Trivial Truth
In order to balance convenience for the programmer with precision, CORE has a notion of when a sentence is trivially true.
Trivially True
Suppose P
is a sentence. To determine whether P
is trivially true, search for the first applicable rule from the beginning in the following list:
If
P
is of the formA -> B
for sentencesA
andB
,P
is trivially true ifA
trivially impliesB
.If
P
is of the formA <-> B
,P
is trivially true ifA
trivially impliesB
andB
trivially impliesA
If
P
is of the formA & B
,P
is trivially true ifA
is trivially true andB
is trivially true.If
P
is of the formA | B
,P
is trivially true ifA
is trivially true orB
is trivially true.If
P
is of the form*X(A(X))
,P
is trivially true ifA
is trivially true.If
P
is of the form~A
,P
is trivially true ifA
is trivially false.If
P
is the sentencetrue
,P
is trivially true.
One should interpret the trivial truth of a sentence with or without unbound variables to mean that no matter which objects are substituted for the unbound variables, the sentence can easily be prove true. A case for the trivial truth of a sentence beginning with an existential quantifier does not exist because not all models may have objects.
Trivially False
Suppose P
is a sentence. To determine whether P
is trivially false, search for the first applicable rule from the beginning in the following list:
If
P
is of the formA -> B
for sentencesA
andB
,P
is trivially false ifA
is trivially true andB
is trivially false.If
P
is of the formA <-> B
,P
is trivially false if eitherA -> B
orB -> A
is trivially false.If
P
is of the formA & B
,P
is trivially false if eitherA
is trivially false orB
is trivially false.If
P
is of the formA | B
,P
is trivially false ifA
is trivially false andB
is trivially false.If
P
is of the form^X(A(X))
or the form*X(A(X))
,P
is trivially false ifA
is trivially false.If
P
is of the form~A
,P
is trivially false ifA
is trivially true.If
P
is the sentencefalse
,P
is trivially false.
One should interpret the trivial falsity of a sentence with or without unbound variables to mean that no matter which objects are substituted for the unbound variables, the sentence can easily be proven false.
Trivial Implication
Suppose P
and Q
are sentences. To determine whether P
trivially implies Q
, search for the first applicable rule from the beginning of the following list. At most one rule is applied, so if a rule is applied and the rule does not give trivial implication, then P
does not trivially imply Q
.
If
P
andQ
have mismatching numbers of unbound variables or predicates, thenP
does not trivially implyQ
.If
P
is of the formA | B
,P
trivially impliesQ
ifA
trivially impliesQ
andB
trivially impliesQ
.If
Q
is of the formA & B
,P
trivially impliesQ
ifP
trivially impliesA
andP
trivially impliesB
.If
P
is trivially false, thenP
trivially impliesQ
.If
Q
is trivially true, thenP
trivially impliesQ
.If
P
is of the form~A
andQ
is of the form~B
,P
trivially impliesQ
ifB
trivially impliesA
.If
Q
is of the form~~A
,P
trivially impliesQ
ifP
trivially impliesA
.If
P
is of the formA -> B
andQ
is of the formC -> D
, thenP
trivially impliesQ
ifC
trivially impliesA
andB
trivially impliesD
.If
P
is of the form*X(A(X))
andQ
is of the form*X(B(X)
, thenP
trivially impliesQ
ifA
trivially impliesB
.If
P
is of the form^X(A(X))
andQ
is of the form^X(B(X))
, thenP
trivially impliesQ
ifA
trivially impliesB
.- If
P
is of the formA <-> B
andQ
is of the formC <-> D
, then apply any of the following rules: If
A
trivially impliesC
,C
trivially impliesA
,B
trivially impliesD
, andD
trivially impliesB
, thenP
trivially impliesQ
.If
A
trivially impliesD
,D
trivially impliesA
,B
trivially impliesC
, andC
trivially impliesB
, thenP
trivially impliesQ
.If
C
trivially impliesD
andD
trivially impliesC
, thenP
trivially impliesQ
.
- If
If
P
andQ
are identical relation sentence terms, thenP
trivially impliesQ
.If
P
andQ
are identical predicate sentence terms, thenP
trivially impliesQ
.- Otherwise, apply any of the following rules:
If
P
is of the formA & B
and eitherA
trivially impliesQ
orB
trivially impliesQ
, thenP
trivially impliesQ
.If
Q
is of the formA | B
and eitherP
trivially impliesA
orP
trivially impliesB
, thenP
trivially impliesQ
.
Regardless of the number of unbound variables of A
and B
, one should interpret “A
trivially implies B
” to mean that no matter which objects are substituted for the unbound variables, one can easily prove that A
implies B
.
Trivial Equivalence
Suppose P
and Q
are sentences. To determine whether P
is trivially equivalent to Q
, determine whether P
trivially implies Q
and Q
trivially implies P
. If so, then P
is trivially equivalent to Q
, otherwise P
is not trivially equivalent to Q
.