These are good days in
Arché: The period from late October to the end of December usually has both
Graham Priest and
Stewart Shapiro visiting to set things straight. The two
Arché Professorial Fellows contributes to most of our seminars, in addition to presenting their own ongoing research. This post is a brief report (with some remarks of my own) of one of these sessions, that is, Graham Priest's recent talk on the relationship between the venerable realism/anti-realism debate and
paraconsistent logics.
Myself I've had some dealings with the
Dummett-Wright-style realism/
antirealism debate before, usually with emphasis on the so-called
revisionary arguments. Crudely put, these are arguments to the effect that there are certain relationships between which logic is the correct one for a domain of discourse (typically classical or
intuitionistic logic) and which metaphysics we should endorse for that domain. The point here, as in Graham Priest's talk, is not to detail the many difficulties with such arguments, but rather to address what the role of
paraconsistent logics could be in this debate.
Graham Priest's answer:
Paraconsistency is neutral in the debate---in its different manifestations it can lend itself to the realist or to the
antirealist. Without rehearsing the old debate, let me just remind the reader that classical logic, with bivalent semantics and law of excluded middle (LEM) as a theorem, is typically thought to fit the realist bill, i.e., the idea, roughly, that there are potentially
verificiation-transcendent truths. On the other hand,
intuitionistic logic, refraining from LEM and its semantic counterpart, suits
antirealist purposes. Now of course, neither of these
logics are
paraconsistent; they both have
explosion---anything follows from a contradiction. The question raised by Priest, then, is whether going
paraconsistent has any interesting consequences for the metaphysical debate.
Again, Priest's answer is that it depends on which
paraconsistent logic you look at. Here we'll only take a look at a
paraconsistent logic that Priest suggests as an addition to the
antirealist's logical instruments.
As a preliminary to the
paraconsistent logics, Priest first introduces the (non-
paraconsistent) logic
N_3, from
Nelson (1949) (
JStor). An interpretation is a structure
, where W is the non-empty set of worlds, R is the reflexive and transitive accessibility relation between them, and ρ is a relation between propostional letters and the truth-values {1,0} for different worlds. ρ is constrained by the so-called exclusion principle:
For any propositional letter p and world w: it is not the case that pρ1 at w and pρ0 at w.
In other words, nothing is both true and false (although something being neither is not ruled out). Additionally, there are two heredity principles:
For every propositional letter p: if pρ1 at w_0 and w_0Rw_1, then pρ1 at w_1; and
for every propositional letter p: if pρ0 at w_0 and w_0Rw_1, then pρ0 at w_1
Truth-conditions for the connectives are defined unsurprisingly as follows:
(A∨B)ρ1 at w
iff Aρ1 at w or Bρ1 at w
(A∨B)ρ0 at w
iff Aρ0 at w and Bρ0 at w
(A∧B)ρ1 at w
iff Aρ1 at w and Bρ1 at w
(A∧B)ρ0 at w
iff Aρ0 at w or Bρ0 at w
(A→B)ρ1 at w
iff for all w' s.t.
wRw', it is not the case that Aρ1 at w' or Bρ1 at w'
(A→B)ρ0 at w
iff Aρ1 at w and Bρ0 at w
¬Aρ1 at w
iff Aρ0 at w
¬Aρ0 at w
iff Aρ1 at w
The truth value 1 is the only designated value, and validity is defined in terms of preservation of designated values at all worlds of all interpretations.
Priest's first observation is that this logic is very similar to
intuitionistic logic, in fact the positive fragments are the same. The action is with negation. In
N_3, negation has regained the nice flip-flop property we know from classical logic, but with the price of deviating from
intuitionistic logic (needless to say this does not bring back classical logic as we do not have the exhaustion principle, i.e., there are truth gaps). Truth and falsity are, in other words, on equal standing in
N_3; whereas truth, through its relation with verification, had the upper hand in
intuitionistic logic.
Priest continues, what do we make of falsity, Aρ0, in N_3? The suggestion is that falsity is to falsification what truth is to verification. Now, what about applications, are there room in the
antirealist picture for showing that something is false without going via the route of showing that something is true, i.e., by verifying something (that a proof of some A can be transformed into a proof of ⊥)? Priest provides perception as an example: One can
see (non-inferentially) that something is
not the case, e.g., that Pierre is not in the room (to use the example Priest borrows from Sartre). If Priest is correct, this is a sort of direct falsification which is independent of verification.
Some worries about such an application: Firstly, buying into the philosophy of perception story told by such examples comes with a cost. In particular, the ensuing ontology, it might be argued, contains negative facts or negative objects, entities that many philosophers (myself included) find dubious. Furthermore, there is also an alternative perceptual story where the type of Pierre-case are cases where perception has inferential content of some sort. Secondly, it is not clear to me what to make of the application to mathematics after the shift from
intuitionistic logic to
N_3. At least it is
unclear to me what sort of mathematics can be done in this logical environment (if you know anything about this, please comment).
However, the logical upshot is quite straightforward. As opposed to
intuitionistic logic,
N_3 gives us double negation elimination (no surprise that we get both directions considering the definition of
constructible negation);
contraposition fails in the same direction as
intuitionistic logic (i.e., it is not the case that A → B gives ¬B → ¬A); but you still won't get LEM.
But recall that
N_3 is not
paraconsistent. Priest brings us to the non-explosive domains by simply omitting Exclusion from
N_3, yielding the logic
N_4. In
N_4 the tie between truth and falsity, or rather verification and falsification, is completely severed. The ρ-relation now allows for pρ1
and pρ0; contradiction will no longer trivialize a theory. As before,
N_4 does not have LEM, so the antirealist, according to Priest, should be fine. All that is required are applications to motivate the antirealist's migration over to the paraconsistent terrain.
This, Priest continues, is readily provided by the sciences. It is not uncommon for sciences to have verification and falsification living side by side. Think about experiments testing, say, the pH value of a liquid. The type of application Priest has in mind is one in which two indicators, e.g., litmus paper and bromothymol blue, give oppsite results---one classifying the liquid as an acid, the other not. By dropping Exclusion you are given a flexibility in the notions of verification and falsification (truth and falsity) that will let you treat such cases. Of course, the cost is an even thinner connection between truth-values and Truth, but then again, the antirealist was never too keen on capital Ts anyway.
Priest sums the situation up by saying that paraconsistency can live happily with antirealism. Although paraconsistency does not belong exclusively on either side of the metaphysical divide, it might have virtues for both.
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