That’s It! Hyperintensional Total Logic

Call a truth complete with respect to a subject matter if it entails every truth about that subject matter. One attractive way to formulate a complete truth is to state all the relevant positive truths, and then add: and that’s it. When the subject matters under consideration are non-contingent, a non-trivial conception of completeness must invoke a hyperintensional conception of entailment, and of the completion operation denoted by ‘that’s it’. This paper develops two complementary hyperintensional conceptions of completion using the framework of truthmaker semantics and determines the resulting logics of totality.


Introduction
Among all the truths concerning a given subject matter we may distinguish between those that constitute merely part of the overall truth concerning that subject matter, and those that exhaust the subject matter: the complete truth or truths about that subject matter. To a first approximation, we might take a truth to be (strongly) complete (with respect to a given subject matter) iff it entails every truth (pertaining to that subject matter). Totality operators, which may informally be glossed by the phrase '. . . and that's it', are an important resource in articulating complete truths. For a toy example, consider the subject matter of my breakfast. Suppose I had porridge and tea, and nothing else. In particular, then, I did not have eggs, bacon, coffee, nor roasted crocodile, . . . In a straightforward sense, these are truths pertaining to the subject matter, so a complete truth about that subject matter would need to entail them. Rather than list them all, it seems, I may say I had porridge and tea, and that's it, and achieve the same effect. The advantages of proceeding in this way-roughly speaking, by conjoining the positive truths about the subject matter, and adding that that's all-do not merely concern efficiency. Let us call a truth weakly complete with respect to a subject matter iff it falls short of completeness only in the way just illustrated, by omitting some negative truths, so that the proposition that , and that's it, is plausibly a strongly complete truth with respect to that subject matter. It seems that we may learn something significant about a given subject matter by determining what kind of truth is weakly complete with respect to it. An attractive first idea for explicating physicalism, for instance, is as the claim that the conjunction of all physical truths is a complete truth. But if physicalism is true, then, presumably, it is also true that there are no angels, and this truth does not appear to be entailed by the conjunction of all physical truths. The sense in which the latter might be complete, accordingly, is the sense of weak completeness rather than strong. 1 In previous work [6], I argued that some applications call for a hyperintensional understanding of these notions, and proposed an explication of the notion of a whole truth within the framework of truthmaker semantics. The present paper employs the same framework to develop a formal semantics for suitable hyperintensional totality operators, and determines the resulting propositional logics of totality. I begin by describing the main motivations for a hyperintensional approach (Section 2). After introducing the general framework of truthmaker semantics (Section 3), I use it to develop a detailed account of two natural and complementary totality operators (Section 4), describe the resulting logics of totality (Section 5), and compare them to the intensional logic of totality studied by [8] (Section 6). An appendix gives proofs of soundness and completeness.

Motivating a Hyperintensional Approach
A strongly complete truth, we said, is a truth that entails every truth. A weakly complete truth still has to entail every positive truth. 2 A totality operator is one that takes any weakly complete truth into a strongly complete one. These notions are of very general application. More or less any question that might be raised can be regarded as determining a subject matter, and thus as determining a collection of all truths, and a collection of all positive truths, pertaining to that subject matter. We can then ask whether a given proposition constitutes a correct and weakly or strongly complete answer to the question according as it entails all elements of either collection. Correspondingly, in any such context, there is application for some form of totality operator turning a weakly complete truth into a strongly complete truth.
Let me give some examples. I have already mentioned the role of totality operators in the formulation of physicalism and related doctrines concerning the kind of truth that is weakly complete. Another example in metaphysics concerns the grounds of true universal quantifications. Suppose that everything is concrete. On one natural account of the grounds of this truth, it is grounded in the truths that is concrete, is concrete, . . . together with the truth that exists, exists, . . . , and that's it. Another context in which totality operators may have application is the theory of non-monotonic reasoning, when inferences are to be based on the totality of one's evidence. Suppose I know that tweety is a bird. If that's all I know about tweety, I may be entitled to infer that tweety can fly, but I am not entitled to that inference if I also know that tweety is a penguin, and that penguins don't fly. There is also both explicit and implicit use being made of totality operators in natural language, especially in question-answer discourse. Suppose you ask me what's in my backpack, and I say: a bottle of water and an apple. Under the most natural interpretation of my utterance, it excludes the possibility of there being additional items in my backpack.
This interpretation thus appears to invoke a tacit totality operator, as in: my backpack contains a bottle of water, an apple, and that's it. 3 Depending on how we understand the notion of entailment invoked in the characterization of completeness, we obtain different versions of our triad of concepts. If we understand entailment in purely modal terms, for example, we obtain a liberal notion of a complete truth as a truth that necessitates every truth. If we invoke a more demanding conception of entailment, we obtain a likewise more demanding notion of a complete truth. I do not want to claim that there is a uniquely correct or best way to go here; which notion of entailment is the most appropriate one may simply depend on our particular purposes and interests. What I do want to claim is that some legitimate and reasonable purposes and interests call for a hyperintensional conception of completeness and hence of entailment.
