ON PRESERVING
GILLMAN PAYETTE, BLAINE D’ENTREMONT, PETER K. SCHOTCH, AND R. E.
JENNINGS
Abstract.
This paper brings together a number of threads in paraconsistent logic and the so-called preservationist approach to inference, including paraconsistent inference. At the heart of these matters is our notion of a level which dates back to the earliest work of Jennings and Schotch.
1.
Introduction
The (classical) semantic paradigm for correct inference is often given the name
‘truth-preservation.’ This is typically spelled out to the awe-struck students by some such unpacking as:
Some inference from a set of premises, say Γ, to a conclusion, say
α , is correct, say valid , if and only if whenever all of the members of Γ are true, then so is α .
This way of unpacking the slogan is surely tried, but there is reason to suspect that it may not be true. It is rather heterogenous for a fact and thus could be no better than a quick and dirty gloss. The chief virtue of the formulation is that of seeming correct to the naive and untutored.
But what of the sophisticates? They might well ask for the precise sense in which truth is supposed to be preserved in this way of unpacking. On the right hand side of the ‘whenever’ we are talking about the truth of a single formula while on the left hand side we are talking about the truth of a bunch of single formulas. Is it the truth of the whole gang which is ‘preserved?’
Of course it is open to the dyed-in-the-wool classicalist to reply scornfully that we need only replace the set on the left with the conjunction of its members. In this way is truth preserved from single formula to single formula as homogenously as anyone could wish.
It is open, but not particularly inviting. We are inclined to think that this business of coding up the valid inferences in terms of their ‘corresponding conditionals’ 1 as an accident of the classical way of thinking about inference and that it is no part of the definition of a correct account of inference. We also notice that on the proposal, we are restricted to finite sets.
Setting aside this unpalatable proposal then, we ask how is this notion of truthpreservation supposed to work? Since there is no gang on the right we seem to be talking about a different kind of truth, individual truth maybe, from the kind we are talking about on the left—mass truth perhaps. Looked at in that somewhat
The authors wish to acknowledge the support of SSHRCC under research grant 410-2005-1088.
1

2 PAYETTE, D’ENTREMONT, SCHOTCH, AND JENNINGS jaundiced way, there isn’t any preserving going on at all, but rather a sort of transmuting.
The classical paradigm really ought to be given by the slogan ‘truth transmutation.’ In passing from the gaggle of premises to the conclusion, gaggle-truth is transmuted into single formula truth. It may be more correct to say that, but it makes the who paradigm somewhat less forceful or even less appealing.
What we need in order to re4scue the very idea of preservationism is to talk entirely about sets. So we shall have to replace the arbitrary conclusion α with the entire set of conclusions which might correctly be drawn from Γ. We even have a sexy name for that set—the (classical) theory generated by Γ or the deductive closure of Γ. In formal terms this is
C
`
(Γ) = { α | Γ ` α }
When ` is the classical relation, the deductive closure is likewise said to be classical.
Now that we have sets, can we say what it is that gets preserved—can we characterize classical inference, for instance, as that relation between sets of formulas and their closures such that the property Φ is preserved?
We can see that gaggle-truth would seem to work here in the sense that whenever
Γ is gaggle-true so must be C
`
(Γ). We are unable to rid ourselves, however, of the notion that gaggle-truth is somewhat lacking from an intuitive perspective.
It may gladden our hearts to hear then, that there is another property, perhaps a more natural one, which will do what we want. That alternative property is consistency .
2.
Making A Few Things Precise
We were interested in how an inference relation might be characterized in terms of preserving some property of sets. We have singled out consistency as a natural property of sets and having done that we can see that preservation of consistency comes very naturally indeed. The time has come to say a little more exactly what we mean by ‘characterized.’ In order to do this we shall be making reference to the following three structural rules of inference.
[R] α ∈ Γ = ⇒ Γ ` α
[Cut] Γ , α ` β & Γ ` α = ⇒ Γ ` β
[Mon] Γ ` α = ⇒ Γ ∪ ∆ ` α
Unless there is a specific disavowal every inference relation we consider will be assumed to admit these three rules.
Let us say that an inference relation, say ’ ` ’, preserves consistency if and only if:
If Γ is consistent (in the sense that not every formula is proved according to ` , by Γ, then so is C
`
(Γ).
We say that ` preserves consistency in the strong sense when the condition given above as necessary, is also sufficient.
Now we are ready to state an important proposition:
Proposition 1.
Let X be a theory of inference which admits the structural rules
[Cut] and [Mon] , then ` preserves X -consistency in the strong sense if and only if
` agrees with X -inference on the class of theorems).

