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CS1502 Formal Methods in
Computer Science
Lecture Notes 3
Consequence Rules
Boolean Connectives
Symmetry of Identity
1)
a=b
2)
a=a
= Introduction
3)
b=a
= Elimination 1, 2
Example Informal Proof
Prove: If a is smaller than b and c is identical to b
then c is larger than a.
Since we are given that a is smaller than b, it follows
that b must be larger than a. Moreover, since c is
identical to b, it follows that c must be larger than a.
QED
Consequence Rules
There are three consequence “rules” in Fitch
– Tautological Consequence (Taut Con)
– First-order Consequence (FO Con)
– Analytic Consequence (Ana Con)
Cons rules are proof seekers that work behind the
scenes. Success is indicated by the beloved blue
check mark . Failure is indicated by the infamous red
X.
Taut Con
Weakest of the three Cons “rules”
Attempts to prove if the current step
follows from the cited statements by virtue
of the truth-functional connectives. E.g., p
^ q can be replaced by p v q.
FO Con
More powerful than Taut Con but weaker
than Ana Con
Attempts to prove if the current step
follows from the cited steps by virtue of the
truth-functional connectives, the
quantifiers, and the identity predicate. E.g.,
a=b ^ b=c can be replaced by a=c.
Ana Con
Most powerful Cons “rule”
Attempts to prove if the current step
follows from the cited steps by virtue of the
truth-functional connectives, the
quantifiers, the identity predicate and the
meanings predicates of Tarski’s World.
E.g., Cube(a) v Tet(a) v Dodec(a) is true under Ana Con
Example Formal Proof
1)
2)
Larger(a,b)
c=b
Prove: c is smaller than a
3) Smaller(b,a)
Ana Con 1
4)
c=c
= Introduction
5)
b=c
= Elim 2, 4
6)
Smaller(c,a)
= Elim 5, 3
Explicit Proof of an Ana Con Step
Proof performed by the system, behind the scenes
Showing Non-consequence
To show Q is not a consequence of premises P1,
P2, …, Pn, create a world where the premises
are all true and the conclusion Q is false.
Invalid Argument
1. SameRow(b,c)
2. SameRow(a,d)
3. SameRow(d,f)
4. LeftOf(a,b)
5. LeftOf(f,c)
Boolean Connectives
Negation 
Conjunction 
Disjunction 
It is not the case that
And, but, moreover
Or
These are truth-functional connectives
Negation
P
T
P
F
F
T
Negation Facts
One translation of P is
It is not the case that P.
(a = b) is equivalent to a  b
 P is equivalent to P
P
P
 P
T
F
T
F
T
F
The Game
Used to understand the truth value of a
complex sentence
Strategy: Given a sentence of the form
   P that you believe to be True
(False) implies   P is False (True)
implies  P is True (False)
implies P is False (True)
Assessment is incorrect
Conjunction and Disjunction
P
Q
PQ
PQ
T
T
T
T
T
F
F
T
F
T
F
T
F
F
F
F
Game Revisited
If you believe P  Q is true you will be asked
to select the disjunct that is true.
If you believe P  Q is false Tarski’s World will
attempt to find a disjunct that is true.
If you believe P  Q is true Tarski’s World will
attempt to find a conjunct that is false.
If you believe P  Q is false you will be asked
to select a conjunct that is false.
Game Example
In lecture
Translation
Both A and B
Either A or B
Neither A nor B
AB
AB
(A  B)
Both c and e are cubes.
Cube(c)  Cube(e)
Either c or e is a cube.
Cube(c)  Cube(e)
Neither c nor e is a cube.
(Cube(c)  Cube(e))
Translation
John entered the car and drove to his home
Entered ^ Drove(john,car(john),house(john))
– OK?
No – the truth functional connectives connect sentences,
not predicates
Entered(john,car(john)) ^ Drove(john,house(john))
– This is OK
English sentences often convey more information than
their straightforward logical meanings, e.g., temporal
order in “John entered the car and drove home”
That’s OK: in this class, you can just give the
straightforward translation into logic
Translation
I will vote for Smith or I will vote for Jones
Straightforward literal translation:
Vote(I,smith) v Vote(I,jones)
Logic “or” is inclusive or.
But “or” in our English sentence is exclusive or – I’ll vote
for one of them but not both
We can define exclusive or using ^ and v:
P xor Q equiv: (P ^ ~Q) v (Q ^ ~P)
(Vote(I,smith) ^ ~Vote(I,jones)) v (Vote(I,jones) ^ ~Vote(I,smith))
Unless the text says otherwise, assume inclusive “or” (even when it
says “Either A or B”)
Avoid Ambiguity
(A
A B)
B
 CC
Both either A or B and C
A  (B  C)
Either A or both B and C
(A
AB
B)CC
Either both A and B or C
A  (B  C)
Both A and either B or C
Either both Max is at home and Claire is tall or Carl is happy.
[Home(Max)  Tall(Claire)]  Happy(Carl)
Avoid Ambiguity
Let’s look at that example again;
Either both Max is at home and Claire is
tall or Carl is happy
[Home(max) ^ Tall(claire)] v Happy(carl)
Home(max) ^ (Tall(claire) v Happy(carl))
Translation:
Both Max is at home and either Claire is
tall or Carl is happy.
Equivalences
Two FOL sentences P and Q mean the same thing (are logically
equivalent, written P  Q) iff they have the same truth value in all
situations. If two sentences are logically equivalent, you can
substitute one for the other.
Identity Laws: P ^ T  P; P v F  P
Domination Laws: P v T  T; P ^ F  F
Idempotent Laws: P v P  P; P ^ P  P
Double Negation: ~~P  P
Commutative Laws: P v Q  Q v P; P ^ Q  Q ^ P
Associative Laws: (P v Q) v R P v (Q v R)
(P ^ Q) ^ R  P ^ (Q ^ R)
Distributive Laws: P v (Q ^ R)  (P v Q) ^ (P v R)
P ^ (Q v R)  (P ^ Q) v (P ^ R)
DeMorgan’s Laws: ~(P ^ Q)  ~P v ~Q
~(P v Q)  ~P ^ ~Q