Edwin
D. Mares displays the problem (if it is a problem) with a purely
formal logic by offering us the following example of a valid
argument:

__The sky is blue.__

∴ there
is no integer

*n*greater than or equal to 3 such that for any non-zero integers*x*,*y*, z,*x*^{n}=*y*^{n}+*z*^{n}.
Mares
says that the above “is valid, in fact sound, on the classical
logician's definition” (609). It's the argument that is valid;
whereas the premise and conclusion are sound (i.e., true). In more
detail, the “premise cannot be true in any possible circumstance in
which the conclusion is false”.

Clearly
the content of the premise isn't connected, semantically, to the
semantic content of the conclusion. However, the argument is valid
and sound.

So
what's the point of the above?

Perhaps
no logician would state it for real. He would only do so, as Mares
himself does, to prove a point about logical validity. But can't we
now ask why it's 'valid' even when the premise and conclusion are
true?

Perhaps
showing the bare bones of this argument will help. Thus:

__P__
∴

*Q*
Do
that look any better? I suppose so. Even though we aren't given the
semantic content, both

*P*and*Q*must be seen to have a truth-value. (In this case, both*P*and*Q*are true.) It is saying:*P is true. Therefore Q is true*. It isn't saying,*Q is a consequence of P*; or that*P entails Q*. Basically, we're being told, by the logic, that two true statements can exist together if they don't contradict each another.********************************************************

There
can be cases in which the premises of an argument are all true, and
the conclusion is also true, yet, as Stephen Read puts it, “there
is an obvious sense in which the truth of the premises does not
guarantee that of the conclusion” (237). Ordinarily, of course, the
truth of the premises is meant to guarantee the truth of the
conclusion. So let's look at Read's example, thus:

i)
All cats are animals

ii)

__Some animals have tails__
iii)
So
some cats have tails.

Clearly,
premises i) and ii) are true. Indeed iii) is also true. (Not all cats
have tails. And, indeed, according to

*some*logicians, saying 'some' also implies saying 'all'.)
So
why is the argument invalid?

It's
invalid not for the truth-values of the premises and conclusion; but
for another reason.

The
reason is that the sets in the argument are, as it were, mixed up.
Thus we have [animals], [cats] and, indeed, [animals which have
tails]. In other words, it doesn't logically follow from “some
animals have tails” that “some cats have tails”. If some
animals have tails it might have been the case that cats were animals
which didn't have tails. Thus iii) doesn't necessarily follow from
ii). (And iii) doesn't follow from i) either.) ii) can be taken as an
existential quantification over [animals]. iii), on the other hand,
is an existential quantification over [cats]. Thus:

ii)
((Ǝ

*x*) (A*x*) iii) (Ǝ*x*) (C*x*))
Clearly
A

*x*and C*x*are quantifications over different classes. It doesn't follow, then, that what's true about [animals] generally is true also of [cats]; even though cats are members of the set [animals]. Thus iii) doesn't follow from ii).
To
repeat: even though both premises and the conclusion are all true,
the above still isn't a valid argument.

Read
himself helps to show this by displaying an argument-form with
mutually-exclusive sets. Namely, [cats] and [dogs]. Thus:

i)
All cats are animals

ii)

__Some animals are dogs__
iii)
So some cats are dogs.

This
time, however, the conclusion is false; whereas i) and ii) are true.
It's the case that the subset [dogs] belongs to the set [animals].
Some animals are indeed dogs. However, because some animals are dogs,
it doesn't follow that “some cats are dogs”. In other words,
because dogs are members of the set [animals], that doesn't mean that
they're also members of the subclass [cats] simply because cats
themselves are also members of the set [animals]. Cats and dogs share

*animalhood*; though they are different subsets of the set [animal]. In other words, what's true of dogs isn't automatically true of cats. (Wouldn't iii) above work better if it were 'some dogs are cats', not 'some cats are dogs'?)
The
importance of sets, and their relation to subsets, may be expressed
in terms of bracketed predicates. Thus:

[animals
[cats [cats with tails]]]

not-[animals [cats [dogs]]]

not-[animals [cats [dogs]]]

**Material and Formal Validity**

Stephen
Read makes a distinction between formal validity and material
validity. He does so using this example:

i)
Iain is a bachelor

ii)
So Iain in unmarried.