ON PRESERVING 3
Proof.
The proof is obvious. Since ` preserves X -consistency in the strong sense, both X and ` agree not only on which sets are consistent but also which sets are not. This means in particular that since the unit sets of all X-theorems are Xinconsistent, so must they be ` inconsistent. But then X and ` have the same set of theorems. Since any inference relation which has [Cut] must preserve consistency, and for any set which is not consistent, neither is its deductive closure, by [Mon].
¤
We can do better than this, if we are allowed sets of formulas on both sides of the inference relation in the manner of Gentzen’s sequent calculi. In such a setting we can relate consistency and provability in general rather than simply the class of theorems. This is because Γ ` α if and only if Γ , ¬ α ` ∅ , which is to say exactly that the latter set is inconsistent in terms of ` .
2
These results go some way to bolstering the claim that, far a great many logics, preserving consistency is characteristic.
3.
On The Other Hand
As a candidate for preservation, consistency, as we have seen, has much to recommend it. Preservation of consistency, in the strong sense, is linked to any inference relation satisfying just a few structural rules—rules which are widely regarded as expressing non-controversial features of any ’genuine’ notion of deductive inference.
There is a problem which we now wish to expose: The classical account of consistency is not adequate in many situations in which we certainly require a notion like it. Given the results above, this is tantamount to saying that there are situations in which classical logic is not adequate. While such an assertion might have enjoyed a certain amount of shock-value in the last century, things have changed since those heady days. Now most people agree that some changes are often necessary in the classical account of inference, though there is nothing approaching wide-spread agreement on the precise nature of those changes.
For beginners, it is important to stress that the ‘problem of inconsistent premises’ is not one of those which have been devised by logicians largely for their own gratification. Although it has intrigued and titillated logic for several decades at least, it grows out of the deeper bafflements of philosophical and ordinary life.
We need only reflect upon the prevalence of reductio in philosophical debate for reminder that inconsistency is abroad. Main paper, refutation, then discussion has become colloquial protocol. But it is not just in the predations of philosophical meetings that inconsistencies are exposed. They are a pervasive, discomfiting and persistent feature of any difficult philosophical enterprize. Witness David Hume:
I find myself in such a labyrinth, that, I must confess, I neither know how to correct my former opinions, nor how to render them consistent.
It is no good telling Hume that if his inconsistent opinions were, all of them, true then every sentence would be true. Even if he could be got to accept this startling claim, it would have brought him no comfort. The most which might be wrung out of it is that not all of his opinions could be true. This, so far from being news to
Hume, was precisely what occasioned much of the anguish which he evidently felt.
What we need in such circumstances is a way to cope.
Nor is it merely in the realm of metaphysics that inconsistent opinions charm us.
In the physical sciences as well we seem to have inconsistent consequences thrust

4 PAYETTE, D’ENTREMONT, SCHOTCH, AND JENNINGS upon us by equally well confirmed hypotheses. And even if, as we reassure ourselves, the inconsistences are not ineluctable, we must live with them whilst awaiting the crucial experiment or the new paradigm or the new metaphor or merely promotion.
Eventualities may render the inconsistencies resolvable or illusory or unimportant.
In the meantime however, we must draw inferences from them which do not depend upon their inconsistency, but which are informed by it.
A more humanistic logic is required. Such a logic would accord with what we must frequently do, namely the best we can with data which, although inconsistent are nevertheless the best data we are able to command. We would like to be able to reflect but also judge the reasonings of ordinary moral and doxastic agents. We wish to offer them standards of correctness which survive without triviality, conflicts of obligation and belief. This humane approach need not pander. It would accept classical consistency as an ideal without the pretense that it is pervasive or even especially common in actual belief sets or set of perceived obligations. It is not self-evident that the problems of conflicting beliefs and conflicting obligations are sufficiently similar that they will both be solved by a single innovation. And in fact our general approach will come in sufficiently many flavors that we may hope to convince the reader of handling at least the majority of the problems. For now we shall consider just the case of belief.
3.1.
The Inconsistency of Belief.
We say nothing controversial when we claim that often, perhaps usually, the set of sentences to which someone will actually assent would, if scrutinized, be discovered to be classically inconsistent. The question is: What are we to make of this observation? There appear to be two distinct reactions. The first is to say that some such apparently inconsistent belief sets are real and constitute a genuine difficulty for the received theory of inference. The second takes such inconsistencies to be mere appearance—a fanciful mask over the underlying consistent reality.
Our choice between these is important, for our ordinary understanding of rationality involves drawing inferences from our beliefs. We are not positively required to draw these inferences but we are required to accept the conclusion once the validity of such an inference has been made known to us. But how is non-trivial reasoning possible in the face of inconsistency? We are like jurors made trusting by cruel penalties. To the extent that we accept the classical account of reasoning, to that extent we resist the view that the set of a person’s beliefs can really be inconsistent in a full-blooded sense. We want to either to acquit or somehow diminish responsibility.
We have recourse to such strategies as distinguishing between active beliefs, which must indeed be consistent and mothballed beliefs which lie strewn about in their cells unregarded and out of mind. For inconsistencies among this disused bricabrac or between them and the brighter ornaments of present thought, we will not keep the agent long in dock.
It is tempting to think of this distinction as being congruent with the distinction sometimes made between explicit and implicit contradiction. We are happy to offer forgiveness if not pardon, for the latter but visit upon the former harshness without mitigation. We must step carefully here for the implicit/explicit distinction can be variously applied to confusing effect. We may wish to say that there is an explicit contradiction in a person’s beliefs when she actively adheres to two or more mutually contradictory opinions. Implicit contradictions, on this score, would be

