Equivalence of
propositions
Definition: Two abstract propositions P and Q are equivalent,
val
notation P = Q, iff they induce the same truth-function
on any sequence containing their common variables
val
Property: The relation = is an equivalence on the set of all
abstract propositions.
Example
Are the following equivalent? b
b
c
0
0
0
1
1
0
1
1
b
c b
b c
b and c
c
c
Example
Are the following equivalent? b
b
c
0
0
1
0
1
1
1
0
0
1
1
0
b
c b
b c
b and c
c
c
Example
Are the following equivalent? b
b
c
0
0
1
1
0
1
1
0
1
0
0
1
1
1
0
0
b
c b
b c
b and c
c
c
Example
Are the following equivalent? b
b
c
0
0
1
1
0
0
1
1
0
0
1
0
0
1
0
1
1
0
0
0
b
c b
b c
b and c
c
c
Example
Are the following equivalent? b
b
c
0
0
1
1
0
0
0
1
1
0
0
0
1
0
0
1
0
0
1
1
0
0
0
0
b
c b
b and c
b c
c
c
Example
Are the following equivalent? b
b
c
0
0
1
1
0
0
0
1
1
0
0
0
1
0
0
1
0
0
1
1
0
0
0
0
b
c b
b and c
b c
c
c
Their truth values are the same, so they are equivalent
val
b
b
c
c
Tautologies and
contradictions
Def. An abstract proposition P is a tautology iff
its truth-function is constant 1.
all tautologies are
equivalent
Def. An abstract proposition P is a contradiction iff
its truth-function is constant 0.
but not all
contingencies!
all contradictions are
equivalent
Def. An abstract proposition P is a contingency iff
it is neither a tautology nor a contradiction.
Abstract propositions
Definition
Basis (Case 1) T and F are abstract propositions.
Basis (Case 2) Propositional variables are abstract propositions.
Step (Case 1) If P is an abstract proposition, then so is (¬P).
Step (Case 2) If P and Q are abstract propositions, then so are
(P ∧ Q), (P ∨ Q), (P Q), and (P Q).
a recursive/inductive
definition
Propositional Logic
Standard Equivalences
Commutativity and
Associativity
Commutativity
P
P
P
Q
val
Q
P
Q
val
Q
P
Q
val
Q
P
Commutativity and
Associativity
Commutativity
P
P
P
Q
val
Q
P
Q
val
Q
P
Q
P
Q
P
0
Q
1
val
val
Q
P
Q
P
P
Q
1
Q
P
0
Commutativity and
Associativity
Commutativity
P
P
P
Associativity
Q
val
Q
P
Q
val
Q
P
Q
val
Q
P
P
Q
P
P
Q
Q
R
val
P
Q
R
R
val
P
Q
R
R
val
P
Q
R
Commutativity and
Associativity
Commutativity
P
P
P
Associativity
Q
val
Q
P
Q
val
Q
P
Q
val
Q
P
P
Q
P
P
P
Q
Q
Q
R
val
P
Q
R
R
val
P
Q
R
R
val
R
val
P
Q
R
P
Q
R
Commutativity and
Associativity
Commutativity
P
P
P
Associativity
Q
val
Q
P
Q
val
Q
P
Q
val
Q
P
P
P
P
Q
R
Q
Q
P
P
Q
P
R
val
P
Q
R
R
val
P
Q
R
R
Q
Q
val
R
val
R
P
Q
R
P
Q
R
P
Q
R
Commutativity and
Associativity
Commutativity
P
P
P
Associativity
Q
val
Q
P
Q
val
Q
P
Q
val
Q
P
P
P
P
Q
1
R
0
Q
Q
P
P
0
Q
P
R
val
P
Q
R
R
val
P
Q
R
R
Q
Q
val
R
val
R
P
Q
R
P
Q
R
P
Q
R
Commutativity and
Associativity
Commutativity
P
P
P
Associativity
Q
val
Q
P
Q
val
Q
P
Q
val
Q
P
P
P
P
Q
1
R
0
Q
Q
P
P
0
Q
P
R
val
P
Q
R
R
val
P
Q
R
R
Q
Q
0
val
R
val
R
P
Q
R
P
Q
R
P
Q
1
R
Idempotence and Double
Negation
Idempotence
P
P
P
val
P
val
P
P
P
P
P
P
val
val
P
P
Idempotence and Double
Negation
Idempotence
P
P
P
val
P
val
P
P
Double negation
P
val
P
P
P
P
P
val
val
P
P
T and F
Inversion
T
val
F
F
val
T
T and F
Inversion
T
val
F
F
val
T
Negation
P
val
P
F
T and F
Negation
Inversion
T
val
F
F
val
T
P
Contradiction
P
P
val
F
val
P
F
T and F
Negation
Inversion
T
val
F
F
val
T
P
Contradiction
P
P
val
F
Excluded Middle
P
P
val
T
val
P
F
T and F
Negation
Inversion
T
val
F
F
val
T
P
Contradiction
P
P
val
F
P
P
val
T
P
F
T/F - elimination
P
P
Excluded Middle
val
P
P
T
val
F
val
T
val
F
val
T and F
Negation
Inversion
T
val
F
F
val
T
P
Contradiction
P
P
val
F
val
P
P
val
T
F
T/F - elimination
P
P
Excluded Middle
P
P
P
T
val
P
F
val
F
T
val
T
F
val
P
Distributivity, De Morgan
Distributivity
P
Q
R
val
P
Q
R
val
P
Q
P
R
P
Q
P
R
Distributivity, De Morgan
Distributivity
P
Q
R
val
P
Q
R
val
P
Q
P
R
P
Q
P
R
De Morgan
P
P
Q
val
P
Q
Q
val
P
Q
Implication and Contraposition
Implication
P
P
Q
Q
val
val
P
P
Q
Q
Implication and Contraposition
Implication
P
P
Q
Q
val
val
P
P
Q
Q
Contraposition
P
Q
val
Q
P
Implication and Contraposition
Implication
P
P
Q
Q
val
val
P
P
Q
Q
Contraposition
P
Q
val
Q
P
P
Q
val
P
common
mistake!
Q
Bi-implication and Selfequivalence
Bi-implication
P
Q
val
P
Q
Q
P
Bi-implication and Selfequivalence
Bi-implication
P
Q
val
P
Q
Self-equivalence
P
P
val
Q
P
Bi-implication and Selfequivalence
Bi-implication
P
Q
val
P
Q
Self-equivalence
P
P
val
T
Q
P
Calculating with equivalent
propositions
(the use of standard equivalences)
Recall…
Definition: Two abstract propositions P and Q are equivalent,
val
notation P = Q, iff they induce the same truth-function
on any sequence containing their common variables
val
Property: The relation = is an equivalence on the set of all
abstract propositions.
Substitution
Simple
Simultaneous
meta rule
Sequential
EVERY
occurrence of P
is substituted!
The rule of Leibnitz
Leibnitz
formula that has
as a sub formula
meta rule
single
occurrence is
replaced!