‘To Pee and not to Pee?’  Could that be the Question?

(Further Reflections of The Dog) ˆ*

 

 

According to Chrysippus, who was certainly no friend of non-rational animals, the dog even shares in the celebrated dialectic.  In fact, the author says that the dog uses repeated applications of the fifth indemonstrable argument-schema when, arriving at a juncture of three paths, after sniffing at the two down which the quarry did not go, he rushes off on the third without stopping to sniff. (Sextus Empiricus, Outlines of Pyrrhonism, in Mates [1997], p 69 )

 

A real live

                                barking

                                                democratic dog

engaged in real

                                free enterprise

with something to say

                                                about ontology

something to say

                                                about reality

                                                                                and how to see it                               

(Lawrence Ferlinghetti, ‘Dog,’ in Ferlinghetti [1974], p 68.)

 

Everything is real and is not real,

Both real and not real,

Neither real nor not real.

This is Lord Buddha’s teaching. 

(N›g›rjuna, MMK XVIII:8)

 

1.             Down to the Crossroads:  Where we left The Dog

In (1990) we left The Dog at the crossroads, defending his rejection of disjunctive syllogism and the adoption of a relevance logic in the context of the epistemically hostile environment we share with him—an environment in which nature (including The Man—see Anderson and Belnap [1975]) places obstacles, both passive and active, in the path along which we pursue the truth.  These obstacles, The Dog noted, include both the prevalence of outright deception, passive sources of error and illusion in our environment and our own cognitive limitations, including preeminently our limited memories, limited deductive powers, propensity to fallacious inference and failure to track the dependencies and sources of our beliefs. 

In such circumstances, The Dog argued, idealizations of epistemic subjectivity according to which we are completely consistent, logically omniscient, etc are irrelevant to epistemology, and the adoption of a logic suited to inferential angels would be irrational.  Rational epistemic subjects, he concluded, adopt less promiscuous logics, logics less likely to lead to spurious error being introduced through inference itself. In Garfield (1990) the Dog argued along the following lines: (1) we inhabit what he called an epistemically hostile environment—an environment in which, due to our own imperfections as well as to outright deception we are likely to acquire a great many false beliefs, and in which we are likely to be without information about many matters of importance to us.  (2) Rationality is an evolutionary adaptation making us fit for such an environment.  (3) Logic is a normative theory of such rationality.  (4) Rational inference in the face of epistemic hostility is non-monotonic and relevant. 

The argument is thus an extension and development of that in Cherniak (1986), but endorsing a specific kind of logic as a canon of rational inference—relevance logic—and involves a rejection of Harman’s (1986) wholesale attack on the centrality of logic to the theory of rationality, while accepting the force of that attack with respect to classical logic.  The account is at the same time normative—and hence non-psychologistic—and sensitive to the psychology of actual rational organisms.  It is an account of the logic appropriate to a Dog, and to any who would count themselves among his best friends.  On those grounds, The Dog spurned classical logic for relevance logic, most prominently eschewing the disjunctive syllogism as providing guidance in his pursuit of his master.[1]  

Belief revision, on this view, is non-monotonic[2] and so must be its logic.  That is why The Dog is right to adopt a relevant, rather than a classical logic.  Moreover, where error is likely or inconsistent belief sets are probable, rational inference is driven by entailment, and not by truth-functional implication.  And epistemic hostility guarantees the possibility of inconsistent belief sets, even if the world is consistent.

Against this background, The Dog now ponders:  Will I pee on yonder tree or will I not?  Moreover, he reasons, if I do pee on yonder tree, I will become famous, inasmuch as I will be The Dog whose pee trumps all other pee.  And if I do not pee on that tree, I will become famous as The Dog who leaves no sign.  He continues to muse:  I have excellent reasons to believe that I will pee:  So many dogs before me have done so, and I always  pee on such trees.  On the other paw, I have excellent  reasons to believe that I will not  pee.  I’ve peed so many times today that my bladder is empty, and I know that The Man is well off down the road and needs me.  I never stop in such situations.  Furthermore, this logician among dogs says to himself, I have overwhelming reason to believe the conjunction of any two beliefs each of which I have good reason to believe. What’s a dog to believe?  Most importantly, will I become famous? These musings lead The Dog to wonder: Should I accept violations of the law of non-contradiction (LNC)?[3]