ON PRESERVING 5 those between two inactive beliefs or between one which is active and another which isn’t. This is not Hume’s predicament.
His problem lay in his active acceptance of contradictory opinions. On the latest account of the distinction, Hume’s contradictions were explicit. Yet he did not recognize whatever harsh penalties later logic would have meted out to him. It did not trivialize his inquires. It simply provided him with ‘a sufficient reason to entertain a diffidence and modesty in all my decisions.’
There is an alternative account of the distinction according to which Hume’s contradictions would count as implicit rather than explicit. On this account an explicit contradiction is a single sentence which is self-contradictory; while for one’s set of beliefs to contain an implicit contradiction one must maintain mutually inconsistent opinions, whether actively or inactively. The two ways of drawing the distinction are not always kept separate and the fear of explicit contradictions of the second sort may make us wary of the first sort also.
The prohibition against explicit contradictions of the second sort may be expressed as the requirement that every member of a set of beliefs must be logically capable of being true. This stricture seems plausible enough but it does not preclude sets of beliefs which contain implicit contradictions of the second sort. Moreover, it seems likely that implicitly inconsistent belief sets of the second sort are quite common.
3
On the other hand the notion of someone believing an explicit contradiction of the second kind seems devoid of content. One might argue that it is part of the root meaning of ‘belief’ that the sentence expressing it must be at least capable of being true. If the classical penalties of triviality are to imposed anywhere, let them fall upon a transgression which no one is capable of committing. We feel no qualms at the prospect of requiring anybody who believed an explicit contradiction to believe everything.
The stronger restriction upon belief sets corresponds to the requirement that such sets be satisfiable, which is to say that all the members be capable of simultaneous truth. It is this condition which flies so directly in the face of our ordinary experience of belief. We have now arrived at a crux.
If we let ordinary experience (not to say common sense) be our guide allowing that our belief sets to contain implicit contradiction (of the second kind) but not explicit contradictions, we must of necessity abandon classical logic. For by means of the classical picture of inference we may always derive an explicit contradiction from a set of beliefs which is implicitly inconsistent. Thus classical modes of reasoning trample what is manifestly a useful distinction.
How far ought this felt need for a non-classical logic to be indulged? Appeals to experience can easily lead us astray. After all, almost anybody who proposes some species of non-classical logic is tempted to argue that since the world is non-classical so should our logic be. To travel to far along this road is to invite some form of the genetic fallacy.
If our arithmetic or our syllogisms are not in accord with the classical canons, the fault might well lie in our inability to calculate or to construct Venn diagrams.
The observation that the reasonings of actual people are sometimes non-classical, in order to bite, must be coupled with evidence that such reasoning is correct, or at the very least not evidently mistaken. If invention is to be mothered let it be through necessity and not through the failure to take suitable precautions.

6 PAYETTE, D’ENTREMONT, SCHOTCH, AND JENNINGS
Nevertheless some of the evidence is compelling. Even when our stock of beliefs is inconsistent we routinely draw a distinction between what follows from it and what does not, and regard certain inferences from our beliefs as improper, in spite of the circumstances. If our procedures were genuinely classical we would not, indeed could not, make such a distinction. There is no classical issue to be taken with any inference from an inconsistent set.
Failure to take this sort of evidence into account seems as misguided as any of the worst excesses of those who would base logical theory democratically upon the actual inferential practice of the proletariat. It is no doubt true that if the rule
If A then B; B; therefore A.
(sometimes known as Modus Morons) were from time to time invoked in country districts or local government, we should nevertheless resist viewing it as a ratiocinative dialect, or alternative inferential lifestyle. But if humanity universally manages to distinguish, at least in some cases, valid arguments from inconsistent premises on the one hand from invalid arguments from those same premises, this is another and weightier matter.
The view that everyone commits a logical error in making such a distinction is an adaptation for logic of the doctrine of original sin.
We have already considered one non-classical view of the matter, namely the view that when we argue from our beliefs we disregard some, reasoning only from the active portion of our belief set. That this is non-classical is evident from the fact that it is non-monotonic. Whatever sense of “inferable” we adopt, it is clear that we do not think of our conclusions as inferable in that significant way from an inconsistent set. The reason for such a non-monotonic stance is precisely that the whole of one’s belief might be seen to be inconsistent if examined, that an inference drawn from the set taken as a whole might be trivial. To escape this consequence we must take the further step of claiming that the only real beliefs are the active ones.
This protestant view, which is really an attempt to deny that beliefs can be inconsistent, suffers from two flaws. The first is that on this doctrine it is nigh on impossible to identify a belief set. For suppose someone’s apparent belief set is inconsistent. Which subset of the set of apparent beliefs constitutes the real belief set? (assuming that the real beliefs are also apparent; otherwise the problem becomes even worse). Even if we insist that the real beliefs form a maximal consistent subset of the apparent beliefs we do not thereby guarantee uniqueness. We must elaborate more sophisticated side conditions to ensure this, perhaps invoking the probability calculus to assist. In this case we must have handy some interpretation of probability other than one of its belief interpretations. The project seems a mare’s nest. But these difficulties would not deter us from the task were it not for the second flaw.
The second flaw is that the basic premise is false. Try any thinking beyond the wallpaper or musak variety of daily life, and inconsistent pairs of beliefs do turn up, the one belief as active as the other.
There is a second non-classical view which is decidedly not protestant. According to it, there can be belief sets in which both a sentence A and its negation not-A appear. What is more, such a belief set will also contain the single conjunction “A and not-A”. Subscribers to this account suppose that the true meaning of “not” is such as to make “A and not-A” capable of being true. This move allows the

