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William Elliot wrote:
> On Tue, 24 Oct 2006, Confutus wrote:
>
> Since your draft isn't posted here, but there for your convenience, my
> comments can not be put under your statements where they will be best
> understood. Some tables weren't clear and you didn't define theorem.
>
> For better scrutiny, post here where I can make comments instead of there
> where comments aren't possible.
I'm sorry the format I'm accustomed to using for truth tables isn't
quite clear. For reference, here they are again, in a slightly
different format.
Unary connectives
(~ P) Negation, "Not P", or "It is not the case that P". The first and
last entries in this table match classical two-valued logic. It's
reasonable to suppose that the negation of a doubtful statement is also
doubtful, although this has hidden implications.
~ T = F; ~U = U; ~F = T
([]P) Certainty. This is roughly equivalent to "Necessarily",
"Certainly", "Definitely", "Unambiguously", "Indubitably", "Provably"
although each of these terms has connotations that don't exactly agree
with the behavior dictated by the theorems.
[]T = T; []U = F; []F = F
(<>) Possibility. As used here, this differs from the meaning more
usually employed in philosophy or modal logic, to mean "Theoretically
possible" or "non-self-contradictory". It has a rather broader sense of
"At least possible", or "not impossible".
<>T = T, <>U = T, <>F = F
(?P) Equivocation . This might be read "P is equivocal". It is a claim
that P has the truth value U. P is "not certain" is an ambiguous phrase
which could be taken to mean either ~[]P or ?P. "Equivocal" is less
ambiguous and better captures the "yes and no" character associated
with the middle truth value.
?T = F, ?U = T; ?F = F
(!P) Dichotomy "P is dichotomous", meaning certainly either T or F,
with no middle ground allowed. In this logic, this is true of some
statements but not of others.
!T = T, !U = F, !F = T
It may be noted that other combinations are possible. There are no
connectives provided that "degrade" T or F to U, although some other
3-valued logics do provide them. The connective [] asserts that
statement has the truth value T. ? asserts that a statement has the
truth value U. There is no single connective that asserts that a
statement has the truth value F, but the combination ~<> accomplishes
the same purpose.
In a complementary fashion, <>P does not state that P has a certain
truth value. it says that its value may be T or U, but it isn't F.
Likewise, !P declares that P's value may be T or F, but it isn't U.
There is no single connective that states that a statement is U or F,
but not T, but the combination ~[] accomplishes the same purpose.
Binary connectives
( P v Q). The logical (inclusive) "or". The value of P v Q is the most
true of the values of P and Q.
T v T = T, T v U = T, T v F = F ,
U v T = T, U v U = U, U v F = U,
F v T = T, F v U = U, F v F = F.
( P & Q) The logical "and". The value of P & Q is the least true of the
values of P and Q
T & T = T, T & U = U, T & F = F ,
U & T = U, U & U = U, U & F = F,
F & T = F, F & U = F, F & F = F.
( P -> Q ) The conditional. This is probably the best equivalent to the
English "if...then..." This is the conditional proposed by Lukasiewicz,
and is not easily defined in terms of any others.
T -> T = T, T -> U = U, T -> F = F ,
U -> T = T, U -> U = T, U -> F = U,
F -> T = T, F -> U = T, F -> F = T.
( P => Q) The strict conditional, "Certainly, if P then Q". "If P then
certainly Q" might be interpreted to mean this, [](P ->Q). or might
also be used for (P -> []Q). This is generally the most useful
conditional in 3VL.
T => T = T, T => U = F, T => F = F ,
U => T = T, U => U = T, U => F = F,
F => T = T, F => U = T, F => F = T.
Two other connectives of interest to logicians are not much used here.
They are not adequate for L3 because they make no provision for the
uncertain or doubtful. The role of the material conditional of
classical two valued logic, (~P v Q) or ~(P & ~Q) is best played by the
strict Lukasiewicz conditional.
The Lewis strict conditional (fishhook) ~<>(P & ~Q), commonly used in
modal logic, is also generally not used, and the strict Lukasiewicz
conditional is also used instead.
(P <-> Q) Biconditional (if and only if). This provides a logical
equivalence.
T <-> T = T, T <-> U = U, T <-> F = F ,
U <-> T = U, U <-> U = T, U <-> F = U,
F <-> T = F, F <-> U = U, F <-> T = T.
( P == Q ) Equivalence. This is a truth-functional equivalence, stating
that the two expressions have the same truth value. It is also the
strict biconditional.
T == T = T, T == U = F, T == F = F ,
U == T = F, U == U = T, U == F = F,
F == T = F, F == U = F, F == F = T.
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