The Man, of course, infamously classical, endorses both the LNC and the law of the excluded middle (LEM).  He reasons as follows:  We could infer from p&¬p that the Dog will realize his ambitions of fame.  But then we could infer anything from p&¬p, so that conclusion would hardly be warranted on that ground.  On the other hand, we know that pv¬p, and if indeed p and ¬p each entail his fame, we can be sure that the Dog will indeed be a dog of renown.  The Dog, however, is already wary of Ex contradictione quodlibet on relevance grounds and all the more so on paraconsistent grounds.  He is also, as we shall see, deeply suspicious of LEM.  What is he to think?

 

2.                Arguments for the truth of contradictions The Dog won’t swallow

We will return to The Dog’s epistemic predicament.  But first, let us be clear about his ideology.  Some (Priest 1987,  2002) argue not only that we are generally prone to inconsistent belief sets but that the world is genuinely contradictory—that there are true contradictions.  Now, if The Dog believed this, the answer to his final question would be obvious, as would the way to resolve his perplexities.  But The Dog will have none of that.  As far as he is concerned, all of the plausible cases of true contradictions are either epistemic in character  (such as the legal dialethias or the knower paradox), spurious consequences of bad metaphysics (such as Priest’s alleged dialethias of motion and change) or merely formal (such as the semantic, set theoretic and logical dialethias).   These contradictions, he thinks, have few implications if any for the empirical world which real objects have or lack real smells.

Now the Dog may or may not be wrong about this.  We set that aside, as he is a proponent of dialethic logic despite his conservative attitude towards the truth of contradictions.  But a few things about his position should be noted:    First, while he dismisses epistemic dialethias as in a certain sense artifactual, we will see that he nonetheless takes them very seriously indeed, and in fact takes them to motivate his use of dialethic logic.  Second, while he thinks that the moment of change is not paradoxical, he agrees with Priest (1987) that this would be the best case for the truth of concrete dialethias.  He just thinks that the argument for Hegelian analysis of change in (1987) over a Russellian analysis is question-begging.  His mind remains open on the question of whether there indeed are any concrete dialethias. But he doubts it, and in any case they play no role in his own use of dialethic logic.[4]  Finally, the Dog indeed does accept the inconsistency of set theory, arithmetic and indeed of any logic adequate to the theory of truth.  And he grants that this would indeed be sufficient to motivate a dialethic logic.  But he recognizes that there are those Men who would do anything to save consistency, including develop baroque semantic hierarchies, tolerate inconsistent metalanguages, etc, and so does not wish to rest his own dialethism on this formal basis.  He would prefer to show that logic itself demands dialethism, even if the paradoxes can be resolved--even if one keeps all four feet on the ground in one’s reasoning, considering only that which one can sniff.

 

3.                Epistemic Hostility and the Purpose of Logic

The Dog believes that Logic is about inference, about what follows from what, and derivatively even about what follows from that.  And the Dog thinks that the reason we care about what follows from what is that we care about realising and preserving truth, that is, about believing what we ought to believe.  If this is the case, the Dog realizes, logic is a branch not of mathematics, but of epistemology.  And the whole point of epistemology is to enable us to generate and to maintain knowledge in an epistemically hostile environment.   The Dog adopted a relevant logic in recognition of epistemic hostility. (Garfield 1990) Recognizing that he might come to believe AvB on the basis of being told A and disjunction-introduction, and recognizing that he might later be told ¬A, given the arbitrariness of B and the difficulty of tracking the sources of his beliefs, inferring B from AvB and ¬A would be downright irrational.[5]

 Several components of epistemic hostility emerge from this brief consideration of disjunctive syllogism:  First, we cannot be sure that our sources of information are veridical, and even in a consistent world, we cannot be sure that our information about that world is consistent.  Second, our own faculties are fallible, and even given reliable sources we may introduce error into our belief sets through perfectly valid inference coupled with forgetfulness, or indeed, through simple inferential error.  It is hence rational to presume that our belief set contains both falsehood and inconsistency and that our quest for truth and for knowledge must always be grounded in a lot of falsehood and error.  Rational belief extension will therefore not, in general, be monotonic, and any logic that employs rules that are non-trivializing (that is, that do not permit arbitrary conclusions) only in consistent contexts will be useless as an epistemic tool.

