout
Homework I
• Memorize the primitive rules
Except ->I and RAA
– Capable of writing the annotation
m vI
– Cite how many premises make up each rule
one premise rule
– Cite what kind of premises make up each rule
can be any kind of wff (i.e., one of the disjuncts)
– Cite what sort of conclusion may be derived
a disjunction
See The Rules Handout
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
“Natural Deduction”
SOLVING PROOFS
Key Lesson Today
P -> Q, Q -> R ⊢ P -> R
(1) Read Conclusion
Valid Argument:
True Premises Guarantee a True Conclusion
(2) Find Conclusion in Premises
Homework II
Ex. 1.4.2
S1 – S10
My Directions
Conclusion
(1) What is the conclusion?
Conclusion in Premises
(2.a)
Is the
conclusion
as aembedded
whole embedded
in any
2) How
is the
conclusion
in the premises?
premise?
If yes, where? Else…
(2.b) Where are the parts that make up the conclusion
embedded in the premise(s)?
Conclusion in Premises
• Example: S16
P -> Q, Q -> R ⊢ P -> R
Conclusion in Premises
• Example: S16
P -> Q, Q -> R ⊢ P -> R
C
1. 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.
“Natural Deduction”
SOLVING PROOFS
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?
Φ
Φ
,
&
Ψ
Ψ
Φ
&
Φ
Ψ
⊢
⊢
• Primitive Rules are Basic Argument Forms
m,n
m
&E&I Ampersand-Elimination
Ampersand-Introduction
– simple valid argument forms
Given two
a sentence
sentences,
that conclude
is a conjunction,
a conjunction
conclude
of
them. either conjunct
• Rule Structure
– One conclusion
– Premises
• Some rules employ one premise
• Some rules employ two premises
Catch-22
You 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
Elimination
Introduction
&E ampersand elimination
&I ampersand introduction
vE
vI
wedge elimination
wedge introduction
->E arrow elimination
->I arrow introduction
<->E double-arrow
<->I double-arrow
elimination
introduction
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)
Proofs
THE TEN “PRIMITIVE” 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
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 Number
the 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
&E, vE, ->E, <->E
–Introduction Rules – make conclusions
&I, vI, ->I, <->I
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(later)
ad absurdum)
= 10 rules
“Natural Deduction”
SOLVING PROOFS
The “annotation”
Doing Proofs
m &E
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
(2)
A
Read the sequent!
P&Q⊢P
1
(1) P & Q
(2) P
A
???
"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
(2) P
A
???
P&Q⊢P
1
(1) P & Q
(2) P
A
1 &E
P&Q⊢P
1
(1) P & Q
(2) P
A
1 &E
P&Q⊢P
1
(1) P & Q
(2) P
A
1 &E
P&Q⊢P
1
(1) P & Q
(2) P
A
1 &E
P&Q⊢P
1
(1) P & Q
1
(2) P
A
1 &E
The “annotation”
Doing Proofs
m,n &I
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
(2)
A
A line of a proof contains four elements:
(i) line number
(number within parentheses)
P, Q ⊢ Q & P
1
(1) P
(2)
A
A
A 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
(2) Q
A
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)
P, Q ⊢ Q & P
1
(1) P
2
(2) Q
A
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
2
(2) Q
(3)
A
A
Read the sequent!
P, Q ⊢ Q & P
1
(1) P
2
(2) Q
(3)
A
A
???
"P & Q" is not embedded in any premise.
We will have to make the conjunction. Hence &I
P, Q ⊢ Q & P
1
(1) P
2
(2) Q
(3)
A
A
?, ? &I
P, Q ⊢ Q & P
1
(1) P
2
(2) Q
(3) Q & P
A
A
?, ? &I
P, Q ⊢ Q & P
1
(1) P
2
(2) Q
(3) Q & P
A
A
1, 2 &I
P, Q ⊢ Q & P
1
(1) P
2
(2) Q
(3) Q & P
A
A
1, 2 &I
P, Q ⊢ Q & P
1
(1) P
2
(2) Q
1,2
(3) Q & P
Don't forget to define
the assumption set!
A
A
1, 2 &I
P, Q ⊢ Q & P
1
(1) P
2
(2) Q
1, 2
(3) Q & P
A
A
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?