Ultralogic

The following post is taken from an online acquaintance.

A highly traditional view of logic is that it is the universal science par excellence. That is, logic has been understood to be the discipline that can be applied anywhere without breakdown. This formulation is a bit ambiguous and has led to several different understandings of what this notion of universality means: for example, that logic applies to all (declarative) sentences, that it applies to all objects, or that it applies to all theories.

Routley’s ultralogic program has as its goal the delineation of a single logic that can satisfy all these traditional understandings of universality (the program thus expresses a strong commitment to logical monism). It may be wondered why there should be only one such logic. That is, one may feel a pull towards logical pluralism. Routley objects to this sort of pluralism (or as he calls it, the “local logics option”) both because of standard regress worries (i.e. which logic should we use to reason about the plurality of ‘correct’ logics? And if that itself is a plurality, then which logic do we use to reason about that plurality? and so on) and because pluralism “soon runs into difficulties … at boundaries, as to how the local logics impinge upon one another and how they combine”.

Though Routley doesn’t single out one definite logic as the ultralogic, it’s quite clear from the essay that an ultralogic will be a paraconsistent contractionless relevant logic with excluded middle and outfitted with neutral quantifiers. If one is already inculcated in the relevant logic tradition then it’s probably fairly clear why he would veer towards this direction; but the choice may seem a bit obscure to those outside this tradition. So I will try my best to outline the steps that led him here.

In particular, I will do my best to clarify the following issues:

1. Why does Routley specifically pick a relevant logic, i.e. a sociative logic where Disjunctive Syllogism P & (~P v Q) →. Q fails?
2. Given Routley’s choice of a relevant logic, why does he rule out the stronger systems of Anderson & Belnap such as R, E, and T? On the other hand, why does he rule out systems weaker than DJ?
3. Why does Routley insist on ultralogic being paraconsistent, i.e. why does he insist on the failure of Material Detachment P, ~P v Q / Q, or equivalently, the Spread Rule P, ~P / Q ?
4. In contrast to the above, why does he suggest that ultralogic should not be paracomplete, i.e. that Excluded Middle should hold?
5. How are we to understand his somewhat obscure remark that a main bonus in favor of relevant logic is that it can distinguish between “a situation’s being lawlike vs a law holding in a situation”, which classical and modal-type logics are unable to do?

So the central claim Routley makes is that the only notion which can allow us to satisfy the various interpretations of logical universality is that of absolute sufficiency. So suppose we have the following argument “As A, therefore B”. We can say that A is absolutely sufficient for B when A alone allows us to conclude B, purely on the basis of logic. Thus, the One True Logic will be the strongest logic that contains only the inferences that meet this criteria of absolute sufficiency in this sense.

To perhaps make this a bit clearer, let’s consider the following blatantly invalid inference:

“As grass is green, therefore the sky is blue”

Clearly, this inference isn’t valid in any coherent sense, and we can immediately recognize that this is because the antecedent doesn’t logically give us the consequent. But compare it to the following inference:

“As grass is green and the sky is blue, therefore the sky is blue”

This inference seems unproblematically valid, in that the antecedent on its own contains all we need to pass by pure logic to the conclusion. We can thus say that the antecedent of this inference is absolutely sufficient for the conclusion, and thus that it is valid.

We can also see why the former inference was invalid; namely, a crucial piece of information needed to logically derive the conclusion (i.e. “the sky is blue”) is suppressed. Thus, we could alternatively say that an inference is genuinely (or logically) valid just when it contains no suppression of any kind. (We might denote this property as “suppression-freedom”)

Routley interprets the notions of absolute sufficiency and suppression-freedom model-theoretically; that is, A is absolutely sufficient for B just when every reasoning situation/deductive situation in which A is true is one in which B is true (we will explain shortly what a reasoning situation or a deductive situation is). So another criterion of adequacy for a truly universal logic is that the class of models which characterizes its valid inferences includes every deductive situation.

(We might note that an almost immediate corollary of these considerations is that each model for the universal logic must satisfy the Maximal Variation Principle =df. “If B is not a consequence of A, then some situation s must be such that A holds in s whilst B does not”).

Given these criteria, it is obvious that classical logic flunks the test; for the models of classical logic don’t allow for any variation whatsoever; much less maximal variation. Consider how the material implication is semantically characterized in classical logic: $P\supset Q$ is true iff either P is false or Q is true. But since the models of classical logic assign one fixed truth-value to a given sentence, this means that a true sentence is implied by any arbitrary sentence $P\supset (Q\supset P)$ and that a false sentence implies any arbitrary sentence $\neg P\supset (P\supset Q)$. Thus in some sense, classical logic doesn’t even allow us to reason about declarative sentences in any very discriminatory way.