ON PRESERVING 7 satisfiability condition to reappear since many more sets (perhaps even every set) will be satisfiable.
But if the previous approach is protestant, this one is profligate; the one squeamishly avoiding contradictory pairs, the other rushing to embrace contradictions. In mincing negation, this scheme avoids the classical consequences of contradiction by changing the meaning of “contradiction”. Why call such a view “profligate”? Not for its boldness of approach which is admirable, and certainly not because its advocates have been irresponsible, for they have not. Both the theory underlying the move and its inferential consequences have been energetically worked out and widely published. But where the former school seeks to deny that an offence has been committed in the meaning of the act, the latter denies that any legislation has been enacted, or that if it has, it is unconstitutional. And where one ought to feel release and fill one’s lungs, one feels a kind of dull unease. For even if classical negation is not primordial in human language, but merely a contrivance of latter days, it is nonetheless with us and even if not all, at least some of our contradictions seem to be on that scale.
The unhappy fact of the matter is that sometimes, through no fault of our own, we are simply stuck with bad data. This can happen in any of the variety of circumstances in which we gather information from several sources. Should the sources contradict each other, we may have a way of measuring their relative reliability in some reasonable way. Equally often however either the means at our disposal will not settle the matter conclusively (as in the case of beliefs) or we simply have no means for the adjudication of conflict between our sources.
The classical theory of inconsistency must retire in such a situation since nontrivial classical reasoning is impossible in the face of inconsistent premise sets. All that the classical logician can advise in these circumstances is to start over again with a consistent set of premises. Sometimes this is wise counsel but more often it is of a piece with what the doctor said when told by a patient: ”It hurts when I do this.”
4.
Definitions
Definition 1.
The notion of an indexed family of sets is defined relatively to some collection I (of indices) by means of the (partial) injective functions
(Where V is the universe of sets)
( .
) i
: V → I .
{ X i
} i ∈ I
= { X | ( ∃ i ∈ I )(( .
) i
( X ) = i ) }
There is no assumption in this definition that the indices in I are ordered in some way. In most of our applications of the idea however, we will require an order.
We shall often use ω (the collection of finite ordinals) with the usual order, to index the families in which we will be interested.
Definition 2.
An indexed family of sets { X i or a covering family of the set Y provided:
COV ( { X i
} i ∈ I
, Y ) ⇐⇒ Y = i
{ X i
}
} i ∈ I i ∈ I is said to be a cover of the set Y
In the special circumstance that all the members of a covering family are disjoint, the cover is usually called a partition of Y .
In all of our work, we shall be concerned with sets of formulas (sometimes called well-formed formulas or well-formed expressions drawn from some formal language

8 PAYETTE, D’ENTREMONT, SCHOTCH, AND JENNINGS appropriate for sentential logic. We assume that the reader is possessed of an adequate notion of well-formedness. We shall use capital Greek letters Γ, ∆, etc.
with or without subscripts to represent these sets.
We also presume that some notion of inference or provability has been given in advance. Such a notion will sometimes be referred to as the underlying one. Mostly we assume that this notion is classical, but the reader should take that as no more than a particularly clear example. We do not intend to bar non-classical underlying accounts of inference and will explicitly to make reference to such alternatives from time to time in the sequel. We usually employ ` to indicate the underlying relation, while ² represents the semantic version.
Definition 3.
Given an underlying inference relation ` , as set Γ of formulas is said to be trivial if, relative to that set, there is no distinction between what follows and what doesn’t. This is unpacked for classical (and most non-classical) logics in the following ways
Γ is left-trivial if and only if Γ ` α for every formula α .
Alternatively, if the underlying inference supports multipleconclusions, when Γ ` ∅ .
If the underlying inference supports multiple conclusions, Γ is right-trivial if and only if ∅ ` Γ .
When the underlying inference supports only single conclusions, the qualifier
“left” is omitted.
This terminology deviates sharply from the usual according to which what we call a left-trivial set, would be called an inconsistent set.
If we assume that the language contains a symbol, ⊥ is the customary one, which depicts an absurd sentence which is to say that {⊥} is trivial, we can recast the definition of triviality (inconsistency) as Γ `⊥ .
The thing about triviality, is that we may be able to find a covering family for a trivial set, none of the members of which are themselves trivial. Triviality, like its specialized version inconsistency, is not preserved by subsets—in the sense that not all subsets of a trivial set can be trivial, unless the underlying inference relation is trivial.
4
We next define a pair of predicates based on this idea.
Definition 4.
Assuming the underlying inference relation supports multiple conclusions:
N T L (Γ , ξ ) ⇐⇒ ( ∃ i ∈ ξ ≤ ω ) : COV ( { X i
} i ∈ ξ
( ∀ i ∈ ξ )( X i
0 ∅ )
N T R (Γ , ξ ) ⇐⇒ ( ∃ i ∈ ξ ≤ ω ) : COV ( { X i
} i ∈ ξ
( ∀ i ∈ ξ )( ∅ 0 X i
)
, Γ) &
, Γ) &
These predicates can, in their turn, be used to define the notion of degrees of triviality, as certain kinds of limits.
Definition 5.
The concept of the level (of triviality) of a set, ∆ of formulas is defined according as the set is a set of premises or (when such a mode is supported by the underlying inference) a set of conclusions.