Perhaps the clearest and most compelling reason to want an alternative, hyperintensional account of strongly and weakly complete truths is that some version of these notions should apply in an appropriately discriminating way to non-contingent subject matters. Obvious examples are various mathematical subject matters. Many statements about mathematical objects are necessarily true if true at all. Objects of pure mathematics are typically held to exist necessarily if at all. Many important questions about their natures are likewise plausibly non-contingent, such as the question whether they are abstract, whether they depend on the mathematical structures they are elements of, and whether they are mind-independent. In the modal sense of entailment, truths about these matters are thus vacuously entailed by any truth, and so any truth whatsoever will be regarded as complete with respect to this subject matter. But there surely is a non-trivial sense in which a correct description of certain mathematical objects may be, or fail to be, complete concerning the questions under discussion. For example, consider the question of the integer solutions of the 3 There is an extensive body of literature in linguistics about this phenomenon, often invoking so-called exhaustivity operators which are somewhat similar to the totality operator studied by Chalmers & Jackson and Leuenberger. Spector [9] provides a helpful overview and comparison of various proposed exhaustivity operators. A proper engagement with this literature is beyond the scope of this paper, but it is worth mentioning that the existing approaches in linguistics are usually set within the intensional framework of possible worlds and thus not suited to address the specific problems that motivate my hyperintensional approach. equation 2 4 0. The answer that 2 and -2 are solutions, and that's it, is correct and complete. The answer that 2 is a solution, though necessarily equivalent, is correct but not complete. 4 In response, one might perhaps consider an interpretation of entailment as logical rather than metaphysical necessitation. But similar problems then arise for logically non-contingent subject matters. Even if the necessity of identity is a logical truth, a correct theory of identity may in a very natural sense fail to be complete if it fails to state the necessity of identity. 5 More could undoubtedly be said about these issues, but the present remarks may suffice at least to motivate interest in the project of developing a more discriminating, hyperintensional theory of the whole truth, and a corresponding totality operator.

Truthmaker Semantics
In truthmaker semantics, a proposition is modelled not by the set of worlds at which it is true, but by the set of states that make it true, as well as the set of states that make it false. 6 Both notions, that of a state and that of truthmaking, require some comment. Unlike a possible world, a state may be partial in the sense that it settles the truth-value of only some propositions. Unlike a possible world, it may also be impossible in the sense that it makes an impossibility, even a contradiction true. Like a possible world, however, a state is required to be (relatively 7 ) specific: it can only make a disjunction of the language true by making one of its disjuncts true. Thus, the truthmakers of the proposition that the ball is red or blue include the state of the ball being red and the state of the ball being blue, but we do not recognize a further disjunctive state of the ball being blue-or-red. 8 In order to make a given proposition true, a state must be wholly relevant to the truth of that proposition. In particular, it is not enough that the state's obtaining necessitates the truth of the proposition. For instance, the state of snow being white does not make true the proposition that 2+2=4, since it is wholly irrelevant to the truth of that proposition. And the state of it being cold and rainy is not a truthmaker for the proposition that it is cold, since it is partially irrelevant to the truth of that proposition, by virtue of containing the irrelevant part of it being rainy. Analogous remarks apply to falsitymaking. I shall also call truthmakers verifiers, and falsitymakers falsifiers, while stressing that any epistemic connotations of these terms should be thoroughly discarded. 4 Thanks to an anonymous referee for the example. 5 These and similar points are developed at greater length in Section 3 of [6]. 6 For a more detailed exposition of the framework, see [4]. 7 We need not understand the specificity requirement in an absolute sense, but may instead take it as relating to the language that we wish to interpret. The requirement is then that each state be specific enough that it makes a disjunction true only if it makes one of its disjuncts true. 8 As indicated in the previous footnote, we may take there to be such a thing as the state of the ball being red as long as the language under discussion does not provide the resources to discriminate between the different shades of red.
The states are taken to form a set ordered by part-whole ( ), and we assume that given any states, we may form their fusion, which will also be a state. More formally, recall that a partial order on a set is any binary relation on that is transitive, reflexive, and anti-symmetric. We call a partial order on complete iff every subset of (including the empty set) has a least upper bound with respect to that order. 9 Then our basic structure is that of a state-space: is a non-empty set 2.
is a complete partial order on If is a state-space and 1 2 , we denote the least upper bound of w.r.t. by or 1 2 and also call it the fusion of (the members of) . When , is called the nullstate, or , which is part of every state. When , is called the fullstate, or , which has every state as part. The greatest lower bound of 1 2 w.r.t. -the greatest common part of the elements of -will be denoted 1 2 10 Definition 2 Let be a state-space.