The legal arena, as Priest (1987) notes, provides ample examples.  One Florida statute, notes the Florida Supreme Court, requires the election to be certified within seven days after polling.  Another permits a challenge to the count up to six days after the poll, and permits a recount if the challenge is well-founded.  Given that no recount can take less than one day, these laws are patently contradictory given a well-founded challenge.  It is both required and forbidden that the election be certified on the seventh day.  If either Bush or Gore had reasoned from this contradiction in the law to the truth of his position, on the grounds that (A&¬A)--> B, he would have properly been laughed out of court.  And this circumstance is familiar in epistemic situations beyond the legal domain.[6]

Logic, then, if it is to play its proper epistemological role, must not only get us from truth to truth but also prevent our going from truth to falsehood, and even more importantly, from unavoidable, justifiedly believed falsehood to spurious falsehood.  Classical logic, the Dog realized long ago, is far too inferentially promiscuous to fit the bill.  Heretofore his tastes have run to relevance logic.  But lately paraconsistent logics have drawn his attention.  For, he has realized, not only must he avoid irrelevant inferences, but he must worry about inconsistent domains, and simple bivalent relevance logics simply don’t have enough truth values to represent the problems posed by his environment.

 

4.             Gaps, Gluts and Gluttony

There is an interesting problem the Dog  faces, though, and recognizing it will show why LP[7] won’t meet his demands, either.  His problem is not simply that he is often told contradictory information about the world, and that he loses track of who told him what and when, but also that there are some matters regarding which he is simply ignorant; things about which the Man has told him nothing; things about which he may have been told something once long ago, but when he was gnawing a bone, and not paying attention. 

Intuitionists notoriously argue from (a part of) epistemic hostility to the necessity of admitting truth value gaps.  Under the influence of intuitionists most Men who have taken epistemic hostility of the world (or, more narrowly, human imperfection) seriously as having consequences for bivalence have taken its implication, if any, to be that we must admit gaps.  (Quine [1981], Dummett [1978])  The Dog, as we shall see, takes this point.  But he regards it as too narrow.  Gluts, as well as gaps, are to be swallowed if we are to come to grips with the full epistemic horror of our existence.  But let us first consider gaps.

LP maps each wff[8] into a non-empty subset of {1,0} and Priest offers arguments for this policy.  The Dog is skeptical of this policy.  He starts from the methodological premise that it is at least prima facie ad hoc once subsets of {1,0} have been identified as the codomain of the valuation function to exclude one of those subsets, viz, ø.[9]  So he thinks that any arguments against truth value gaps had better be pretty good.  Here is the only positive argument Priest offers:

In a nutshell, if there is no Fact that makes a true, there is a Fact that makes ¬a true, viz., the Fact that there is no Fact that makes a true…. Suppose that a is a sentence, and suppose that there is nothing in the world in virtue of which a is true; no fact, no proof, no experimental test.  Then this is the Fact in virtue of which ¬a is true.  We may not know that this fact obtains, but this is irrelevant. (Priest [1987], p 83, italics mine)

 

I italicize the last sentence, because this is where the Dog sees that the question is begged against him.  This lack of knowledge is only ‘irrelevant’ if epistemology is irrelevant to logic.  The intuition on which Priest trades in dismissing gaps is this:  Logic is about the relation between Truths, where a Truth is a sentence representing a Fact.  If A is not a Truth in this sense, ¬A must be because either A or ¬A represents a Fact.  But this mistakes the place of Logic in human cognitive activity:  Logic is the canon of inference—the science of what follows from what—and its role is the regulation of the extension of knowledge.   From the fact that we have no right to assert A or to use it as a premise in reasoning nothing whatever follows about our right to assert or to make inferential use of ¬A.  It is often the case that it is rational to refrain from assigning a determinate truth value of {1} or {0} to a wff, and hence to assign ø.  The epistemic understanding of logic—an understanding Priest curiously otherwise endorses (1987, pp 77-80)—hence  forces us to swallow gaps.