ON PRESERVING 9
`
L
(∆) =

 min
ξ
( N T L (∆ , ξ )) if this limit exists

F otherwise
`
R
(∆) =

 min
ξ
( N T R (∆ , ξ )) if this limit exists

F otherwise
We shall omit the subscript on ` when it is clear from the context which version of the function is intended.
One of the interesting things about this notion is that we can distinguish two sorts of classically consistent sets: those of level 1, which we might think of as the usual sort, and those of level 0. The latter level is the one we conventionally assign to the empty sets, both on the left and on the right. This assignment makes intuitive sense when we realize that the union of any classically consistent set with the empty set, must be consistent (on the left), whereas the union of two (arbitrary) classically consistent sets might be classically inconsistent (on the left). Precisely the same situation, mutatis mutandis on the right—any non-trivial conclusion set remains non-trivial when the empty set is added, but not in general upon the addition of an arbitrary non-trivial set.
∅ is the only such ’ambidextrous’ set—i.e. the only one which has the same level on the left as on the right.
Finally, we use these notions to define an inference relation. The basic motivation for the definition is the preservationist one: We aim to construct an inference relation which doesn’t increase the level of premise (and conclusion) sets. We begin with defining relations which allow only singleton sets on the right and left respectively. First some preliminary notions.
Definition 6.
Without loss of generality, we confine our attention to partitions of premise (respectively conclusion) sets in which meet the condition N T L (respectively
N T R ) for some ξ , if any coverings can.
Π(Γ) indicates the set of partitions π of Γ such that π has ` (Γ) cells each of which is classically consistent.
5.
Preserving Level
The level of (triviality of) a set (no matter which side it’s on) is a measure of how bad things are. On that ground, it is a prime candidate for preservation. We want inference to preserve level because, whatever other properties it has, it shouldn’t make things worse.
But what does it mean, strictly speaking, to say that an inference relation preserves level? We shall say, as we did before that to preserve some property Φ of sets, is for the deductive closure of all Φ-sets to have that same property.
It is relatively easy to see that classical logic does preserve the levels 0 and 1— the level of classically consistent sets, but not any other levels. This will come as no surprise to anybody familiar with the classical rules of inference who examines the definitions of level.
Under the (tacit) assumption that premise sets are consistent, classical logic meets this preservationist criterion but not when that assumption is unfounded.

10 PAYETTE, D’ENTREMONT, SCHOTCH, AND JENNINGS
Alas, neither does any other of the well-known accounts of inference. So if we wish to use a level preserving inference relation, we shall have to roll our own.
Definition 7.
First, where Γ is a set and α a formula
Γ ° α ⇐⇒ ∀ π ∈ Π(Γ) , ∃ p ∈ π : p ` α
In words: some formula α is a consequence (in this sense) of the set of premises Γ
(assumed to have have level ` (Γ)) if and only if in every ` (Γ)-fold partition of Γ for which all of the cells p are classically consistent, at least one of the cells classically proves α . Γ ° Σ is usually read “Gamma level forces Sigma” where either Γ or Σ is a singleton.
Definition 8.
And in the “other direction”
α ° ∆ ⇐⇒ ∀ π ∈ Π(∆) , ∃ p ∈ π : α ` p
Similarly, for a formula to prove a set in this sense is for it to classically prove at least one cell of every non-trivial `
R
(∆)-fold partition of ∆.
The general idea of the inference relations we shall examine is, that the underlying relation will be a guide to the constructed relation so long as certain sets are involved. These sets we call singular. To begin with:
Definition 9.
S (Σ) ⇐⇒ Σ = ∅ or Σ = { α } for some formula α
In words, A set is singular if and only if it is either empty or the unit set of some formula.
[Pres] Γ ` α = ⇒ Γ ° α if S (Γ)
Since it will prove useful at several points below and since somebody might believe (as a reviewer once did) that this account comes down to nothing more than the classical consequences of singular sets, we shall provide a counterexample to that assertion.
Suppose that Γ is a set of premises such that ` (Γ) = n . It is easy to verify by
“pigeonhole reasoning” that the following principle must hold:
[
£
° ∧ ]
Γ
£
Γ
°
°
α
1
& . . .
& Γ
( α i
1 ≤ i = j ≤ n +1
£
° α n +1
∧ α j
)
Where the formula on the right in the consequence of this rule represents the disjunction of all pairwise conjunctions of distinct α i
’s.
It is similarly easy to see that this disjunction cannot be a classical consequence of ∅ nor of any of the individual member of Γ (nor any of the individual α i
).
It is also worth-while to note the special case in which ` (Γ) = 1. In such a case we have:
Γ
£
°
Γ
£
( α
°
1
α
∧
1
α
& Γ
2
) ∨
£
(
°
α
α
2
2
∧
But, of course, the conclusion collapses into Γ
α
1
)
° α
1
∧ α
2
, which is just the classical rule of conjunction introduction on the right. The latter rule is simply a special case of our more general rule.
[R] α ∈ Γ = ⇒ Γ ° α

