Re: Equivalence



George Dance wrote:

herbzet wrote:
BTW, I have "invented" for myself the definition "trivial implication"
for the cases of A -> B when A is a contradiction or B is a validity.
I have no idea if this is a common convention or not, but I think
it's a very useful distinction.

IMO it's an extremely useful distinction; your 'trivial' implication is
close to the concept of 'vacuous' (empty) implication, but with an
important difference.

I'd suggest using the term (or your defining your term as) "logically
vacuous" implication - as opposed to materially, but not logically,
vacuous implications like "If pigs had wings, they'd fly."

I think a better example would be one with clearly declarative
sentences as antecedent and consequent, e.g., "If pigs have wings,
then pigs fly", thereby avoiding all the ancillary problems of
dealing with sentences in the subjunctive mood.

Even better would be an example such as "If pigs have wings,
then Kerry is president", in which the lack of linguistic
linkage helps bring out the lack of logical connection between
antecedent and consequent. Less confusion is likely to ensue,
and believe me, there's a huge amount of confusion on this
subject already.

The difference being that a materially vacuous would be vacuous in at
least one interpretation or state of affairs, while a logically vacuous
interpretation would be vacuous in all interpretations or states of
affairs.

You should understand that a statement that is just false, but not
contradictory, can imply other statements in a perfectly straight-
forward manner. There is nothing vacuous about that.

E.g., "Margaret Thatcher is a man, and all men are politicians" is
false, and compounded of two falsehoods, but it implies "Margaret
Thatcher is a politician" in a perfectly substantial way.

If we say that it "vacuously" implies "Pigs have wings" because it
is false, this is really misleading. It does not imply "Pigs have
wings" at all. It "materially implies" "Pigs have wings", but
that is just a phrase: material implication is not a sort of
implication -- it is not implication at all, the term "material
implication" is misleading -- one ought to don a bio-hazard suit
before using it, it's toxic to the understanding.

--
hz
.

Relevant Pages

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