In
the following I will argue against the paraconsistent acceptance of
inconsistencies. I also question the acceptance of logical explosion and logical triviality by classical logicians.

As C.I. Lewis once claimed (quoted by Bryson Brown), no one “'really accepts contradictions'”. From that it can be said that the prime motivation for paraconsistency - as can be gleaned from some paraconsistent logicians themselves (at
least implicitly) - is entirely epistemological and, sometimes, also based on positions and findings within science (specifically physics).

The following positions are also seemingly at one with the position of David Lewis
who argued that it's impossible for a statement and its negation to
be true at one and the same time. Having said that, all this depends on what's said about the embracing of both

*A*and ¬*A;*as well as how it's defended.
A
related objection is that negation in paraconsistent logic isn't
(really?)

*negation*; it's merely, according to B.H. Slater, a “subcontrary-forming operator”. Bryson Brown and Graham Priest also call*negation*(or*it*) “denial”. Indeed Priest explicitly states that paraconsistent negation isn't (Boolean) negation.
Thus
if the acceptance of inconsistencies is an epistemological move (as I
argue), then that move isn't “really” an acceptance of both

(All the above may or may not be ironic when one bears in mind that relevance logic is a sub-branch/type of paraconsistent logic.)

*A*and its negation at one and the same time.(All the above may or may not be ironic when one bears in mind that relevance logic is a sub-branch/type of paraconsistent logic.)

**The Acceptance of**

**A****∧**

**¬**

**A**
As already stated, Bryson
Brown says that “a defender of [C.I.] Lewis's position might argue
that we never really accept inconsistent premises” (630).
I'm prone,

*prima facie*, to agree with that. Indeed Bryson immediately follows that statement with a defence of inconsistency which doesn't seem to work. Brown says that “[a]fter all, we are finite thinkers who do not always see the consequences of everything we accept”. Perhaps C.I. Lewis's reply to that might have been that we don't “accept inconsistent premises” that we*know*- or that*think*we know - to be inconsistent. Sure, having finite minds is a limit on what we can know. Nonetheless, we still don't accept the conjunction*A***∧**¬*A*; at least not in its (empty) symbolic form. Though it needn't always be entirely a problem of symbolic autonyms. Paraconsistent logicians don't accept*1 = 0*either; and we don't accept the disjunctive statement “John is all black and John is all white”.
As
for not seeing consequences of our premises: no, we don't – not of

*every*premise (or proposition) we accept or even know. Though we do know the consequences of*some*of the premises we know! The finiteness of mind doesn't stop us accepting certain premises - or even whole arguments - either. Still, Brown may only be talking about premises which we're not sure about. In such cases then, sure, the limitations of our minds is salient – we can't know*all*the consequences of*all*the premises we accept and we can't know if they're*all*truly consistent.
Similarly,
do we accept inconsistent premises for pragmatic reasons, as Bryson
suggests? Would C.I. Lewis, again, have also said that even in this
case “we never really accept inconsistent premises”? Brown goes
on to say that “[i]nference is a highly pragmatic process involving
both logical considerations and practical constraints of salience”.
Talk of

*pragmatics*and “salience” is surely bound to make us*less*likely to accept inconsistent premises, rather than the opposite. Take salience. Not only will inconsistent premises throw up problems of salience/relevance, such salience/relevance will also (partly) determine our choice between two contradictory premises. What's more, talk of “how best to respond to our observations and to the consequences of what we have already accepted” (Brown's words) will, again, make it less likely that we would accept inconsistent premises, not more likely. That is,*p*may have observational consequences radically at odds with the observational consequences of ¬*p*. Thus why would we accept both – even provisionally? (Unless accepting both*p*and ¬*p*is a way of hedging one's bets.)**Logical Explosion**

Dale
Jacquette
puts the paraconsistent position when he says that “logical
inconsistencies need not explosively entail any and every
proposition” (7). What's more, “contradictions can be tolerated
without trivialising all inferences”. There we have the twin
problems (for paraconsistent logic) of logical explosion ( or