A unilateral proposition on is any non-empty subset of A bilateral proposition on is any pair P P P of unilateral propositions We think of the members of a unilateral proposition as its verifiers, and of the members of P (P ) as the verifiers (falsifiers) of a bilateral proposition P.
We turn to the definition of the boolean operations. We take a disjunction, as one might expect, to be verified by exactly those states that verify one of the disjuncts. A conjunction we take to be verified by exactly those states that may be obtained by taking the fusion of a verifier of the one conjunct and a verifier of the other conjunct. Dually, we take a disjunction to be falsified by the fusions of falsifiers of the disjuncts, and a conjunction to be falsified by the falsifiers of the conjuncts. Negations are verified by the falsifiers, and falsified by the verifiers, of their negatum.

Definition 3 Let
be unilateral propositions and P Q bilateral propositions on . and 9 An upper bound of is any state that has every member of as part. A least upper bound of is any upper bound of that is part of every upper bound of . By anti-symmetry, least upper bounds are unique if they exist. 10 That greatest lower bounds always exist follows from the fact that every subset of , including the empty set, has a least upper bound. Indeed, we can always form the greatest lower bound of a subset of by taking the fusion of all the lower bounds, i.e. all the states that are part of every member of the given subset of .

967
That's it! Hyperintensional Total Logic P P P P Q P Q P Q P P P Q P Q So far, we have a very liberal conception of propositions. 11 Note, for instance, that we have not excluded the possibility of one and the same state both verifying and falsifying a proposition. Intuitively, verifiers and falsifiers of a proposition should not only be distinct, but incompatible. To capture this fact, we need to incorporate a distinction between possible and impossible, or consistent and inconsistent states within our state-space.
is a non-empty subset of such that whenever for We call the members of the possible or consistent states, and we say that some states 1 2 are compatible iff their fusion is a possible state. We call a state a (possible) world iff it is a possible state that has every state it is compatible with as a part. A modalized state-space is called a W-space iff every possible state is part of a world. Our interest henceforth will be in W-spaces.

Definition 5
Let be a W-space and P a bilateral proposition on that Wspace.
P is exclusive iff no member of P P is consistent P is exhaustive iff every world contains some member of P P It may be shown that the properties of exhaustivity and exclusivity are preserved under the boolean operations. Moreover, say that P is true (false) at a world iff contains some verifier (falsifier) of P as a part. Then for exclusive and exhaustive P and Q, at any world , P is true or false but not both, and the boolean operations behave classically: P is true at iff P is not, P Q is true at iff both P and Q are, and P Q is true at iff either P or Q is (cf. [4, pp. 665f]).
There are a number of natural relations of entailment that may be defined between bilateral propositions. For present purposes, the most important ones are those of inexact and loose entailment, respectively.

Definition 6
Let P Q be bilateral propositions on some W-space. P inexactly entails Q iff every verifier of P contains a verifier of Q P loosely entails Q iff every world containing a verifier of P contains a verifier of Q 11 In many applications of truthmaker semantics, requirements are imposed to the effect that the set of verifiers of a given proposition be closed under non-empty fusion-so that for every non-empty subset of -and convex-so that whenever and . In the present context, for reasons that will become clear later, it seems preferable to me not to impose these requirements.
It is known that the propositional logic of loose entailment over exclusive and exhaustive propositions is classical, while that of inexact entailment is the logic of first-degree entailment (cf. [4, p. 669]).

Totality operators
The central constraint on our intended totality operator is that it turn all and only weakly complete truths into strongly complete ones: P is a strongly complete truth iff P is a weakly complete truth.
It is helpful to contrast this constraint with a slightly different one: This constraint would motivate an interpretation of P as simply saying that P is a weakly complete truth. Plausibly, such an interpretation would also satisfy ( S). 12 But as we shall see, there are natural interpretations of which satisfy ( S) but not ( T). To make the constraint precise, we need to explicate the notions of a strongly and a weakly complete truth within the truthmaker framework. To do this, it may be useful to have an intensional explication available for comparison. Suppose, then, that we adopt the familiar intensional conception of propositions as sets of possible worlds, and let @ stand for the actual world. Then a proposition is true iff @ is a member of it, and it entails every truth iff it is the singleton set @ . So the only sensible understanding of a strongly complete truth here is as the proposition @ . To characterize weakly complete truths, a natural approach-first suggested by [1] and studied in detail in [8]-is to appeal to a relation of outstripping among possible worlds. Roughly speaking, the idea is that a world outstrips another iff it contains more positive facts. A proposition may then be regarded as a weakly complete truth iff is a truth, and is false at every world that is outstripped by @-if it takes all of @, as it were, to make true. is a weakly complete truth, therefore, iff the actual world is a minimal -world.