But of course that does not mean that we do not also find ourselves chewing on a few gluts.  When it is rational to accept A and also rational to accept ¬A, it is often, at least ceteris paribus, rational to accept A&¬A.   And (bracketing for now, as the Dog will, semantics, logic and set theory, as well as quantum mechanics) in any epistemically hostile environment we will occasionally be forced into such situations.  Logic hence must admit contradictions whose truth values include 1, even if A is a prima facie reason to believe ¬A and vice versa.  The rational Dog hence adopts a policy of logical gluttony:  He accepts both truth value gaps and gluts, and in doing so accepts all subsets of {1,0} as falling in the codomain of the valuation function for his logic. 

Fortunately this is not hard.  For the extensional connectives, there is a natural extension of the truth tables for LP:

 

&

{1}

{0}

{1,0}

ø

v

{1}

{0}

{1,0}

ø

`X

¬

{1}

{1}

{0}

{1,0}

ø

{1}

{1}

{1}

{1}

{1}

{1}

{0}

{0}

{0}

{0}

{0}

{0}

{0}

{1}

{0}

{1,0}

ø

{0}

{1}

{1,0}

{1,0}

{0}

{1,0}

ø

{1,0}

{1}

{1,0}

{1,0}

ø

{1,0}

{1,0}

ø

ø

{0}

ø

ø

ø

{1}

ø

ø

ø

ø

ø

 

A few remarks are in order about these extended tables (where {1} and {1,0} are designated values as in LP):  First, it is easy to verify that the standard distribution laws, DeMorgan laws, etc are preserved with only the obvious changes.  Nothing important really changes regarding disjunction and conjunction.  Second, negation deserves some comment.  Here the ‘top’ and ‘bottom’ values {1,0} and Ø each function as fixed points, and not as duals of one another as one might expect. For when we consider the negation of dialethias such as the Liar sentence or the assertion that the Russell set is not a member of itself, we find that these are dialethias as well, as Priest of course notes.  But similarly, when we negate truth-valueless sentences we encounter more truth-valuelessness.  Consider, for instance the Goldbach conjecture and assume that intuitionists are right to deny it a truth value.  If they are, then, according to our matrices, its negation, viz, that there is  an even number that is not the sum of two primes, must also lack a truth value; and this seems right, since, if it were assigned any truth value, truly unacceptable paradox would ensue:  If it were true, the conjecture would be false; if false, the conjecture would be true.

A somewhat more controversial consequence of this approach concerns the lack of logical truth and falsehood.  Because of the tolerance of gaps (valuations of ø) there are no purely extensional logical truths (or logical falsehoods) in this system.  Even sentences which cannot ever be false, such as (pv¬p) can have truth-value gaps (as when p has a gap).  On the other hand, there is some consolation in the fact that these turn out to be logical non-falsehoods, which, for those who care about such things, must be almost as good.  Moreover, there are many entailments that are logically true. This will disturb those who believe that the subject matter of logic is logical truth and logical falsehood, and that there is lots of it around—that is, that besides entailments, there are sentences that are logically true or false.  The Dog is not troubled, though.  He recognizes that logic is not about truth and falsity, but about what follows from what,  that is, about entailment.  If there are logical truths, they are entailment truths.  Beyond that, all truth is empirical, and it is not a matter of logic to decide when at least one of a pair of literals is true, let alone regarding which one.[10]

Entailment, the Dog thinks, is at the heart of Logic.[11]   He takes as a basic account of entailment that of Priest (1987).  Besides providing a compelling and intuitive account of entailment in the context of paraconsistency, it avoids Curry paradoxes.  Only one fiddle is necessary to get the LP account of entailment in line with this four-valued version:  1e(A-->B)w only if  for some W’ such that W’RW v(A)w’≠ ø.[12]  That is, formulae with no truth values anywhere don’t entail anything.   And, of course, we should introduce one more fiddle in order to get the system just right:  Since, as the Dog discovered long ago (Garfield 1990), the rational dog is the relevant dog, LP entailment is weakened by adding normal relevance restrictions.  Adding these restrictions weakens the LP entailment relation somewhat, but in the right direction, and because LP is taken as the core, contraction fails, and so this relevant entailment does not fall prey to the Curry paradoxes to which standard relevant entailment accounts succumb.