*ex contradictione sequitur quodlibet*) and logical triviality.
To
be honest, I never really understood the logical rule that “if
someone grants you (or anyone) [inconsistent] premises, they should
be prepared to grant you anything at all (how could they object to

*B*, having already accepted*A*and ¬*A*?)” (Brown, 628). I was never sure how this worked. I was never sure of the logic - or is it the*philosophy*? - behind it. In other words, how does*anything*follow from an inconsistent pair of premises (or propositions), let alone*everything*? An inconsistent pair surely can't have*any*consequences – at least not any obvious ones. (You can derive, I suppose, trivialities such as ¬¬A**∧****¬A and similar conclusions or premises.) In terms of truth conditions (if we take our symbols - or logical arguments – to have truth conditions or extensional interpretations), how could we derive***anything*from the premises “John is a murderer” and “John is not a murderer”? Again, there are (indirect) trivial consequences; though none of interest. (E.g. that “John is a person”?)
In
terms of the logic of explosion, let's take it step by step so it can
be shown where the problems are. One symbolisation can begin in this
way:

i)
If

*P*and its negation ¬*P*are both assumed to be true,
then

*P*is assumed to be true.
So
far, so good (at least in part). If the conjunction

*P*∧ ¬*P*is*assumed*to be true (which paraconsistent logicians accept, though others don't), then of course*P*(on its own) must also be true. Here, the inference itself is (kind of) classical; even though the conjunction of*P*and its negation isn't.
Following
on from that, we have the following:

ii)
From i) above, it follows that at least one of the claims,

*P,*and some other (arbitrary) claim*A,*are true.
This
is where the first problem (apart from the conjunction of
contradictories) is found. It can be said that

*some*proposition or other must be the consequence of*P*; though how can that consequence (*A*) be*arbitrary*? An arbitrary*A*doesn't follow from*P*. Or, more correctly,*some*arbitrary*A*may well follow; though not*any*arbitrary*A*(this is regardless of whether or not*A*, like*P*, is true).
Thus
perhaps this isn't about

*consequence*.*A*, instead, may just sit (i.e. be consistent or coherent) with

*P*without being a

*consequence*of

*P*;

*or without it*

*following*from

*P*. Thus if

*A*isn't a consequence of

*P*(or it doesn't follow from

*A*), then the only factor of similarity it must have with P is that it's true. Though, if that's the case, why put

*A*with

*P*at all? Why not say that

*P*is

*arbitrary*too? If there's no

*propositional*

*parameter*between

*P*and

*A*, and if

*A*doesn't follow as a consequence of

*P,*then why state (or mention)

*A*at all? Moreover, why does

*A*

*follow*from

*P*?

Then
comes the next bit of the argument for explosion. Thus:

iii)
If we know that either

*P*or*A*is true, and also that*P*is not true (that ¬*P*is true), we can conclude that*A -*which can be anything - is true.
This
is where the inconsistent conjunction is found again. Here there's a
(part) repeat of ii) above. That is,

*P*is*both true and also not true; and again we conclude**A*. In other words,*A*follows the conjunction*P**∧***¬***P*. This can also be seen as*A*following*P*and*A*following*¬P*(i.e. separately). Though, again, why an arbitrary*A*? Instead of*any A*following from an inconsistent conjunction, why not say that*no A**can follow from an inconsistent conjunction*? However, the gist seems to be that because we have*both**P*and ¬*P*, it's necessarily (or automatically) the case that any arbitrary*A*must follow from any inconsistent conjunction.^{1}**Logical Triviality**

Rather
than explosion, we now encounter logical triviality; to which it is very
similar. Instead of dealing with

*any*(arbitrary) proposition/theorem following from an inconsistent conjunction, we now have*every*proposition/theorem (in a theory) doing so. It goes as follows:
Thus
if a theory contains a single inconsistency, it is trivial — that
is, it has every sentence as a theorem.

