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worlds, whereas constructive logic would be given when cases are
taken to be possibly incomplete bodies of information or warrants or constructions,
and relevance logic would be given when cases are taken to be
possibly incomplete or inconsistent (or both) ways the world might or might
not be. Thus, there could be different collections of inferences valid in all
cases, for they could be valid in all cases but of different kinds.
This pluralist reply seems not to be a good one, for then ‘ all the cases ’
does not mean “ all the cases ” and makes logic dependent on the content
or particularities of the case under consideration, which goes against the
generality and topic - neutrality expected from logic. Moreover, the inferences
valid in all the (different kinds of) cases would be regarded as the real
valid inferences, for they are indeed valid in all cases, do not vary from case
to case, and hence hold independently of the particularities of each case.
Another pluralist option, not very well developed yet, is to bite the bullet,
to take the pre - theoretical notion of validity at face value and then try to
show that it might be inapplicable. The logical monist assumes that the
collection of valid inferences, defi ned as inferences holding in all cases, is
not empty. We have seen in the preceding paragraph that a logical monist
might insist on the existence of one true logic, claiming that the inferences
valid across all the cases of every kind are the real valid inferences. This
move rests on the third premise below. But what if it were false; that is,
what if there were no inferences valid in all cases (of all kinds)? Would there
be no logic at all? Some arguments by trivialists and possibilists seem to
imply that there are no inferences holding in all cases. However, this hardly
entails the inexistence of any logic at all. Even though there were no inferences
valid in all of them, cases might need special inferences as inferential
patterns ruling right reasoning in them. To complicate things, premise 3
requires further an “ enough ” number of valid inferences, for even though
if the collection of valid inferences were not empty – if it consisted of, say,
only one or just few inferences – it would be vacuous in practice to call
“ logic ” to such a small number of valid inferences. However, the greater
the collection of inferences, the more likely that they could not hold together
in all cases.
Logical Monism 113
It seems, then, that logic should be better characterized as an inferential
device and the universal quantifi er on the notion of validity should be
explicitly restricted:
An inference X => Y is k - valid if and only if it holds in all k - cases. As it
is, this notion of validity is compatible with both the existence of one
true logic (since it does not prevent the nonemptiness of the case of all
cases) and the idea that logics may be inferential devices for specifi c
domains.
Priest rejects the idea that, in practice, every principle of inference – or
at least a large amount of them so as to make speaking of a logic vacuous
– fails in some situation. His argument for this, premise 3, is that to the
extent that the meanings of connectives are fi xed, there are some principles
that cannot fail. The discussion of this reply would lead us quite far from
our present concern, though, for it introduces the problem of the meaning
of logical connectives.
The pluralist replies considered hitherto tried to provide a special account
of the phrase ‘ all cases (or domains) ’ or attempted to give reasons to reject
premise 3. There is an additional way of challenging logical monism, not
necessarily incompatible with the former and just recently being taken into
account in the specialized literature. It consists of challenging premises 1
and 2, that is, challenging at least the uniqueness of the pre - theoretical
notions of holding in a case and validity. For example, the following characterizations
of validity turn out to be equivalent in classical logic, which
has just two, sharply separable truth values (true and false), but in general
they are not:
V1. An inference X => Y is valid if and only if in all cases in which X is
true then Y is true too.
V2. An inference X => Y is valid if and only if in all cases in which X is
not false then Y is true.
V3. An inference X => Y is valid if and only if in all cases in which X is
true then Y is not false.
These different notions of validity may give rise to different collections
of valid inferences and hence to a plurality of logics with very different
properties. This last pluralist strategy surely has its shortcomings, but in
order to discuss it in detail, it is necessary to introduce further and more
technical remarks on truth values and the ways the collections of truth
values can be partitioned. However, I hope this brief note is helpful for
anyone looking to enter the fascinating problem of whether there is only
one correct logic.
114 Luis Estrada-González
Priest expresses his logical monism in the following terms:
Is the same logical theory to be applied in all domains, or do different
domains require different logics? [ … ] Even if modes of legitimate inference do
vary from domain to domain, there must be a common core determined by the
syntactic intersection of all these. In virtue of the tradition of logic as being
domain - neutral, this has good reason to be called the correct logic. But if this
claim is rejected, even the localist must recognise the signifi cance of this core.
Despite the fact that there are relatively independent domains about which we
reason, given any two domains, it is always possible that we may be required
to reason across domains. (Priest, 174f; emphasis in the original)
I hereby present a version of the argument using valid inferences, but it
can be easily turned into an argument about logical truths. ‘ X => Y ’ is read
“ Y is inferred from X. ” I use also the word ‘ case ’ , but you can read ‘ domain ’
if you prefer.
P1. An inference X => Y holds in a case if and only if, in that case if X is true,
then Y is true (the pre - theoretical notion of holding in a case).
P2. An inference X => Y is valid if and only if it holds in all cases (the pre -
theoretical notion of validity.)
P2 ′ . X => Y is not valid if and only if it does not hold in all cases (contraposition,
P2).
P3. There is at least one collection of (enough) inferences holding in all
cases (existence of a logic).
P4. If two collections of all inferences holding in all cases are different, then
there is at least one inference X => Y such that it belongs to a collection
but not to the other (extensionality of collections).
P5. There are at least two different collections of all inferences holding in

worlds, whereas constructive logic would be given when cases are

taken to be possibly incomplete bodies of information or warrants or constructions,

and relevance logic would be given when cases are taken to be

possibly incomplete or inconsistent (or both) ways the world might or might

not be. Thus, there could be different collections of inferences valid in all

cases, for they could be valid in all cases but of different kinds.

