Substitution
Simple
Simultaneous
meta rule
Sequential
EVERY
occurrence of P
is substituted!
The rule of Leibniz
Leibniz
formula that has
as a sub formula
meta rule
single
occurrence is
replaced!
Strengthening
and
weakening
We had
Definition: Two abstractval propositions P and Q are equivalent,
notation P = Q, iff
(1) Always when P has truth value 1,
also Q has truth value 1, and
(2) Always when Q has truth value 1,
also P has truth value 1.
if we relax this,
we get
strengthening
Strengthening
Definition: The abstractval proposition P is stronger than Q,
notation P = Q, iff
(1) Always when P has truth value 1,
also Q has truth value 1, and
(2) Always when Q has truth value 1,
also P has truth value 1.
Q is weaker
than P
Strengthening
Definition: The abstract proposition P is stronger than Q,
val
notation P = Q, iff
always when P has truth value 1,
also Q has truth value 1.
always when P is true,
Q is also true
Q is weaker
than P
Properties
Lemma E1:
iff
Lemma EW1:
Lemma W2:
Lemma W3:
Lemma W4:
is a tautology.
iff
and
.
val
P |ù P
val
val
val
If P |ù Q and Q |ù R then P |ù R
iff
is a tautology.
Standard Weakenings
and-or-weakening
Extremes
Calculating with weakenings
(the use of standard weakenings)
Substitution
just holds
Sequential
Simple
val
r⇠{P s |ù
Simultaneous
r⇠{P s
EVERY
occurrence of P
is substituted!
The rule of Leibniz
Leibniz
formula that has
as a sub formula
does not hold
for weakening!
The rule of Leibniz
Leibniz
Monotonicity
does not hold
for weakening!