The articles and essays in this blog range from the short to the long. Many of the posts are also introductory (i.e., educational) in nature; though, even when introductory, they still include additional commentary. Older material (dating back mainly to 2005) is being added to this blog over time.
Wednesday, 14 October 2015
Four Brief Thoughts on Validity & Soundness
Modal Logic From the Beginning?
An argument is valid if the truth of the premises
guarantees the truth of the conclusion. So how does that actually work?
Regardless of the truth (or otherwise) of the premises and conclusion,
what is the relation (in a valid argument) between premises and
conclusion? Is it a necessary connection? Is it semantic? Is it
syntactic? Or is it logical – full stop?
what precisely is meant by the word “guarantee” (it doesn't seem
like a word from logic)?
with the word “impossible” (as in “it's impossible for the
premises to be true and the conclusion false”)? What does the modal
word “impossible” mean in this context? Is it natural
impossibility? Or, again, purely logical (i.e., syntactic)?
how do we recognise the soundness and validity of arguments? Again,
through semantic connections or through logical (syntactic) form alone?
More interestingly, does the logical (syntactic) run entirely free of
Modal logic is implied by propositional and predicate logic.
That is, with words such as “necessarily” and “possible”,
aren't we moving beyond propositional and predicate logic?
example, if we say,
couldn't be possible for the premises to be true and
the conclusion false.
introduces possibility. (Indeed even the world “couldn't”
the premises are true, then necessarily the conclusion must be
we can say:
is impossible for
the premises to be true and the conclusion to be false.
again, introduces possibility.
The Cogito and Implicit Premises
arguments may only have one premise. Thus we move from that single
premise to a conclusion.
is the case with Descartes' Cogito: “I think. Therefore I am.”
follows, then, that in order for the argument to valid (if not true),
the single premise may - or must - have hidden content.
That's certainly the case with the Cogito.
“I think” leads to the conclusion “Therefore I am” because
that “I think” has implicit premises (or hidden content). So
what is its hidden content?
this: “Anything that thinks, must exist.” Then it can be said
that “I think. Therefore I am” is effectively a tautology in that
the “I think” itself contains the notion of the speaker's (or
thinker's) necessary existence. In other words, existence is implied
in the premise - “I think”. Thus:
a living and existing being, think.
the implicit premise can be even more detailed or broad. Thus:
a thing thinks, then it must exist/live.
we have two conditionals (or one conditional within another
If a thing thinks,
then it must exist.
Therefore I exist.
are other examples of a one-premise argument.
world is flat. Therefore the world is not mountainous.
is a gay. Therefore, he's not heterosexual.
is because, again, there are implicit premises involved. Thus in the
is a bachelor. Therefore he's an unmarried man.
implicit premise is “No bachelor can also be married”. Similarly
with 'gay' and 'heterosexual', as well as with 'flat' and
Whereas 'bachelor' and 'unmarried man' can be deemed synonyms, that's
not the case with 'flat' and 'mountainous'. In this case we have
antonyms rather than synonyms. However, it isn't really the case the
'mountainous' is the antonym of 'flat'. A more accurate antonym
of 'flat' would be, say, 'bumpy'. Or, more logically, the purest
antonym of 'flat' is, in fact, 'not flat' (except, of course, that antonyms don't
usually simply negate the source of the antonym).
Validity Without Soundness
invalid argument can have a true conclusion.
put it simply, if the conclusion doesn't follow from the premises,
then it doesn't matter if it's true or false because, well, it
doesn't follow from the premises.
That argument itself works as a
If a conclusion doesn't follow from the premises of an argument,
then it doesn't matter – logically - if the conclusion is either
true or false.
a conclusion genuinely follows from false premises, then the
conclusion can come out false. Again, that would only be the case if
the logical moves from the premises to the conclusion are valid. In other
words, in this scenario falsity is passed on from premises to
about the case in which the premises are true yet the argument is
invalid? In that case, false premises can lead to a true conclusion
if the argument is invalid because any conclusion (as already stated)
can follow an invalid argument.
obvious point to make is that because content (or even truth) is
unimportant when it comes to recognizing a logical form, you can
create bizarre arguments which are nevertheless valid (though not
corbetts are bricks.
bricks can solve equations.
all corbetts can solve equations.
The importance of this lack of a connection between premises and
conclusion (or between validity and soundness) can be shown with the
example of a true conclusion which follows an invalid argument. Or,
more likely, one may not immediately believe that the conclusion is
true because of the invalid argument. Thus one may look for a flaw in
the argument which led to it. However, even if the argument is
invalid, the conclusion can still be true.
following argument is valid because it's impossible for the premises
to be true and the conclusion to be false:
Either Corbett eats Cornflakes or he eats Ready
Brek. ii) Corbett doesn't eat Cornflakes. iii) Therefore
Corbett eats Ready Brek.
course the obvious question is: Why is this an either/or case?
Couldn't Corbett eat neither Cornflakes nor Ready Brek? Sure.
However, that would be a factual matter and not the concern of logic.
Corbett may eat neither Cornflakes nor any other cereal. Again, that
would be irrelevant from a logical point of view. What matters here
isn't content or fact, but logical form. More precisely, it's the
relation between a disjunctive premise (as in “...or...”) , a
premise which is a existential negation (“... does not...”) and a