This document contains lecture notes on logic and proofs. It introduces logical arguments with premises and conclusions. It explains valid and sound arguments. It also introduces Fitch notation for representing logical proofs and shows examples of basic rules like introduction and elimination rules for identity. It discusses what constitutes a proof and provides examples of proofs in Fitch notation. It also discusses proving that conclusions do not follow from premises by providing counterexamples.

1. Truth, Deduction,
Computation
Lecture 3
The Logic of Atomic Sentences
Vlad Patryshev
SCU
2013

2. Introducing Arguments...
Premise1, premise2… conclusion!
Or: conclusion - because premise1,...
E.g.
● All men are mortal; Superman is a man, hence
Superman is mortal
● Pavlova is a man: after all, Pavlova is mortal,
and all men are mortal

3. Introducing Arguments...
Premise1, premise2… conclusion!
Or: conclusion - because: premise1,...
E.g.
● All men are mortal; Superman is a man,
hence Superman is mortal
● Pavlova is a man: after all, Pavlova is mortal,
and all men are mortal

4. Arguments
● Valid arguments (true, assuming premises
are true)
● Sound arguments (valid, and premises are
true)

5. Fitch Notation (LPL version)
All cactuses have needles
Prickly pear is a cactus
Prickly pear has needles
Fitch Bar
Conclusion
Premises

6. What is a Proof?
Definition. Proof is a step-by-step
demonstration that a conclusion follows from
premises.
Counterexample:
I ride my bicycle every day
The probability of an accident is very low
I will never have an accident

7. Good Example of a Proof
1. Cube(c)
2. c=b
3. Cube(b)
= Elim: 1,2

8. Elimination Rule
Aka the Indiscernibility of Identicals
Aka Substitution Principle (weaker than Liskov’s)
Aka Identity Elimination
If P(a) and a = b, then P(b).
E.g.
x2 - 1 = (x+1)*(x-1)
x2 > x 2 - 1
x2 > (x+1)*(x-1)

9. Introduction Rule
Aka Reflexivity of Identity
P1
P2
…
Pn
x = x

10. Symmetry of Identity
If a = b then b = a
a = b
a = a
b = a

11. Transitivity of Identity
If a = b and b = c then a = c
a = b
b = c
a = c

12. Other relationships may be transitive
If a < b and b < c then a < c
a
b
c
a
<
<
<
<
b
c
d
d

13. F-notation (specific to LPL book)
(Has nothing to do with System F)
We include in intermediate conclusions
For example:
P1
P2
…
Pn
S1
S2
…
Sm
S
1. a = b
2. a = a
3. b = a
= Intro
= Elim: 2, 1

14. Introduction Rule in Fitch
P1
P2
…
Pn
x = x

15. Introduction Rule (= Intro) in F
= Intro
x = x

16. Elimination Rule in F
= Elim
P(n)
…
n = m
…
P(m)

17. Reiteration Rule in F
= Reit
P
…
…
…
P

18. “Bidirectionality of Between” in F
Between(a,b,c)
…
…
…
Between(a,c,b)

19. Now, How Does It Work?
From premises SameSize(x, x) and x = y, prove
SameSize(y, x).
1. SameSize(x, x)
2. x = y
…
?. SameSize(y, x)

20. Now, How Does It Work? (take 2)
From premises SameSize(x, x) and x = y, prove
SameSize(y, x).
1.
2.
…
?.
?.
SameSize(x, x)
x = y
y = x
SameSize(y, x)
= Elim: 1, ?

21. Now, How Does It Work? (take 3)
From premises SameSize(x, x) and x = y, prove
SameSize(y, x).
1.
2.
…
3.
4.
5.
SameSize(x, x)
x = y
y = y
y = x
SameSize(y, x)
= Intro
= Elim: 3, 2
= Elim: 1, 4

22. Analytical Consequence in Fitch
This is something like a rule, but is based on “common
sense” and external knowledge. E.g.
Cube(a)
SameShape(a, b)
Cube(b)
=Ana Con
(“because we know what Cube means”)
Can be used to prove anything as long as we believe in our rules. It’s okay.

23. Proving Nonconsequence
E.g.
Are all binary operations associative?
Addition is, multiplication is, even with matrices or
complex number
1. op(a, b) = x
2. op(b, c) = y
?. op(a, y) = op(x, c)

24. Proving Nonconsequence
E.g.
Are all binary operations associative?
Addition is, multiplication is, even with matrices or
complex number
1. op(a, b) = x
2. op(b, c) = y
?. op(a, y) = op(x, c)
No!!!
Take binary trees. Take terms (from Chapter 1)

25. Proving Nonconsequence
Given premises P1,...,Pn, and conclusion Q.
Q does not follow from P1,...,Pn if we can provide a
counterexample.

26. References
What Fitch actually is:
Fitch Online:
http://en.wikipedia.org/wiki/Fitch-style_calculus
http://logik.phl.univie.ac.at/~chris/gateway/formular-uk-fitch.html
LPL software online (in Java Applets)
http://softoption.us/content/node/339