Turning back to the hyperintensional framework of truthmaker semantics, we first need to define a notion of truth for propositions in a given W-space. To that end, we designate one of the worlds in a given W-space as the actual world @. For a proposition to be true is then for one of its verifiers to be part of @. As before, we want a strongly complete truth to entail every truth. But now there are several ways we can understand this condition, depending on which entailment relation we take to be at issue. If we appeal to loose entailment, we obtain an intensional criterion, since intensionally equivalent propositions agree on their loose consequences. Since we are aiming for a hyperintensional account, a more appropriate choice is inexact entailment. For surely a truth verified only by @ should count as strongly complete.
Any such truth will indeed inexactly entail every truth, and inexact entailment is the only natural hyperintensional entailment relation within truthmaker semantics in which a truth verified only by @ stands to every truth.
Note, though, that a truth may have more verifiers than just @ and still inexactly entail every truth, as long as every one of its additional verifiers has @ as a part. (Any such state will then be inconsistent.) Thus, suppose P is verified just by @, and suppose that there are no ghosts in the actual world. Then the proposition P P there are ghosts) also inexactly entails every truth. Still, since it is distinct from P, and since there is a non-actual way for it to be true-albeit an inconsistent one-it seems more natural to deny it the title of the whole truth. So we shall count a truth as strongly complete iff it is verified by @, and only by @. 13 We can give an equivalent definition in terms of inexact entailment and truth: 14 (SCT): A proposition P is a strongly complete truth iff: P is true, and P inexactly entails every truth, and P is inexactly entailed by every truth that inexactly entails every truth.
We now turn to the notion of a weakly complete truth. Given the way we have introduced the notion, just as we want a strongly complete truth to entail every truth, we want a weakly complete truth to entail every positive truth. To make this condition precise, we need not give a definition of a positive proposition or even a positive truth. Let us simply assume as given some part @ of @ as the complete positive part of @. It seems very plausible that a proposition verified only by @ should count as a positive truth, and that a proposition is a positive truth only if some part of @ is one of its verifiers. It is then easy to see that a proposition P inexactly entails every positive truth iff every verifier of P contains @ as a part, and some verifier of P is part of @. For similar reasons as before, we should still exclude some truths of this sort. For suppose P is weakly complete, and that @ contains no ghosts. Then the truth that P P there are ghosts) inexactly entails every positive truth. But it shouldn't count as weakly complete. Roughly speaking, it does not just fail to entail the negative truth that there are no ghosts, it explicitly allows for the possibility that there are ghosts. So I propose that we count a truth as weakly complete iff it is verified by some part of @, and all its verifiers contain @ and are part of @. There is no straightforward way to give an equivalent definition parallel to (SCT), but once we have characterized a suitable totality operator that conforms to ( S), we will be able to give an equivalent definition in terms of and strong completeness.
So let us now turn to developing a suitable truthmaker account of . To give such an account, we need to characterize the truthmakers and the falsitymakers of a proposition P given the truthmakers and falsitymakers of P. The issues raised on the side of the truthmakers are largely separate from those on the side of the falsitymakers, so we shall begin by considering just the truthmakers. First of all, it seems very plausible that if the proposition P has just a single truthmaker , P will also have just a single truthmaker, which we may think of as the state that ( obtains, and that's it). Intuitively, there is nothing disjunctive about it being the case that obtains, and that's it, so we are justified in taking this condition to correspond to a single state. So we may take there to be a function on the states, which maps any given state to the state to the effect that obtains, and that's it. The task of obtaining an account of the truthmakers of P then divides into two subtasks: that of determining the properties of the completion operation on the states, and that of identifying the truthmakers of P when P is disjunctive, i.e. has more than one verifier. We address these tasks in turn, before we consider the matter of the falsifiers.

State-Completions
What can we say about the properties of ? Part of what it is for it to be the case that obtains, and that's it, is for it to be the case that obtains. So it is plausible to hold that always . -Containment: Now since is the state that obtains, and that's it, in some sense, must not contain anything in excess of . Of course, we cannot demand that strictly contain nothing beyond what is contained in , since then would always be identical with . Rather, the idea has to be that contains no positive state that is not already contained in . Indeed, any positive state not contained in would constitute a counter-example to the claim that obtains, and that's it, so must be incompatible with any such state.
One strategy for capturing these ideas formally is to assume as given a division of the states into positive and negative, and simply lay down as a further requirement on that every positive state that is part of also be part of , and that every other positive state is incompatible with . Other things being equal, however, it would seem preferable not to rely on a multitude of primitive notions-the function and a positive/negative divide-within our semantics for , and to stick to a single primitive if possible. So for now, I propose to instead appeal to a distinction between positive and negative states only in an informal, heuristic capacity to motivate constraints on that can be stated without appeal to that distinction. Below, we will see that there is a natural way to define the notion of a positive state in terms of .