(One
doesn't ordinarily find an argument with only a single premise.)

The
above is

*materially*valid because there's enough semantic material, as it were, in i) to make the conclusion acceptable. After all, if*x*is a bachelor, he must also be unmarried. Despite that, it's still*formally*invalid because there isn't enough content in the premise to bring about the conclusion. That is, one can only move from i) to ii) if one already knows that*all bachelors are unmarried*. We either recognise the shared semantic content or we know that the word 'unmarried man' is a synonym of 'bachelor'. Thus we have to add semantic content to i) in order to get ii). And it's because of this that the overall argument is said to be formally invalid. Nonetheless, because of what I've already said, it is indeed*materially*valid.
The
material validity of the above can also be shown by its inversion,
thus:

i)
Iain is unmarried

ii)
So Iain is a bachelor.

Read
makes a distinction by saying that its “validity depends not on any
form it exhibits, but on the content of certain expressions in it”
(239) Thus, in terms of logical form, it is

*invalid*. In terms of content (or the expressions used), it is*valid*. This means, obviously, that the following wouldn't work as either a materially or a formally valid argument. Thus:
i)
Iain is a bachelor.

ii)
So Iain is a footballer.

There's
no semantic content in the word 'bachelor' that can be directly tied
to the content of the word 'footballer'. Iain may well be a
footballer; though the necessary consequence of him be a footballer
doesn't follow from his being a bachelor. As it is, the conclusion is
false even though the premise is true.

Another
way of explaining the material, not formal, validity of the argument
above is in terms of what logicians call a “suppressed premise”.
This is more explicit than talk of

*synonyms*or*shared contents*. What the suppressed premise does, in this case, is show the semantic connections between i) and ii). The actual suppressed premise would be the following:
All
bachelors are unmarried.

Thus we would actually have the following argument:

i)
Iain is a bachelor.

ii)
All bachelors are unmarried.

iii)
Therefore Iain is unmarried.

It
may now be seen more clearly that

i)
Iain is unmarried.

ii)
So Iain is a bachelor.

doesn't
work formally; though it does work materially.

What
about this? -

i)
All bachelors are unmarried.

ii)
So Iain is unmarried.

To
state the obvious, this is clearly a bad argument. (It's called an

*enthymeme.*) Indeed it can't really be said to be an argument at all. Nonetheless, this too can be seen to have a suppressed premise. Thus:
i)
All bachelors are unmarried.

[Iain
is a bachelor.]

ii)
So Iain is unmarried.

Now
let's take the classic case of

*modus ponens*:*A*, if

*A*then

*B*/ so

*B*

That
means:

*A*, if

*A*is the case (or true), then

*B*is the case (or true).

*A*is the case, so

*B*must also be the case.

The
obvious question here is:

*What connects A to B (or B to A)?*In terms of this debate, is the connection material or formal? Clearly if the content of both*A*and*B*isn't given, then it's impossible to answer this question.
We
can treat the above as having the aforesaid suppressed premise. Thus:

[Britain's
leading politician is the Prime Minister.]

i)
Theresa May is Britain's leading politician.

ii)
So she is Prime Minister.

In
this instance, the premise and conclusion are both true. Yet they're
only contingently, not necessarily, connected.

***************************************************

*)
Stephen Read makes the formalist position on logic very clear when he
states the following:

“

**Logic is now seen – now redefined – as the study of***formal*consequence, those validities resulting not from the matter and content of the constituent expressions, but from the formal structure.” (240)
We
can now ask:

*What is the point of a logic without material or semantic content?*Would all premise, predicate, etc. symbols - not the purely logical**symbols****- simply be autonyms or self-referential in nature? (Thus all the***p*'s,*q*'s,*x*'s,*F*'s,*G*'s etc. would be self-referential/autonyms.) And what would be left of logic if this were the case? Clearly we could no longer really say that it's about argumentation – or could we? That is, we can still learn about argumentation from schemas/argument-forms which are purely formal in nature. The dots don't always - or necessarily - need to be filled in.**References**

Mares,
Edwin D. (2002) 'Relevance
Logic'.

Read,
Stephen. (1994) 'Formal
and Material Consequence'.