 

5.             The Rational Dog in the Real World

So the Dog does reject LNC, but also accepts gaps and so rejects LEM as well, and does so because of his cognizance of the role of logic in canine intellectual life.  His rejection can be understood in two ways:  On one level, he notes that neither of these laws expressed in the object language, is a logical truth.  No surprises there, as there are no extensional logical truths.  And since they are not logical truths, there is no reason that they should be accepted on logical grounds.  But this is not the best way to understand the Dog’s attitude.  His is a metalogical commitment:  LNC and LEM are first and foremost semantic principles—principles which together assert that every statement is either true or false, and that none is both.  This metatheory, the Dog reckons, is ill-advised for anyone reasoning in the real world, and the laws understood at this level fail to describe the logic he here proposes, just as they fail to be logical truths in the system.

As Man’s best friend, he recommends this policy to humans as well, though he takes no position on the question of the actual consistency of the world.  Whether that world is inconsistent or not, and whether or not the paradoxes of set theory, semantics and logic are resolvable through consistent means, our knowledge is always going to be inconsistent and gappy, and logic is always going to be our cognitive tool for extending our belief set through reasoning.  If that is so, our logic must be relevant, must tolerate truth-value gaps, and must tolerate contradictions.  No stronger logic is adequate to our epistemic predicament.[13]

 

What are the implications of all of this for epistemic practices in which logic figures so prominently?  Very few.  All of this can be seen as a kind of reconstruction of our actual epistemic practices.  Everything we actually do is to be left in place, replacing only an unfortunate classical idealization of epistemic practices with a relevant, paraconsistent idealization.   To the extent that there are implications at the ground level, they may be in mathematics, which might benefit from a good paraconsistent reconstruction.  That task is already underway. (Mortensen 1995, Priest 2002)

And so back to our friend the Dog and his musings.  The Dog, of course, rejects LEM and so will never reason from pv¬p to anything entailed by either unless he has independent reason for either p  or ¬p.    But here he has good reasons for both.  And so good reason to endorse p&¬p.  Now, as we have seen, for the Man, this would undermine his warrant for anything.   But not for the Dog.  From p&¬p he can infer p (or ¬p, for that matter) and from p, not just anything, but surely his undying fame.    Will he in fact pee and not pee?  He doubts it every bit as much as does the man.  But in his reasoning is he entitled to assert that he will do both?  Absolutely, for in reasoning assertion is often provisional, in full awareness of our own fallibility.  And, with the Man as his best friend, the Dog is certainly acutely aware of his own fallibility.  And that is why, ever since his discovery by Chrysippus, the Dog has been renowned among logicians.

 

References

Anderson, A R and N D Belnap, Jr (1975).  Entailment: The Calculus of Relevance and Necessity.  Princeton: Princeton University Press.

Beall, J C and G Restall (2000).  ‘Logical Pluralism,’ Australasian Journal of Philosophy 78: 4,  pp 475-493.

Cherniak, C. (1986).  Minimal Rationality.   Cambridge: MIT Press.

Dummett, M.  (1978).  ‘The Philosophical Basis of Intuitionistic Logic,’ in Dummett, Truth and Other Enigmas.  Oxford:  Duckworth Press.

Ferlinghetti, L. (1974). A Coney Island of the Mind. San Francisco: City Lights.

Garfield, J. (1990).  ‘The Dog: Relevance and Rationality,’ in J M Dunn and A Gupta, eds, Truth or Consequences: Essays in Honor of Nuel D Belnap, Jr.  Dordrecht:  Kluwer Academic Publishers.

Mates, Benson. (1997).  The Skeptic Way: Sextus Empiricus’ Outlines of Pyrrhonism.   Oxford: Oxford University Press.

Mortensen, C. (1995).  Inconsistent Mathematics.  Dordrecht: Kluwer Academic Publishers.

Priest, G. (1987).  In Contradiction.   Dordrecht: Kluwer Academic Publishers.

Priest, G. (2002).  Beyond the Limits of Thought (second edition).  Oxford:  Oxford University Press.

Quine, W.V.O. (1981).  ‘What Price Bivalence?,’ The Journal of Philosophy 79, 69-75.