We begin by considering some modal properties of . First of all, we can argue that -Completeness: is a world if consistent.
For recall that P is to be strongly complete if P is weakly complete. Moreover, it is natural to think that this should hold of necessity, so if P is a weakly complete truth with respect to any world , then P is a strongly complete truth with respect to . Now suppose P has a single verifier , which is consistent and which contains the entire positive part of some given world . Then P inexactly entails every positive truth at , and so P must inexactly entail every truth at . So in this case, must be identical to . Note that we are assuming, as seems plausible, that for any positive state, there is at most one world whose positive part is that state.
Suppose now that is consistent but does not contain the positive part of any world. In that case, must be regarded as inconsistent. For suppose it is consistent. If it is a world, then contains a positive state not contained in , contrary to our informal desiderata. If it is a proper part of a world, it is compatible with the part of that world, which is by assumption not contained in , which is likewise incompatible with our desiderata. Finally, if is inconsistent, must also be inconsistent by -Containment. It follows that is a world if consistent.
From the principle that no two worlds share their positive part, we may also draw some conclusions about the conditions under which and are identical. Firstly, if is consistent, and , then . For if is consistent, and hence a world, contains the positive part of . Since , so does . So is consistent and contains the entire positive part of , and so . Secondly, suppose is a world. Then both and contain the positive part of that world, hence so does . So is part of a world and contains the entire positive part of that world, so is that world. And since there is at most one world with a given positive part, it is the same world as and .
For both principles, it seems very plausible to take them to apply in the same way if is not consistent. For consistent or not, goes beyond in a purely negative way, merely excluding the obtaining of any positive state not already in . If is between and , then , too, goes beyond at most in excluding the obtaining of some positive states not already in , and goes beyond only in excluding the obtaining of any positive state not even in . But the positive states not contained in are exactly those not contained in , and so there would seem to be no basis for any distinction between and . 15 Moreover, if is inconsistent, we may still infer that and have the same positive part, since otherwise and would differ with respect to what positive states they exclude. In general, we may think of as dividing into (i) the positive part of , (ii) the part excluding any positive state not contained in , as well as (iii) any other non-positive components of . 16 The positive part of and must be the 15 In [6], I suggested that if and have the same positive part, but this is plausible only under the assumption that and are consistent. 16 This assumes a picture whereby the completion of a state is formed by fusing it with certain negative states that would normally be disjoint from that state. An alternative picture is that completions of a state always sit immediately on top, as it were, of the completed state in the part-whole ordering, so that every proper part of is a part of . While this would violate the mereological principle of weak supplementation, the general constraints on state-spaces do not exclude this possibility, and it does not seem out of the question that the mereology of states should be non-standard in this kind of way. If one adopted this alternative picture, the identity principles would of course still hold, since we would then have that implies .-It is worth noting that if we wish to allow for the existence of subjectmatter-restricted completions, that is some reason not to adopt this alternative picture. For consider a state , and some restricted subject-matter . Then the state that ( obtains, and that's it with respect to ) is plausibly regarded as a part of that is typically disjoint from . same, given that , and so it must also be the same as the positive part of . Consequently, the part of excluding any positive state not contained in must also be the same as the corresponding part of and . Now consider any other nonpositive components of . Since they are also to be found in , they must be among the other non-positive components of , and hence also parts of . We may thus infer that . - Finally, it is plausible that in certain cases, the application of will be redundant. In particular, if is already a world, must be taken to be the same as . By -Containment, the only alternative would be for to be inconsistent. Yet if is a world, then it is surely not inconsistent that obtains, and that's it. More generally, it is very plausible that . For goes beyond at most in saying that no positive state not contained in obtains. But since already says that no positive state not contained in obtains, this condition is already imposed by .
-Redundancy (1): if is a world -Redundancy (2): There are perhaps other constraints that might plausibly be imposed on . As we shall see, though, these constraints suffice to render sound a natural propositional logic for . Inspection of the completeness proof will show, moreover, that a number of further plausible constraints on have no effect on the logic obtained (although this may change, of course, when we consider languages with more expressive resources such as other modal operators).
We are now in a position to argue that ( S) holds for the case of propositions with just a single verifier, using two extremely plausible assumptions about the positive part @ of @. The first is that the completion of @ is @, i.e. @ @. The second is that @ is the smallest state whose completion is @, i.e. @ implies @ . 17 Let P have just a single verifier . As we have argued, P then has just the single verifier . Now suppose P is to be a weakly complete truth. Then its sole verifier has to contain the positive part @ of @, and also to be part of @. But @ @, and so by -Identity(1), @. So P is a strongly complete truth. Conversely, suppose P is a strongly complete truth, hence that @ @ . By -Containment, @. By the second assumption, @ . So P is a weakly complete truth.
