The Logic of Boolean Connectives

1
The Logic of Boolean Connectives
Language, Proof and Logic
Chapter 4
2
Tautologies and logical truth
4.1.a
A sentence is Logically possible if it could be
true. Bob is funny or Bob is not funny
Bob is funny Bob arrived from the Earth to
Pluto within 1 min Logically necessary (logical
truth) if it could not be false. Bob is
funny or Bob is not funny aa if cd
and c is a grumber, then d is a grumber How
about Tet(a) ?
Cube(a) ? Dodec(a)
SameRow(a,a)? Tautology if every row of its
truth table has a T.
You try it, page 100
3
Tautologies and logical truth
4.1.b
Tarskis world necessities
Logical necessities
Tautologies
4
Logical and tautological equivalence
4.2
Two sentences S1 and S2 are Logically
equivalent if it is impossible that S1 is true
while S2 is false, or S1 is false while S2 is
true. Tautologically equivalent if S1 and S2
have identical truth values in their joint truth
table. Construct such tables for ?(A?B) vs.
?A??B A vs. B ?((A?B) ? ?C) vs. (?A??B) ?
C ab ? Cube(a) vs. ab ? Cube(b)
5
Logical and tautological consequence
4.3
A sentence S is a logical consequence of
sentences P1,,Pn if it is impossible that S is
false while each of P1,,Pn is true.
A sentence S is a tautological consequence of
sentences P1,,Pn if in every row of the joint
truth table, whenever each of P1,,Pn is true,
so is S.
Is Cube(a) a logical consequence of Cube(b), ab?
Is it also a tautological consequence?
Is A?B a logical consequence of B?A? Howbout
vice versa?
Is B a logical consequence of A?B, ?A?
6
Tautological consequence in Fitch
4.4
Ana Con vs. FO Con vs. Taut Con You try it, p.
117
7
Pushing negation around
4.5.a
The principle of substitution of logical
equivalents If P ? Q, then S(P) ?
S(Q). Negation normal form (NNF) negation is
applied only to atoms.
De Morgans laws allow us to bring any formula
down to NNF
? (A?B)???C
? ((A?B)??C) ?
? ? (A?B)?C
? (?A??B)?C
8
Commutativity, idempotence, associativity
4.5.b
(A?B)?C?(?(?B??A)?B) ? (A?B)?C?((??B???A)?B)

? (A?B)?C?((B?A)?B)
?
(A?B)?C?(B?A?B)
? (A?B)?C?(A?B?B)
? (A?B)?C?(A?B)
?
(A?B)?(A?B)?C
? (A?B)?C
Commutativity P?Q ? Q?P P?Q ? Q?P
Idempotence P?P ? P P?P ? P
Associativity P?(Q?R) ? (P?Q)?R ? P?Q?R
P?(Q?R) ? (P?Q)?R ? P?Q?R
Chain of equivalences
9
Conjunctive and disjunctive normal forms
4.6.a
Distribution of ? over a?(bc) a?b
a?c Hence, e.g., (ab)?(cd) (ab)?c
(ab)?d
a?c b?c (ab)?d
a?c b?c a?d
b?d Disjunctive normal form (DNF) disjunction
of one or more conjunctions of literals Conjuncti
ve normal form (CNF) conjunction of one or more
disjunctions of literals
What are these? (A?B??C)?(A?D)??B
(A??B)?C
(A?(B?C))?D A?B
10
Conjunctive and disjunctive normal forms
4.6.b
Distribution of ? over ? P?(Q?R) ?
(P?Q)?(P?R) Allows us to bring any NNF to DNF
Distribution of ? over ? P?(Q?R) ?
(P?Q)?(P?R) Allows us to bring any NNF to CNF
(A?B)?(C?D) ? (A?B)?C ? (A?B)?D
? (A?C) ? (B?C) ? (A?B)?D
? (A?C) ? (B?C) ? (A?D) ? (B?D)
(A?B)?(C?D) ? (A?B)?C ? (A?B)?D
? (A?C) ? (B?C) ? (A?B)?D
? (A?C) ? (B?C) ? (A?D) ? (B?D)