ON PRESERVING 11
[Cut] [Γ ∪ { α }
£
° β & Γ
£
° α ] = ⇒ Γ
£
° β are correct for the single conclusion case, and that the obvious mutations of these hold for single premise-multiple conclusion case.
This leaves the rule of monotonicity which is of great interest if only because many have identified it with deduction. In other words, many have said “nonmonotonic logic must be non-deductive”. We have never been able to grasp the arguments in favour of this position (assuming that there are such arguments— which is certainly not true in all cases). Perhaps that has been of help to us in would be better to say that the relation ( ° ) is not unrestrictedly monotonic.
We certainly recognize special cases. These for instance:
[Mon*](i)
{ α }
£
°
£
β & α ∈ Γ
Γ ° β
∅
£
Γ
£
° α
° α
Here again we are restricting our attention to the ‘usual case’ of multiple premises and single conclusions but the other two rules we would have to add to recognize the single-premise multiple conclusion case are clear. In this we recognize that we can dilute any inference which begins with a singular set on the left. For all other inferences, we are only allowed to dilute 5 on the left so long as level of the diluted set suffers no increase in level.
[Mon*](ii)
Γ
£
° α
£
⊆ ∗
Γ 0 ` α
Γ 0 where “Θ ⊆ ∗ appropriate).
Λ” abbreviates “Θ ⊆ Λ & `
L
(Θ) = `
L
(Λ)” (or `
R
(Θ) = `
R
(Λ) as
6.
Generalized Level Functions
We have already given various bits of foreshadowing, but we have in mind a more general account of level functions. Here are some definitons.
Definition 10.
A General Level Function
A level function ` is an ordinal or {∞} -valued function that measures the level of
P -ness of a set of sentences Γ where P is a predicate of sets of sentences.
The way that ` (Γ) is calculated is by defining another predicate ∂ (Γ , ξ ) if and only if, there is a family of sets F
ξ
P (∆
β
= [∆ i
: 1 ≤ i ≤ ξ ] such that for every β ∈ ξ
). The ξ subscript here is to say that F is a family over the ordinal ξ . This,
, one could call a ( P )-cover. We can then define the level function as follows:
½
` (Γ) = min { ξ | ∂ (Γ , ξ ) } if it exists
∞ othewise
In the case of forcing ( ° ) we will be using P as meaning that the set is classically consistent and we will use Q to mean: for each subset ∆ of Γ, which is either a unit set (e.g.
{ ψ } ) or the empty set, if ∆ ` ψ then Γ ° ψ . But that is another story.
The restrictions that must be put on `, P and Q are as follows:
(1) ` is downward monotonic, i.e. if Γ ⊆ Γ 0
(2) P is downward monotonic, i.e. if Γ ⊆ Γ 0 then ` (Γ) ≤ ` (Γ 0 then ` (Γ) ≤ ` (Γ
).
0 ).

12 PAYETTE, D’ENTREMONT, SCHOTCH, AND JENNINGS
(3) The extensions of Q and P are not disjoint. That is, the classes of sentences which satisfies these predicates are not disjoint when we want to consider a Q .
One can actually prove the condition 1 above. Consider if there were a subset with a larger level than the whole set. If there were a cover of the whole set of size, say ξ then there would be a cover of the subset of size ξ , which contradicts the assumption that the subset has a larger level.
7.
Compactness
The concept of compactness is something that must underlie any system that is going to call itself a finitary form of proof.
The Next phase is to prove a compactness theorem for level. First we must consider what compactness means in this sense of level. Using the classical logic concept of satisfaction we can say that a logic is compact when the satisfiability of every finite subset Γ 0 of a set Γ(finite satisfiability) carries up to the satisfiability of the set Γ (satisfiability). That is to say that Γ is consistent if, every finite subset of Γ is consistent. In classical logic of course the two concepts of finite satisfiability and satisfiability are equivalent. Using level we relativize this to say that the level of the whole set is less than or equal to a certain number if and only if, each finite subset’s level is also less than or equal to that number. The one issue however is that we can only make sense of this, when the level is finite. If we consider
` (Γ) = ω , then we mean that ω is the size of the “smallest” possible ( P )-cover of
Γ. This does not mean that ` (Γ) = ∞ that is reserved for only certain types of sets, viz. self-inconsistent ones like { ψ ∧ ¬ ψ } . In the general case ` (Γ) = ∞ means there are no covers of Γ. There being no covers just means that at least one of the unit sets is not a P set. Returning to the example, if we consider equivalent formulations of compactness, viz. if ` (Γ) = ξ then there is a finite subset of Γ with level ξ , then we may have a problem if we are trying to consider a set with level
ω . Any finite set, which does not have level ∞ can have a level of at most a finite number. Since all of the unit sets are P -sets the largest level is going to be when any pair sets are not P -sets, so that is the level which is the size of the set, which is finite. Thus, if a set Γ has level ω no finite subset of Γ can have level ω . Therefore, the two variations of compactness cannot be equivalent for infinite level. That does not mean that such a concept cannot be given but when considering forcing we only want finite levels so we can leave that for some other paper.
Theorem 1.
[D’Entremont [3] Theorem 10]
The following are equivalent when ` (Γ) < ω :
(1) ` (Γ) ≤ n ⇐⇒ ∀ Γ 0
(2) If ` (Γ) = n then ∃ Γ 0
(3)
⊆ Γ
⊆ that are finite,
Γ
If there is a finite subset Γ ∗
` (Γ 0 ) ≤ n .
which is finite, such that ` (Γ 0 ) = n .
of Γ such that for any other finite Γ 0
` (Γ 0 ) ≤ ` (Γ ∗ ) then ` (Γ ∗ ) = ` (Γ) .
⊆ Γ ,
PROOF: We proceed by showing the equivalence in a triangle. All n, m, ketc.
are elements of ω .
1 ⇒ 2 Assume 1 and assume for reductio that ` (Γ) = n and that there is no finite subset of Γ which has level n . We know by monotonicity of level that all of the subsets of Γ must have level less than that of Γ, so there is an upper bound. Call this upper bound m . This m is strictly less than n because otherwise there would