It is worth noting that when P has just a single verifier , even ( T) holds. For then P has just the single verifier , and so if P is to be true at all, must be @, so P is again a strongly complete truth.

Examples of C-Spaces
It is clear that there are C-spaces. Indeed, there are very small ones. Take any object , and let , , , and let . Then is readily seen to be a C-space. Any unmodalized state-space may be extended into a C-space in a trivial way by letting and for all . Still, such C-spaces are not very natural. Since we have and is supposed to always extend in a purely negative way, in these C-spaces we are, in effect, regarding all states (except, perhaps, the nullstate) as negative.
But there are also more interesting and natural C-spaces. We specify two methods for constructing such spaces. Let 0 0 be any unmodalized state-space. We may construct a C-space by regarding every member of 0 as wholly positive, and adding for each state in 0 a distinct -state immediately 'on top'. First, with each pair of states with 0 , we associate a distinct item that is not a member of 0 . Intuitively, we may think of as saying that obtains, and that no positive state not contained in obtains. We pick an additional item to play the role of the fullstate in the C-space to be constructed. Let -the new states-be the set comprising and the values of , and let 0 . We define the partial order on indirectly, by first specifying the fusion operation and then letting iff . For , we distinguish three cases. First, if 0 then 0 . Second, if has at least two members that belong to , . Third, if is the sole member of in , then 0 . It is routine to show that as defined from is a complete partial order on , with the least upper bound of . Now let be if 0 , and otherwise, and let Instead of regarding all the elements of 0 and their -completions as consistent, we can also pick any non-empty subset of 0 and let be the closure under part of . The result will then still be a C-space.
There is also a natural way to construct a C-space from a given set of atomic sentences (short: atoms) and a division of the atoms into positive and negative ones. Let be some fixed set of sentence letters, and let . The set of literals is . The converse of a literal is if is an atom, and if . For , a set is said to be -neutral iff neither nor are members of .
Let . Let -the set of positive literals -be , and let -the set of negative literals -be . Note that . For a literal, we let and be the positive and negative members of , respectively.  (2): From the fact that is -neutral for no .

Disjunction
We have given an account of the verifiers of P when P has just one verifier, in the form of various principles about the state-completion function . We now need to extend the account to the case in which P is disjunctive in the sense that it has several verifiers. What should we take the verifiers of P to be in this case? Unfortunately, it is much less clear what we should say here. Most paradigm uses of 'that's it' and its ilk seem to be applications to non-disjunctive statements. A typical disjunctive statement explicitly leaves open, to an extent, how matters stand: are they as described in the one disjunct, or are they as described in the other disjunct? As a result, when a disjunctive statement has been made, there is normally something more to be said, namely which disjunct is true. So there is typically a degree of oddity to following up an explicitly disjunctive statement by stating: and that's it. At any rate, the application of to disjunctive arguments raises distinctive issues of interpretation that do not come up when is applied to propositions with a single verifier.
I want to consider two natural approaches to the problem. One is to take the verifiers of P to be exactly the states of the form when verifies P. 18 On this view, distributes over disjunction in the sense that P Q and P Q have exactly the same verifiers. I will thefore call it the disjunctive conception of . P will then be true just in case some verifier of P has the actual world as its completion.
This view conforms to ( S). To see this, suppose P is strongly complete, so it is verified by @, and only by @. Let be a verifier of P. Then @ @ , and thus by the same reasoning as above, @ @. Conversely, suppose P is weakly complete, so all its verifiers are between @ and @. As before, for any such state , @, so P is verified only by @. The view does not conform to ( T), on which P might be read as saying that P is a weakly complete truth. For if P is a weakly complete truth with a single verifier, then P is a strongly complete truth and therefore true. But then P Q is also true, and by the distributivity principle, so is P Q . ( -T) would allow us to infer that P Q is weakly complete. But Q was arbitrary, and so might be verified, for example, by some small proper part of @ , rendering P Q clearly not weakly complete.
On another approach, we take P to be verified by the fusion of all states of the form when verifies P. On this view, distributes over disjunction in the sense that P Q and P Q have exactly the same verifiers. I will thefore call it the conjunctive conception of . P will then be true just in case every verifier of P has the actual world as its completion. This view conforms to ( T) as well as to ( S), and thus fits with a reading of P as saying that P is a weakly complete truth. For note that P will always have exactly one verifier. That verifier will be @ iff @ for every verifier of P and thus if and only if P is weakly complete. Otherwise it will be the fusion of several -states, and hence inconsistent, so P will be false.