The move to intuitionistic logic doesn’t help any, since these considerations apply to it as well. Minimal logic doesn’t have the result that a false sentence implies every sentence, but it does have the result that a false sentence implies every false sentence $\neg P\supset (P\supset \neg Q)$. The Brazilian paraconsistent systems at least don’t have this noxious result, but they are still such that a true sentence is implied by every sentence whatsoever; so they won’t work either.

A natural move here would be to extend our notion of a model, to allow for variations in truth-value assignments for the propositional parameters. If we do this in such a way that a model includes every logically possible situation (i.e. every situation in which the laws of logic hold), we arrive at the notion of strict implication. Does this do the job we want it to do?

While strict implication avoids the above mentioned paradoxes, we still have the terrible results that a necessarily true sentence is implied by every sentence $P\to (Q\vee \neg Q)$ and that an impossible sentence implies every sentence $(P\wedge \neg P)\to Q$. So while strict implication allows us to reason about contingent sentences in a discriminating fashion, it does not allow us to do so for necessary or impossible sentences.

So it seems like the issue is that the models for strict implication do not vary widely enough. In particular, the problem seems to have arisen in our restriction of models to logically possible worlds; i.e. we have left out situations in which laws of logic may fail and situations in which logical falsehoods may hold.

So this suggests that we should widen our notion of model even further, to encompass all deductive situations. As alluded to before, the notion of a deductive situation is a bit fuzzy, but we may intuitively understand it as any situation that has a ‘logical structure’. So let’s assume that $(P\wedge Q)\to Q$ is a law of logic. If some given deductive situation s is logically impossible, then this law might not hold in s (in fact, it may be that no logical laws hold in s); but nevertheless, if $P\wedge Q$ holds in S, then Q will also hold in s.

(As an aside, this helps to explain Routley’s distinction between a situations’ being lawlike vs a law holding in a situation. A situation s is lawlike just when s is closed under the laws of logic, but this does not mean that the laws of logic hold in s. What it does mean though is that the laws of logic hold over s. So we may say that the deductive situations are all the situations that the laws of logic hold over, whilst the logically possible situations are all those situations that the laws of logic hold in).

So when we expand the situations that define our models, we can avoid all the above mentioned problems (indeed, doing so allows us to avoid all the standard paradoxes of implication. This is fairly straightforward to check and is left as an exercise to the reader). This is enough to give us a sociative logic, meaning that a truly universal logic must at a minimum be sociative.

Now the reader might ask why we don’t expand our notion of situation even further, to include non-deductive situations that aren’t logically closed. Routley does accept that a fully specified logic will have to take these situations into due consideration (for instance in providing an account of hyperintensional operators such as “believes that”, “hopes that”, “knows that”, etc), but it cannot use them in determining what the laws of logic are. Since non-deductive situations can be completely anarchic, our logic couldn’t have any theorems beyond $P\to P$. While this type of logic might be able to avoid breakdown in the sense of not “trivializing” in any situation, it wouldn’t actually permit us to draw any non-trivial inferences.

So we now know that ultralogic is at least minimally sociative. For example, the principle of explosion $(P\wedge \neg P)\to Q$ must be invalid in a truly universal logic. But Routley also believes that a truly universal logic can’t be just a sociative logic, it must also be a relevant logic; i.e. the disjunctive syllogism $(P\wedge (\neg P\vee Q))\to Q$ cannot be a theorem. Why is that?

Well, Routley believes that if we zero in on the intended meanings of conjunction, disjunction, and negation, we can come up with the following minimal intuitive criteria:

$\neg A$ is true iff A is false $\neg A$ is false iff A is true

$A\wedge B$ is true iff A is true and B is true $A\wedge B$ is false iff A is false or B is false

$A\vee B$ is true iff A is true or B is true $A\vee B$ is false iff A is false and B is false

Given these constraints, the principle of addition $P\to (P\vee Q)$ must be a law of logic. But if this is the case and the disjunctive syllogism is a theorem, then we can still derive any arbitrary sentence from a contradiction (even in a logically impossible situation). So given this fact (and given the fact that non-additivity would distort the meaning of extensional inclusive disjunction), the disjunctive syllogism cannot be a genuine law of logic.

So the argument so far delineates a ballpark for where the one true logic must land: it must be at least the logic DW (on the low end) and at most Ackermann’s systems of “Strenge Implikation” (on the high end). But Routley believes that we can safely rule out Strenge Implikation (and indeed any system of relevant logic that isn’t paraconsistent) by appealing to the Liar Paradox. I won’t needlessly go over well-trodden territory here, but Routley believes that the principles of truth involved in the paradox argument are such that they cannot be rationally rejected. But since the rule of material detachment is admissible in the system of Strenge Implikation, this means that we can derive anything we want from the Liar Sentence. Hence, the One True Logic cannot just be relevant, it must also be paraconsistent.

