Mathematical Structures
for Computer Science
Chapter 1.2
Copyright © 2006 W.H. Freeman & Co.
MSCS Slides
Formal



Propositions or statements are sentences that
are either true or false.
In logic, propositions are represented by the
letters A, B, C, …
Propositions can be combined into more
complicated expressions known as wellformed formulas, or wffs.
◦ The connectives: and (), or (), implies (), and
negation ()

Wffs are represented by the letters P, Q, R, …
Propositional Logic
Section 1.2
1
o
o
A system for determining whether or not a particular
logical argument is valid, using rules of inference (also
known as propositional calculus.)
Definition of Argument:
o A sequence of statements in which the conjunction of the initial
statements (called the premises, or hypotheses) is said to imply
the final statement (called the conclusion).
o Presented symbolically as
(P1 Λ P2 Λ ... Λ Pn)  Q
where P1, P2, ..., Pn represent the hypotheses and Q represents the
conclusion.
Propositional Logic
Section 1.2
2
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What is a valid argument? When does Q logically follow
from P1, P2, ..., Pn?
Informal answer: When the truth of the hypotheses
guarantees the truth of the conclusion.
Note: We need to be sure the hypotheses are related to
the conclusion; not just any knowledge we might have
about the conclusion Q.
Example:
◦ P1: Neil Armstrong was the first human to step on the
moon.
◦ P2 : Mars is a red planet
And the conclusion
◦ Q: No human has ever been to Mars.
◦ P1, P2 , and Q are all true, but Q isn’t a logical consequence
Propositional Logic
Section 1.2
3
Definition of valid argument:

In a formal logical system an argument is valid if,
whenever the hypotheses are all true, the conclusion
must also be true. i.e. (P1 Λ P2 Λ ... Λ Pn)  Q is a
tautology.
◦ You must be able to logically deduce Q from the hypotheses

How to arrive at a valid argument?
◦ Using a proof sequence
Propositional Logic
Section 1.2
4
Definition of Proof Sequence:


A proof sequence is a sequence of wffs in which each
wff is either a hypothesis or the result of applying one
of the formal system’s derivation rules to earlier wffs in
the sequence.
Derivation rules allow you to rewrite a hypothesis in a
more useful form, based on the equivalence relations
from section one, or to derive new wffs from the old
ones.
Propositional Logic
Section 1.2
5

Derivation rules for propositional logic
Equivalence Rules
Inference Rules
Allows individual wffs to be
rewritten
Allows new wffs to be derived
Truth preserving rules
Work only in one direction
Propositional Logic
Section 1.2
6


Certain pairs of wffs are equivalent, hence one can be
substituted for the other with no change to truth values.
The set of equivalence rules are summarized here:
Let R and S be statement variables
Expression
Equivalent to
Abbreviation for rule
PVQ
PΛ Q
QVP
QΛP
Commutative
(comm)
(P V Q) V R
(P Λ Q) Λ R
P V (Q V R)
P Λ (Q Λ R)
Associative
(assoc)
(P V Q)
(P Λ Q)
P Λ Q
P V Q
De Morgan’s Laws
PQ
P V Q
Implication (imp)
P
(P)
Double Negation (dn)
PQ
(P  Q) Λ (Q  P)
Equivalence (equ)
Propositional Logic
Section 1.2
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

Inference rules allow us to add a new wffs to aproof
sequence. If one or more wffs match an existing part of
the proof sequence (this is the “from” part) you can
deduce the second part and add to the proof.
From
Can Derive
Abbreviation for
rule
R, R  S
S
Modus Ponens- mp
R  S, S
R
Modus Tollens- mt
R, S
RΛS
Conjunction-con
RΛS
R, S
Simplification- sim
R
RVS
Addition- add
Note: Inference rules do not work in both directions,
unlike equivalence rules.
◦ E.g., if you know R V S you can’t deduce R (or S).
Propositional Logic
Section 1.2
8

Example for using Equivalence rule in a proof sequence:
◦ Simplify (A V B) V C
1. (A V B) V C
2. (A Λ B) V C
3. (A Λ B)  C

1, De Morgan
2, imp
Example of using Inference Rule
◦ If it is bright and sunny today, then I will wear my sunglasses.
Modus Ponens
(given P, P S, conclude S)
It is bright and sunny today. Therefore, I will wear my
sunglasses.
Modus Tollens
(given P S, S, conclude P)
I will not wear my sunglasses. Therefore, it is not bright and
sunny today.
Propositional Logic
Section 1.2
9

Prove the argument
A Λ (B  C) Λ [(A Λ B)  (D V C)] Λ B  D
First, write down all the hypotheses.
1. A
2. B  C
3. (A Λ B)  (D V C)
4. B
Use the inference and equivalence rules to get at the conclusion
D.
5. C
2,4, mp
6. A Λ B
1,4, con
7. D V C
3,6, mp
8. C V D
7, comm
9. C  D
8, imp
and finally
10. D
5,9 imp
The idea is to keep focused on the result and sometimes it is
very easy to go down a longer path than necessary.
◦
Propositional Logic
Section 1.2
10



Used to prove an argument of the form
P1 Λ P2 Λ ... Λ Pn  R  Q
Deduction method allows for the use of R as an
additional hypothesis and thus prove
P1 Λ P2 Λ ... Λ Pn Λ R  Q
Suppose you want to prove
[(A  B) Λ (B  C)]  (A  C)
Propositional Logic
Section 1.2
11

Prove
[(A  B) Λ (B  C)]  (A  C)
Using deduction method, prove (A  B) Λ (B  C) Λ A  C
1.
A  B hyp
2.
B  C hyp
3.
A
hyp
4.
B
1,3 mp
5.
C
2,4 mp