Both views, I believe, correspond to useful and legitimate conceptions of a totality operator, suited for slightly different purposes. They agree on the paradigm applications of totality operators to non-disjunctive arguments, and differ drastically on the applications to disjunctive arguments. For disjunctive P, the conjunctive conception makes it almost impossible for P to be true, while the disjunctive conception makes it fairly easy for P to be true.
The conjunctive conception, as highlighted, fits a reading of as expressing the notion of weak completeness. 19 Especially in the kind of metaphysical contexts described in the introduction, that notion is of central interest, and so it is useful to have an operator expressing it, and worthwhile studying its logic. The disjunctive conception, on the other hand, seems to fit better than the conjunctive one with many ordinary language applications of 'that's it' to disjunctive arguments. Under this conception, P Q is true exactly when P or Q is true. Taken as an account of 'that's it', the view thus predicts that utterances of the form ' or , and that's it' will be felicitous just when ' , and that's it, or , and that's it' will sound felicitous (modulo the awkward repetitiveness of the second formulation). And that prediction seems to be borne out in many cases. Suppose I describe my breakfasting habits by saying: I always have porridge and tea or porridge and coffee, and that's it. Under the most natural reading of my statement, it is equivalent to, and sounds just as good 19 I would like here to thank an anonymous referee whose criticial comments on a previous version of the paper prompted me to formulate and develop the conjunctive conception in addition to the disjunctive one. The attractiveness of a conception of in line with ( T) is also emphasized by Kit Fine in his response to my [6]. Fine's proposal for such a notion is slightly different than mine; a comparison will have to wait for another occasion. as, the claim that I always have porridge and tea, and that's it, or porridge and coffee, and that's it. 20 Of course, such disjunctive answers do not always sound good. Suppose I am asked what I had for breakfast, and I answer that I had cereal or toast, and that's it. This answer seems less than ideal, and invites the question which it was. But that is just what we would expect also for the disjunction 'I had cereal, and that's it, or I had toast, and that's it'. It is normally safe to assume that the speaker will know which disjunct obtains, and so their answer is less informative than it could be and seems called for.
Relatedly, when the hypothesis that the speaker doesn't know which disjunct obtains seems more reasonable, disjunctive 'that's it' statements sound fine. Suppose Bob asks how much Bill had to drink last night. Bill responds: 'I just had two or three beers, and that's it!' Under the most natural interpretation, what Bill said is true iff he either had two beers, and that's it, or he had three beers, and that's it, and the disjunctive answer is fine assuming Bill isn't quite sure whether it was two or three. So there seems to be a systematic pattern in our uses of 'that's it' in application to disjunctive arguments which seems to conform very well to the disjunctive interpretation proposed for . It may be worth making explicit that the usage pattern here does not fit well with any interpretation of that sustains ( T). On such an interpretation, Bill's 'I had two or three beers, and that's it' is equivalent to 'That I had two or three beers is a weakly complete truth'. But that statement is incompatible with Bill having had three beers, since that would be a positive truth not entailed by the truth that he had two or three beers.
Admittedly, there are also some 'that's it' statements that are counted as true under the proposed account, which do sound quite bad. For a proposition may have a state between @ and @ among its verifiers without being particularly informative. The reason is that while exact truthmaking is subject to a relevance constraint, it is not subject to a minimality constraint, and even very large states can be wholly relevant to very weak propositions. For example, it is plausible that @ is among the verifiers of the proposition that something exists, or that something is the case. As a result, the claim that something exists, and that's it, comes out true, bizarre though it sounds. 20 In some contexts, it may be more natural to read ' or , and that's it' as also allowing for the possibility that and , and that's it. One way to accommodate this datum would be to adopt the so-called inclusive clause for disjunction, on which where is verified by the verifiers of , the verifiers of , and any fusion of such verifiers. However, there are also contexts in which it is more natural to read ' or , and that's it' as not allowing for the possibility that and . For instance, I might describe the very minimal breakfast enjoyed by another hotel guest by saying: he just had some coffee or tea, and that's it. (Assume I didn't see whether it was tea or coffee.) The most natural interpretation excludes the possibility that the person had both tea and coffee. So there are good reasons, in the present context, not to rely on the inclusive conception of disjunction across the board. (Note that some other natural language phenomena, to do especially with counterfactuals, seem also to speak against the inclusive conception; cf. [2] and [5].) A better view, then, seems to be that ' or ' is sometimes, but not always, used in accordance with the inclusive conception-equivalently, as shorthand for ' or or both'-and to regiment such utterances accordinly as ' '. On this way of accommodating the datum, we can stick to our original, non-inclusive clause for disjunction as well as the proposed clause for . Thanks here to an anonymous referee for discussion and the references.