ON PRESERVING 13 be a finite subset of level n which there isn’t. With 1 we get ` (Γ) ≤ m < n , which is a contradiction.
2 ⇒ 3 Assume 2 and the existence of a Γ be, by 2, a finite Γ 0 ⊆ Γ with ` (Γ 0
∗ as in the antecedent of 3. There must
) = ` (Γ) thus, ` (Γ) ≤ ` (Γ ∗ ). By monotonicity of
` we get ` (Γ ∗ ) ≤ ` (Γ) hence, ` (Γ ∗ ) = ` (Γ).
3 ⇒ 1 Assume 3. The only if direction of 1 follows from monotonicity of ` thus, assume that for every finite Γ 0 ⊆ Γ ` (Γ 0 ) ≤ n . Let m = max { k | ` (Γ 0 ) = k & Γ 0 bound, viz.
n . Let Γ ∗ = Γ conditions for 3 thus, ` (Γ ∗ ) =
0
⊆ Γ f inite } This must exist since there is an upper
` such that
(Γ) = m ≤
` (Γ n .
0 ) = m . Then we have satisfied the
Q.E.D.
For the next theorem we need a definition that goes something like maximal extension from every other logic and further an extension lemma like that from every other area of logic.
Definition 11.
[D’Entremont [3]] Γ +
1
Γ
1
⊆ Γ over Γ if and only if;
(1) Γ
1
(2) ` (Γ
⊆
+
1
Γ +
1
⊆ Γ
) = ` (Γ
1
) and,
(3) for any formula ψ , if ` (Γ
+
1
∪ { ψ is a maximal level preserving supper set of
} ) = ` (Γ
+
1
) then ψ ∈ Γ
+
1
.
As one would expect we can produce one of these Γ + sets by the following construction. Note that the language that we are using is countable.
Lemma 1.
[Lindenbaum’s Level Lemma]
Let Γ
1
⊆ Γ have level k ∈ ω then it can be extended to a Γ
+
1
.
PROOF: Let Γ
1 meration of Γ \ Γ
1
⊆ Γ have level m ∈ ω . Then Let ψ
0
, ψ
1
, ..., ψ k
, ...
be an enu-
. This enumeration could stop at a ξ < ω , or could continue to ω either way I will refer to the size of the enumeration by ξ for generality. Form sets
Σ n for n ∈ ξ by:
Let Γ +
1
Γ
1
⊆
=
S n ≤ ξ
Σ n
Σ n
=
½
Σ n − 1
Σ n − 1
∪ {
Γ to start with, we have Γ
1
ψ n
}
⊆ Γ
Σ
+
1
0 if
= Γ
`
1
(Σ n − 1 othewise
∪ { ψ n
} ) = ` (Γ
1
)
. Claim: this set is a Level preserving maximal extension of Γ over Γ. Since nothing other than elements of Γ were added to Γ
1 to make Γ
⊆ Γ. It should be clear that ` (Γ +
1
+
1
1 and
) = ` (Γ
1
) by the recursive construction. It can be proved by an easy induction. Lastly, suppose ϕ ∈ Γ, then either a) ϕ ∈ Γ
1
` (Γ ϕ
+
1
∪ { ϕ } ) = ` (Γ +
1
) then ϕ ∈ Γ was considered for membership at stage change its level, adding ϕ to Σ n
+
1 or b) ϕ = ψ n a fortiori n for some
. If b) and ` (Γ +
1 does not change the level of Σ n n
∈ ξ . If a) then if
∪ { ϕ } ) = ` (Γ
, and since adding it to Γ
+
1
+
1
), then does not
, thus ϕ was added at stage n . Therefore, ϕ ∈ Γ
+
1
.
Q.E.D.
This lemma will allow us to prove the result which is the crucial step in proving finite level compactness.
The concept of cover is generalized in this paper to a more topological notion. In particular settings it may not be the case that a cover actually contain the elements of the set that it covers. The definition of cover used in [1] to define ° only requires that every element of the set being covered be classically proved by some member of the covering family. Along that definition we can consider the set
Γ = { C ∧ D, A ∧ B, ( A ∧ B ) ⊃ ¬ ( C ∧ D ) }