ˆ This paper is dedicated to the memories of Alice Ambrose Lazerowitz and W V O Quine, each of whom would be appalled by the extent to which their ideas have led me to these.

* Thanks to JC Beall for many very helpful comments on earlier drafts of this paper.  He should not be taken to agree with much of what I say here, but much of this paper has evolved in dialogue with him.    Thanks also to Mark Colyvan for very helpful suggestions and to two anonymous referees for suggestions that have made the argument clearer.

[1] See especially pp 99-104.

[2] That is, once a belief is added to our belief set, its place is not secure.  Further revisions of the belief set may include deletions of previous beliefs as well as additions of new ones.

[3] The Dog puts it this way because he might well, following Priest 1987 (pp 84-85) recognize that the law of non-contradiction is true—that is, that its instances are theorems—yet still believe that some of its violations might be true (though also false).

[4] Vagueness, as Mark Colyvan (personal communication) points out, is another plausible source of dialethias concerning the empirical world.  The Dog, however, being rather fuzzy himself, is persuaded that a consistent fuzzy logic will probably sort vagueness out and so that our reasoning about vagueness can proceed without recourse to paraconsistency.  He’s not sure of this, of course, and his mind is open.  But for now, he would like to be conservative, and continues to argue that even on the assumption that the world is consistent, including the vague parts of the world,  it is rational to reason paraconsistently.

[5] See Garfield (1990) pp 99-104 for a more extended argument for an epistemic understanding of the role of logic, and see pp 104-105 for a detailed discussion of the status of disjunctive syllogism as a legitimate argument form.  Finally, see pp 105-106 for a consideration of the rejoinder.

[6] Now one might argue that there is in fact no dialethia here, but rather two classically consistent statements:  It is obligatory that P and it is obligatory that not-P, and block the inference from OP and OQ to O(P&Q), or argue that it does not follow from OP and PàQ that OQ.  And indeed some deontic logics weaken classical inference schemas in just such ways.  But to accept such a weakened deontic logic is already halfway to accepting the Dog’s point:  A classical logic won’t be adequate to deontic inference.  The Dog will then argue that he can propose a logic that explains the failure of classical logic in these contexts and adequately captures the relevant inferential relations.  (See Priest 1987).

[7] The paraconsistent logic of Priest (1987).

[8] Well formed formula (pronounced /woof/ [McLeod, personal communication]).

[9] JC Beall (personal communication) responds that there is nothing more ad hoc about selecting these three subsets from among the four in the codomain for the range of the evaluation function than there is about selecting only {1,0} from the entire set of natural numbers for the set of classical truth values.  There is an enormous disanalogy, though, and there is ad hocery in the one case and not in the other.  In the case of the selection of the classical truth values, any pair of values would do, as they merely code for T and F.  To be sure, 1 and 0 are arbitrary choices, but so would any pair of numbers.  T and F are the real choices, and there is nothing arbitrary about that choice, misguided though it may be in other respects.  In the case of the LP choice to exclude only ø from the power set of {1,0} there is  an element of arbitrariness, for here only one  of a set of four possibilities is excluded, and that despite at least prima facie  reasons for its inclusion.

[10] The Dog also thanks JC Beall for pressing on him the significance of this point, and notes that on his view it must come out that all mathematical truths are, au fond, entailments.

[11]  Indeed, the Dog thinks that this is why logical pluralism (Beall and Restall [2000]) is correct: Given that Logic is about entailment, and that there are different accounts of entailment appropriate to different contexts, there are multiple logics.  So the Dog emphasizes here, lest his claim be understood as stronger than he intends it to be, the account of entailment the Dog recommends is that appropriate to inference in an epistemically hostile environment, such as the empirical world as inhabited by fallible beings.  Other domains and other purposes might well demand other logics. 

[12] That is, truth is an element of the valuation set of an entailment sentence at a world only if at some accessible world the valuation set of the antecedent is not empty.

[13] Some may object to calling any inconsistent body of belief knowledge, but not the Dog.  He thinks, after all, that knowledge is Justified True Belief plus a bit of Gettier.  And there is no reason, if your logic is paraconsistent, to conclude that not all of a set of inconsistent beliefs are true, even though some may also be false.  The truth part is the aim of belief, and of knowledge.