There
are two problems here, both related to the points I've already made
about logical explosion. Why does an “inconsistency” have “every
sentence as a theorem”? Sure, if this is true, then one can see the

*triviality*of the*situation**.*Nonetheless, I simply can't see how the conjunction*P**∧**¬P*generates every sentence as a theorem; or, indeed, even a*single*sentence. Strictly speaking, the conjunction*P**∧**¬P*generates*nothing*!
This
isn't to say that inconsistencies aren't a problem for theories. Of
course they are. Though saying that the conjunction

*P***∧***¬P itself*generates every sentence as a theorem is another thing entirely. Or is it?...
At
the beginning of the last paragraph I confessed to rarely having seen
a logical defence of logical explosion – only bald statements of it. However, Bryson
Brown does present C.I. Lewis's proof of logical triviality (the
bedfellow of explosion). Nonetheless, before that Brown says that
“this defence [of Triv] is just a rhetorical dodge”. And that's
how I see it. It seems that the logical rule that “from any
inconsistent premise set, every sentence of the language follows” is rhetorical in nature.

In
terms of the logical notion of the unsatisfiable nature of premise
sets, things seem to be much more acceptable. Firstly, this is
Brown's formulation:

“A
set Γ is inconsistent iff its closure under deduction includes both
α and ¬α for some sentence α; it is unsatisfiable if there is no
admissible valuation that satisfies all member of Γ.”

Unlike
Triv, this seems perfectly acceptable. Of course there's “no
admissible valuation” of α & ¬α! (At least not in my book.)

**Logical Relevance**

If
relevance logic is a type of paraconsistent logic, then that may well
be

*relevant*to some - or many - of the points raised above about explosion and triviality. My point is that if relevance is a logical stance, then*nothing*explodes from accepting both*P*and**¬***P*because it's not the case than an arbitrary*A*can follow from a conjunctive inconsistency. Nor does it follow that if both*P*and**¬***P*are part of a theory, it trivially brings about every sentence as a theorem.
On
the other hand, if we accept relevance, then the very acceptance of a
conjunctive contradiction (or inconsistency) may also be problematic. If
both

*P*and**¬***P*are accepted, it's hard to see relevant derivations or consequences which follow from contradictory propositions. For example, what follows from the propositions “The earth is in the solar system” and “It is not the case that the earth is in the solar system”? Taken individually, of course, much follows from either*P*or**¬***P*; though not when taken together as jointly true.
In
symbols, the above is expressed in the following way:

If

*A*→*B*is a theorem,
then

*A*and*B*share a non-logical constant**(sometimes called a***propositional**parameter*).
On
the other hand, that (indirectly) means that

(

*A*∧ ¬*A*) →*B*
Therefore

*B*.
can't
be a argument in relevance logic.

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

**Note**

**1**To show how radically non-relevant the principle of explosion is, let's deal with everyday statements rather than - possibly misleading - symbolic letters. Thus:

i)
Jesus H. Corbett is dead.

ii)
Jesus H. Corbett is not dead.

iii)
Therefore Geezer Butler is a Brummie.

This
isn't the classical logical point that two true premises necessarily
engender a true conclusion regardless of the propositional parameters
of the premises and conclusion . In the classical case, the premises
can be genuinely true, along with the conclusion. In addition,
there's no contradiction even if the premises and conclusion aren't
semantically related.

Now
take logical explosion. In this case, the premises above engender all
statements or theorems because they're contradictory. The
propositional parameters are irrelevant, only the truth-values of the
premises and conclusion matter. Not only that: we have “proved”
that

*Geezer Butler is a**Brummie*from the premises “Jesus H. Corbett is dead” and “Jesus H. Corbett is not dead”.**References**

Brown,
Bryson. (2007) 'On
Paraconsistency'.

Jacquette,
Dale. (2002)

*A Companion to Philosophical Logic,*Introduction.
Lewis,
David (1998) [1982]. 'Logic
for Equivocators',

Slater,
B. H. (1995). 'Paraconsistent
Logics?', *Papers in Philosophical**Logic*. Cambridge: Cambridge University Press. pp. 97–110.*Journal of Philosophical Logic*. 24 (4): 451–454.

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