Transcript Propositional Logic

Propositional Logic
Copyright © Peter Cappello
Sentence Restrictions
Building more precise tools from less precise tools
• Precise use of natural language is difficult.
• We want a sublanguage suited to precision.
• Restrict discussion to sentences that are:
• declarative
• either true or false but not both.
• Such sentences are called propositions.
Copyright © Peter Cappello
Examples of Propositions
Which of the sentences below are propositions?
• “Mastercharge, dig me into a hole!”
• “Peter Cappello thinks this class is fascinating.”
• “Do I exist yet?”
• “This sentence is false.”
Copyright © Peter Cappello
Not Operator
• Not ( ~ ): p is true exactly when ~p is false.
• Let p denote “This class is the greatest
entertainment since Game of Thrones.”
• ~p denotes “It is not the case that this class is the
greatest entertainment since Game of Thrones.”
p ~p
T F
F T
Copyright © Peter Cappello
Or Operator (Disjunction)
Or (  ): proposition p  q is true exactly
when either p is true or q is true:
p q pq
T T T
T F T
F T T
F F F
Copyright © Peter Cappello
And Operator (Conjunction)
And (  ): proposition p  q is true exactly
when p is true and q is true:
p q pq
T T T
T F F
F T F
F F F
Copyright © Peter Cappello
If and Only If Operator (IFF)
If and only if (  ): proposition p  q is true
exactly when (p  q) or (~ p  ~ q):
p q p« q
T
T
F
F
T
F
T
F
T
F
F
T
Copyright © Peter Cappello
Exclusive-Or
Exclusive-or (  ) is the negation of  .
p q pÙ q p Å q
T T T
T F F
F
T
F T F
F F F
T
F
Copyright © Peter Cappello
Implies Operator (If … Then)
• Implies (  ): proposition p  q is true
exactly when p is false or q is true:
p q p q
T
T
F
F
T
F
T
F
T
F
T
T
Copyright © Peter Cappello
If … Then ...
• Example: “If pigs had wings they could fly.”
• In English, implies normally connotes a
causal relation:
p implies q means that p causes q to be true.
• Not so with the mathematical definition!
If 1  1 then Peter hates Family Guy.
Copyright © Peter Cappello
Converse & Inverse
The converse of p  q is q  p.
The inverse of p  q is ~p  ~q.
Compare the truth tables of the converse & inverse.
The contrapositive of p  q is ~q  ~p.
If p  q then which, if any, is always true:
• Its converse?
• Its inverse?
• Its contrapositive?
Use a truth table to find the answer.
• Describe contrapositive in terms of the converse & inverse.
•
•
•
•
•
Copyright © Peter Cappello
p  q may be expressed as
•
•
•
•
•
•
•
•
•
•
p implies q
if p then q
q if p
q follows from p
q provided p
q is a consequence of p
q whenever p
p is a sufficient condition for q
p only if q
(if ~q then ~p)
q is a necessary condition for p (if ~q then ~p)
Copyright © Peter Cappello
Abstraction
Capture the logical form of a Proposition in English
• Let g, h, and b be propositions:
• g: Grizzly bears have been seen in the area.
• h: Hiking is safe on the trail.
• b: Berries are ripe along the trail.
• Translate the following sentence using g, h,
and b, and logical operators:
If berries are ripe along the trail, hiking is safe on
the trail if and only if grizzly bears have not been
seen in the area.
Copyright © Peter Cappello
1. If berries are ripe along the trail, hiking is
safe on the trail if and only if grizzly bears
have not been seen in the area.
2. If b, ( h if and only if  g ).
3. b  ( h   g ).
Copyright © Peter Cappello
Truth Table of a Compound Proposition
b
T
T
T
T
F
F
F
F
h
T
T
F
F
T
T
F
F
g g hg b(hg)
T
F
T
F
T
F
T
F
Copyright © Peter Cappello
System Specification
• Systems are increasing in complexity.
• e.g., software, hardware, workflow, security, legal
• Can we know that a system works as intended?
1. Specify a set of desired system properties
Each property is expressed as a compound proposition.
2. Verify that such a system is feasible.
All compound propositions are simultaneously satisfiable.
• Z specification language
• Allow: http://alloy.mit.edu/alloy/
Copyright © Peter Cappello
Knights & Knaves
• An island’s only inhabitants are knights (truth
tellers) & knaves (liars).
• You are approached by 2 inhabitants, A & B.
• Determine, if possible, what A & B are, if B says
nothing & A says:
1. “At least 1 of us is a knave.”
2. “I am a knave or B is a knight.”
3. “We are both knaves.”
Copyright © Peter Cappello
Look at 4 cases for A and B
1. “At least 1 of us is a knave.”
A is a knight; B is a knave.
2. “I am a knave or B is a knight.”
A & B are knights
3. “We are both knaves.”
A is a knave; B is a knight.
Copyright © Peter Cappello
Google Search Operators
http://support.google.com/websearch/answer/136861?hl=en
Query: “US states” “income tax rate”
Beatles: “Taxman” (Query: Beatles Taxman)
Let me tell you how it will be
There's one for you, nineteen for me
'Cause I'm the taxman, yeah, I'm the taxman
Should five per cent appear too small
Be thankful I don't take it all
'Cause I'm the taxman, yeah I'm the taxman
Don't ask me what I want it for
If you don't want to pay some more
'Cause I'm the taxman, yeah, I'm the taxman
Now my advice for those who die
Declare the pennies on your eyes
'Cause I'm the taxman, yeah, I'm the taxman
And you're working for no one but me.
If you drive a car, I'll tax the street,
If you try to sit, I'll tax your seat.
If you get too cold I'll tax the heat,
If you take a walk, I'll tax your feet.
Copyright © Peter Cappello