14 PAYETTE, D’ENTREMONT, SCHOTCH, AND JENNINGS
Then consider the cover F
3
= [ { C, D } , { A, B } , { ( A ∧ B ) ⊃ ¬ C } ]. Although this is not a smallest cover there is a member which proves some member of Γ between which they exhaust Γ and ( ∪ F
3
) ∩ Γ = ∅ . For our crux theorem we need a particular kind of cover. What we will call a minimal cover.
S
F
ξ
= Γ
A Minimal Cover of
. Further, if ∆ i and ∆ k
(
Γ i, k is a cover
≤ ξ )
F
ξ such that ξ = are distinct members of
` (Γ)
F
ξ and then
∆ i
∩ ∆ k
= ∅ .
This definition is really just a partition of the set Γ. It is the kind of partition in which each cell is a P -set. Such a thing is not unique in the general case but there is always the possibility of having such a partition.
Theorem 2.
[Minimal Cover Lemma]
If ` (Γ) = n then given a cover F n of Γ there is a minimal cover F 0 n construct from F n
.
which we can
PROOF: Assume that ` (Γ) = n then by definition there is a cover F n
[∆
1
, ..., ∆
∆ 0 i
= ∆ define i n
=
]. We shall construct the minimal cover out of this one. First, let
∩ Γ for all 1 ≤ i ≤ n . Let ∆ n [ k ij
= ∆ 0 i
∩ ∆ 0 j where i = j . Further,
∆ ∗ k
= (∆ 0 k
− (∆ 0 k
∩ ∆ 0 k + m
)) m =1 for 1 ≤ k ≤ n . Claim: These ∆ and ∆ ∗ i
∩ ∆ ∗ j
= ∅ . Then, ψ ∈ ∆
∗ k
0 i
’s are all disjoint and Γ =
∩ ∆ 0 j since ∆ ∗ i
∩ ∆ ∗ j
⊆ ∆ 0 i
S
∩ k ≤ n
∆ 0 j
∆ ∗ k
. Suppose
. Thus, ψ ∈ ∆ ∗ i i
∩
=
∆ j
∗ j
⇔ ψ ∈ ∆
∗ i
& ψ ∈ ∆
∗ j
⇔ ( ψ ∈ (∆
0 i
− ∆ ij
))( ψ ∈ (∆ which is a contradiction. It is clear that
S
The only worry is when ψ is in a one of the ∆ ij which it is in. Thus, if ψ ∈ ∆
Since, ∆ 0 j
− (∆ 0 j
∩ ∆ 0 i ij
0 j k ≤ n
’s, otherwise it is just left in the ∆ then without loss of generality i < j and, ψ ∈ ∆ which is beyond i by assumption. Therefore, Γ ⊆ of Γ because it
− ∆
∆ ij
∗ k
))
⊆ k ≤ n
⇔
∆ ∗ k
(
∗ j
ψ 6∈ ∆
0 i
∩ ∆
0 j
)( ψ
Γ so assume that
6∈
ψ
∆
∈
0 i
∩
Γ.
∗ j
∗ i because it starts at j
. This is clearly a cover
.
∆
0 j
) ⇔ ψ 6∈ ∆
0 i
∩ ∆
0 j
(1) contains Γ
(2) since P is monotonic each member of F 0 n is a P set,
Thus, it satisfies all of he conditions that a minimal cover must.
Q.E.D.
Finally with this result for certain we can then proceed with the primary goal of the paper, the proof of the finite compactness theorem. Like a lot of theorems we can get it more simply via another theorem.
Theorem 3.
[D’Entremont [3] Theorem 11]
Suppose P is compact and Γ is a set of sentences which has finite level. If a finite subset Γ
1 of Γ is such that every finite subset Γ 0 of Γ has ` (Γ 0 ) ≤ ` (Γ
1
) then
` (Γ
1
) = ` (Γ) .
PROOF: Suppose P is compact and Γ is a set of sentences and ` (Γ) ≤ ω say m . Γ is either finite or infinite. If it is finite then Γ itself satisfies the condition, so suppose that Γ is infinite. Let Γ
1 be a finite set who’s level is not exceeded by the level of any other finite set. Suppose for reductio that to Γ
` (Γ
1
+
1 as in lemma 1. For some ψ ∈ Γ ψ 6∈ Γ
+
1
) = ` (Γ) by definition of Γ +
1
, which it isn’t. Let F n
` (Γ
1
) < ` (Γ
1
). Extend Γ
1
. If there were not then Γ = Γ
+
1 be a minimal cover for Γ so
+
1
.

NOTES 15
If any ∆ i
∈ F n same size, viz.
we had n thus,
P
`
(∆
(Γ
+
1 it is not the case that P (∆ i i
∪{ the existence of a finite set ∆ 0 i
ψ }
⊆
) then we could cover Γ
∪ { ψ } ) = ` (Γ
∆ i
+
1
+
1
∪{ ψ } with a cover of the
) which it is not. Hence for any ∆
∪ { ψ } ). By the compactness of P we are guaranteed
, ∀ 1 ≤ i ≤ n such that, not P (∆ by compactness, take the finite sets for each i = j with ∆ ij
⊆ ∆ i
0 i
∪ { ψ } and ∆ ji i
∈ F n
) Also
⊂ ∆ j such that not sets ∆
F n +1
But ` (
∗ i
= ∆
F
0 i
∗ i n +1
P
∪ (
(∆ n ij j =1
∪
) > ` (Γ
1
∆
∆ ij ji
) which we get since is a smallest cover for the set is finite and F n +1 is also finite,
F n
S
F n +1
F n +1
F is a minimal cover. Define the
= [ { ψ } , ∆ ∗ i
| 1 ≤ i ≤ n ]. This n +1 is, and further
S
F
) which is contrary to hypothesis. Therefore, ` (Γ
1 n +1
) = `
⊆ Γ.
(Γ).
Q.E.D.
Thus, it has finally come to an end. We can now draw as a corollary to the combination of this last theorem and Theorem 10;
Corollary 1.
[D’Entremont [3] Theorem 9 Finite Level Compactness]
If Γ is a set of sentences with ` (Γ) < ω and P is compact, then ` (Γ) ≤ n if and only if, For every finite subset Γ 0 of Γ , ` (Γ 0 ) ≤ n .
PROOF: Theorem 1 and theorem 3.
Q.E.D.
Notes
1 We think this usage was coined first by Quine.
2 Of course this assumes that the inference relation has the classical rules for negation introduction on the left and the right.
3 Indeed some have argued that to be rational one must have such a belief set. See [2].
4 Which is to say that every pair < Γ , ∆ > belongs to the relation (where ∆ may be restricted to singletons for single conclusion style inference).
5 This would be our translation of the German word used by Gentzen which is usually translated
”Thin”.
References
[1] B. Brown and P. K. Schotch. Logic and Aggregation.
Journal of Philosophical
Logic , 28(396):265–287, 1999.
[2] Richmond Campbell. Can inconsistency be reasonable?
Canadian Journal of
Philosophy , 11(2):245–270, 1980.
[3] Blaine H. D’Entremont. Inference and level. Philosophy, Dalhousie University,
Halifax, Nova Scotia, September 1982.
E-mail address : GPAYETTE@DAL.CA
E-mail address : DENTREMO@GOV.NS.CA
E-mail address : PETER.SCHOTCH@DAL.CA
E-mail address : JENNINGS@SFU.CA