This pluralist reply seems not to be a good one, for then ‘ all the cases ’

does not mean “ all the cases ” and makes logic dependent on the content

or particularities of the case under consideration, which goes against the

generality and topic - neutrality expected from logic. Moreover, the inferences

valid in all the (different kinds of) cases would be regarded as the real

valid inferences, for they are indeed valid in all cases, do not vary from case

to case, and hence hold independently of the particularities of each case.

Another pluralist option, not very well developed yet, is to bite the bullet,

to take the pre - theoretical notion of validity at face value and then try to

show that it might be inapplicable. The logical monist assumes that the

collection of valid inferences, defi ned as inferences holding in all cases, is

not empty. We have seen in the preceding paragraph that a logical monist

might insist on the existence of one true logic, claiming that the inferences

valid across all the cases of every kind are the real valid inferences. This

move rests on the third premise below. But what if it were false; that is,

what if there were no inferences valid in all cases (of all kinds)? Would there

be no logic at all? Some arguments by trivialists and possibilists seem to

imply that there are no inferences holding in all cases. However, this hardly

entails the inexistence of any logic at all. Even though there were no inferences

valid in all of them, cases might need special inferences as inferential

patterns ruling right reasoning in them. To complicate things, premise 3

requires further an “ enough ” number of valid inferences, for even though

if the collection of valid inferences were not empty – if it consisted of, say,

only one or just few inferences – it would be vacuous in practice to call

“ logic ” to such a small number of valid inferences. However, the greater

the collection of inferences, the more likely that they could not hold together

in all cases.

Logical Monism 113

It seems, then, that logic should be better characterized as an inferential

device and the universal quantifi er on the notion of validity should be

explicitly restricted:

An inference X => Y is k - valid if and only if it holds in all k - cases. As it

is, this notion of validity is compatible with both the existence of one

true logic (since it does not prevent the nonemptiness of the case of all

cases) and the idea that logics may be inferential devices for specifi c

domains.

Priest rejects the idea that, in practice, every principle of inference – or

at least a large amount of them so as to make speaking of a logic vacuous

– fails in some situation. His argument for this, premise 3, is that to the

extent that the meanings of connectives are fi xed, there are some principles

that cannot fail. The discussion of this reply would lead us quite far from

our present concern, though, for it introduces the problem of the meaning

of logical connectives.

The pluralist replies considered hitherto tried to provide a special account

of the phrase ‘ all cases (or domains) ’ or attempted to give reasons to reject

premise 3. There is an additional way of challenging logical monism, not

necessarily incompatible with the former and just recently being taken into

account in the specialized literature. It consists of challenging premises 1

and 2, that is, challenging at least the uniqueness of the pre - theoretical

notions of holding in a case and validity. For example, the following characterizations

of validity turn out to be equivalent in classical logic, which

has just two, sharply separable truth values (true and false), but in general

they are not:

V1. An inference X => Y is valid if and only if in all cases in which X is

true then Y is true too.

V2. An inference X => Y is valid if and only if in all cases in which X is

not false then Y is true.

V3. An inference X => Y is valid if and only if in all cases in which X is

true then Y is not false.

These different notions of validity may give rise to different collections

of valid inferences and hence to a plurality of logics with very different

properties. This last pluralist strategy surely has its shortcomings, but in

order to discuss it in detail, it is necessary to introduce further and more

technical remarks on truth values and the ways the collections of truth

values can be partitioned. However, I hope this brief note is helpful for

anyone looking to enter the fascinating problem of whether there is only

one correct logic.

114 Luis Estrada-González

Priest expresses his logical monism in the following terms:

Is the same logical theory to be applied in all domains, or do different

domains require different logics? [ … ] Even if modes of legitimate inference do

vary from domain to domain, there must be a common core determined by the

syntactic intersection of all these. In virtue of the tradition of logic as being

domain - neutral, this has good reason to be called the correct logic. But if this

claim is rejected, even the localist must recognise the signifi cance of this core.

Despite the fact that there are relatively independent domains about which we

reason, given any two domains, it is always possible that we may be required

to reason across domains. (Priest, 174f; emphasis in the original)

I hereby present a version of the argument using valid inferences, but it

can be easily turned into an argument about logical truths. ‘ X => Y ’ is read

“ Y is inferred from X. ” I use also the word ‘ case ’ , but you can read ‘ domain ’

if you prefer.

P1. An inference X => Y holds in a case if and only if, in that case if X is true,

then Y is true (the pre - theoretical notion of holding in a case).

P2. An inference X => Y is valid if and only if it holds in all cases (the pre -

theoretical notion of validity.)

P2 ′ . X => Y is not valid if and only if it does not hold in all cases (contraposition,

P2).

P3. There is at least one collection of (enough) inferences holding in all

cases (existence of a logic).

P4. If two collections of all inferences holding in all cases are different, then

there is at least one inference X => Y such that it belongs to a collection

but not to the other (extensionality of collections).

P5. There are at least two different collections of all inferences holding in