This rules out the systems of Strenge Implikation; but we still have the Anderson-Belnap systems. Routley believes that these systems can be ruled out by a similar appeal to the Curry Paradox. Since the Anderson-Belnap systems all contain variations upon Contraction

$(P\to (P\to Q))\to (P\to Q))$, including the principle of Pseudo-Modus Ponens

$(P\wedge (P\to Q))\to Q$

this means they trivialize in the presence of the Curry Sentence.

So this rules out the Anderson-Belnap systems and leaves the upper limit currently at system T-W (i.e. the system of Ticket Entailment with Contraction removed). But T-W still has the Exported Syllogistic laws:

$(Q\to R)\to ((P\to Q)\to (P\to R))$ and

$(P\to Q)\to ((Q\to R)\to (P\to R))$

These are sometimes thought to be necessary for capturing the transitivity of inference, but this idea has been criticized by a number of people (indeed, C.I. Lewis roundly criticized these laws on grounds that had nothing to do with absolute sufficiency or relevance; this is actually the main reason why he preferred system S2 over S3). For Routley (and indeed for Lewis), what actually captures this transitivity aspect of inference is the law of Conjunctive Syllogism:

$((P\to Q)\wedge (Q\to R))\to (P\to R)$

This correctly tells us that when can make these sorts of transitivity moves if we assume that both antecedent conjuncts hold; but the Exported Syllogistic laws incorrectly suggest that the mere assumption that only one of the antecedent conjuncts holds is enough for us to draw the transitivity inference. In effect, Exported Syllogistic permits a sort of suppression: for they tell us that if the two antecedent conjuncts together permit a transitivity inference, then we can just arbitrarily select one of them and suppress the other by moving it to the consequent. But if this type of reasoning is invalid, then systems with the Exported Syllogistic laws must be ruled out.

Where does this leave us now? Well the new upper limit for the One True Logic is system DL, i.e. what we get when we remove the Exported Syllogistic laws from T-W and add the Reductio axiom, $(A\to \neg A)\to \neg A$. In fact, the above considerations rule out DW as a universal logic (this is because Conjunctive Syllogism doesn’t hold in DW, meaning that it can’t capture the transitivity aspect of logical inference). So this means that the new lower limit for a universal logic is system DJ (i.e. DW + Conjunctive Syllogism).

(This is actually already quite a narrow band of systems: in fact, this leaves us with only DJ, DK, DR, and DL as viable options).

Routley doesn’t stop there though; since in the essay he states that ultralogic cannot be any weaker than DK (i.e. DJ + LEM). Routley actually doesn’t argue for this point so much in this essay in particular, but he does give us indications for why he thinks this in other writings. In particular, if our logic doesn’t have LEM as a theorem then Routley doesn’t think the logic has a genuine negation operator.

The negation of DJ satisfies all the demorgan, contraposition, and double negation properties that we would expect from the intuitive FDE-style criteria we gave above; but it isn’t a contradictory-forming operator. That is, the DJ negation doesn’t somehow assert that no conjunction of a proposition and its negation are simultaneously true. But in order for a logics’ negation to have this property, the LNC must hold in every logically possible world (in Routleyan terminology, the LNC must be a ‘logical fact’. It can’t properly be called a ‘logical law’, since it doesn’t have anything to do with drawing inferences. Thus it may fail in logically impossible worlds, but that is to be expected). Thus, DJ is ruled out.

(We should note that Ross Brady, who otherwise follows along with Routley’s argument up until now, criticizes the notion that a negation must be a contradictory-forming operator. Indeed, he roundly criticizes the notion of ‘logical facts’ simpliciter and asserts that logic properly conceived deals only with laws. Hence why he lends his support to system DJ).

So with all these criteria in mind, we can see that Ultralogic is one of either DK, DR, or DL. To explain the differences between them, DL has the Reductio axiom:

$(P\to \neg P)\to \neg P$, (which is also a stronger version of LEM)

DR weakens the Reductio axiom to LEM, but keeps the Rule Counterexample:

$P⊢\neg (P\to \neg P)$

DK tosses this rule but keeps LEM

So we can see that given some very traditional starting assumptions about the universality of logic, we can follow a quite natural progression to a very narrow band of so-called ‘depth relevant logics’. And what’s more, at no point in the progression of the argument did we ever make any appeal to relevance at all!