The above is called the rule of Hypothetical Syllogism or
hs for short.
Many such other rules can be derived from existing
rules which thus provide easier and faster proofs.
Propositional Logic
Section 1.2
12

These rules can be derived using the previous rules.
They provide a faster way of proving arguments.
From
Can Derive
Name / Abbreviation
P  Q, Q  R
PR
Hypothetical syllogismhs
P V Q, P
Q
Disjunctive syllogismds
PQ
Q  P
Contraposition- cont
Q  P
PQ
Contraposition- cont
Propositional Logic
Section 1.2
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
Additional inference rules yield shorter proofs, but more
rules to remember.
From
Can Derive
Name / Abbreviation
P
PΛ P
Self-reference – self
PVP
P
Self-reference – self
(P Λ Q)  R
P  (Q R)
Exportation – exp
P, P
Q
Inconsistency – inc
P Λ (Q V R)
(P Λ Q) V (P Λ R)
Distributive - dist
P V (Q Λ R)
(P V Q) Λ (P V R)
Distributive - dist
Propositional Logic
Section 1.2
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
Prove that (P  Q)  (Q P) is a valid argument (called
Contraposition – con).
◦
◦

Hence prove, (P  Q) Λ Q  P (using deduction method).
The above is true using the modus tollens inference rule.
Prove P Λ P  Q
1.
2.
3.
4.
5.
6.
7.
8.
P
P
PVQ
QVP
(Q) V P
Q  P
(Q)
Q
(called Inconsistency)
hyp
hyp
1, add
3, comm
4, dn
5, imp: (A V B)  (A  B)
2, 6, mt: from (P S, S ) get P
7, dn
Propositional Logic
Section 1.2
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
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
(A Λ B) Λ (C Λ A) Λ (C Λ B)  A is an argument
(A Λ B)
(C Λ A)
(C Λ B)
A V B
B V A
B  A
(C) V A
C  A
C V (B)
(B) V C
B  C
B  A
(B  A) Λ (B  A)
hyp
hyp
hyp
1, De Morgan
4, comm
5, imp
2, De Morgan
7, imp
3, De Morgan
9, comm
10, imp
8, 11, hs
6, 12, con
Propositional Logic
Section 1.2
16

At this point, we have now to prove that
(B  A) Λ (B  A)  A

1.
2.
3.
4.
5.
6.
Proof sequence
B  A
B  A
A  B
A  A
A V A
A
hyp
hyp
1, cont
3, 2, hs
4, imp
6, self
Propositional Logic
Section 1.2
17
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To convert a verbal argument to a symbolic form so it can be
tested for validity:
“Symbolize” the argument using propositional wff’s
Use the derivation rules for propositional logic to prove the
argument is valid
Example:
“If interest rates drop, the housing market will improve.
Either the federal discount rate will drop or the housing
market will not improve. Interest rates will drop. Therefore
the federal discount rate will drop.”
Let
I
represent
Interest rates drop
H represent
The housing market will improve
F
represent
The federal discount rate will drop
Then
(I  H)  (F  H)  I  F
Propositional Logic
Section 1.2
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
1.
2.
3.
4.
5.
6.
7.
A proof sequence for
(I  H)  (F  H)  I  F
IH
hyp
F  H
hyp
I
hyp
H  F
2, comm
HF
4, imp
IF
1, 5, hs
F
5, 3, mp
Propositional Logic
Section 1.2
19

Russia was a superior power, and either France was not strong or
Napoleon made an error. Napoleon did not make an error, but if
the army did not fail, then France was strong. Hence the army
failed and Russia was a superior power.
Converting it to a propositional form using letters A, B, C and D

Combining, the statements using logic

Combining them, the propositional form is

A: Russia was a superior power
B: France was strong
B: France was not strong
C: Napoleon made an error
C: Napoleon did not make an error
D: The army failed
D: The army did not fail
(A Λ (B V C))
C
(D  B)
(D Λ A)
hypothesis
hypothesis
hypothesis
conclusion
(A Λ (B V C)) Λ C Λ (D  B)  (D Λ A)
Propositional Logic
Section 1.2
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

Prove (A Λ (B V C)) Λ C Λ (D  B)  (D Λ A)
Proof sequence
1. A Λ (B V C)
2. C
3. D  B
4. A
5. B V C
6. C V B
7. B
8. B  (D)
9. (D)
10. D
11. D Λ A
hyp
hyp
hyp
1, sim
1, sim
5, comm
2, 6, ds
3, cont
7, 8, mp
9, dn
4, 10 , con
Propositional Logic
Section 1.2
21

Prove the following arguments
◦ (A´  B´) Λ (A  C)  (B  C)
◦ (Y  Z´) Λ (X´  Y) Λ [Y  (X  W)] Λ (Y  Z)  (Y  W)


If the program is efficient, it executes quickly. Either the
program is efficient, or it has a bug. However, the
program does not execute quickly. Therefore it has a
bug. (use letters E, Q, B)
The crop is good, but there is not enough water. If there
is a lot of rain or not a lot of sun, then there is enough
water. Therefore the crop is good and there is a lot of
sun. (use letters C, W, R, S)
Propositional Logic
Section 1.2
22


Write down the propositional form of the following
argument:
If my client is guilty, then the knife was in the drawer.
Either the knife was not in the drawer or Jason Pritchard
saw the knife. If the knife was not there on October 10,
it follows that Jason Pritchard didn’t see the knife.
Furthermore, if the knife was there on October 10, then
the knife was in the drawer and also the hammer was in
the barn. But we all know that the hammer was not in
the barn. Therefore, ladies and gentlemen of the jury,
my client is innocent.
Propositional Logic
Section 1.2
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