In defence of the disjunctive account, we may note that it actually predicts that an utterance of this claim, though true, should seem bizarre. For on this account, P is true iff some verifier of P both obtains and contains every positive fact (pertaining to the subject matter). So what one does by applying to an argument P-or by appending 'and that's it'-is in effect to exclude any positive state that does not help verify P. But when P is a proposition that is partially verified by any positive state, no matter how big, then the application of can't serve its purpose, as it excludes nothing. So we should expect an application of 'that's it' to such statements to sound bizarre. 21

Falsifiers
We now turn to the question of the exact falsifiers of totality statements. An exact falsifier of a given statement needs to be 'big enough' to bring about that the statement in question is false, and 'small enough' so as not to include any parts irrelevant to the falsity of that statement. In addition, our account of the falsifiers needs to render totality statements exclusive and exhaustive. That is, we need it to be the case that every verifier is incompatible with every falsifier, and that every possible world contains either a verifier or a falsifier.
First of all, since P seems to relate to P much like a conjunct relates to a conjunction, it is very plausible to take any state falsifying P to also falsify P. But of course falsifying P cannot be the only way to falsify P, another way is to merely exclude what P says beyond P. Let us again begin by considering the simple case in which P has only a single verifier , so P likewise has just a single verifier . A natural suggestion is to then take any -state distinct from to falsify P. Any such state is of course incompatible with , so the condition of exclusivity is met, and it seems wholly relevant to making P false. Roughly speaking, tells us what the world is like in every single respect, and any distinct -state answers the same question, but in a different and therefore incompatible way. Moreover, including all these -states as falsifiers also ensures exhaustivity, since every world is either identical to or to some other -state. One might also plausibly regard some smaller states as falsifiers of P. For example, by invoking the notion of a positive state that I define below, one might take any positive state not contained in also to falsify P. It turns out, however, that such an extension of the set of falsifiers does not change the propositional logic of our totality operators, given the very plausible assumption that any falsifier of P is either a falsifier of P or part of some -state distinct from . For present purposes, it is therefore more convenient to stick to the simpler proposal.
It remains to extend our account to the general case with multiple verifiers for P. Recall that in general, the falsifiers of a conjunction are obtained by taking the disjunction of the falsifier-sets of the conjuncts, and the falsifiers of a disjunction are obtained by taking the conjunction of the falsifier-sets of the disjuncts. So a natural strategy is to apply the same idea to the conjunctive and disjunctive interpretations of as well. Call a state an incompletion of a state iff is a -state distinct from . Let be the set of 's incompletions, and let us also write for the singleton set of 's completion . Then we define the disjunctive and conjunctive -operations as follows: Definition 10 Let P be a bilateral proposition on a C-space . Then P P P P P P P P Lemma 3 Let P be an exclusive and exhaustive bilateral proposition on a C-space . Then P and P are also exclusive and exhaustive.
Proof Exclusivity of P: Let verify P and let falsify P. If is inconsistent, then is trivially incompatible with . If is consistent, for every verifying P. If falsifies P, by exclusivity of P, is incompatible with , and hence with . Otherwise, is an incompletion of , and hence a -state distinct from , and therefore again incompatible with .
Exhaustivity of P: Let be a world. Suppose does not contain a falsifier of P. So is not an incompletion of any verifier of P. Since is a -state, it follows that for every verifier of P, and hence that verifies P. Exclusivity of P: Let verify P and let falsify P. Then for some verifying P. If falsifies P, by exclusivity of P, is incompatible with and hence with . If does not falsify P, is either inconsistent or an incompletion of , so again, is incompatible with .
Exhaustivity of P: Let be a world and suppose does not contain a verifier of P. Then is an incompletion of every verifier of P, and hence a falsifier of P.
Exclusivity and exhaustivity ensure that negation has the classical modal profile when applied to . The central remaining question for our logics of totality is how negated -statements behave within the hyperintensional context of another occurrence of , i.e. under what conditions might be true. Under the disjunctive interpretation, this will be so exactly when is true. For if is true, then it is verified by-perhaps among other things-the actual world, and then so is . Note that this would still be the case if we were to add further falsifiers to . Under the conjunctive interpretation on the other hand, it will almost be impossible for to be true. Specifically, is false whenever there is at least one world in our state-space which is non-empty in the sense that . For in that case, there are at least three -states-, , and -of which at most one will verify . So at least two -states will verify , and hence will be verified by a fusion of distinct -states, which is bound to be inconsistent. Again, the point is stable under the addition of more falsifiers.
In models with just two -states, and therefore just one, empty world @, holds iff does. For suppose holds. Then is verified only by parts of @, and since every verifier of verifies , it follows that is verified only by parts of @. Since @ is empty, any such state has @ as its completion, so holds. Conversely, if holds, is verified only by parts of @, so is verified only by states not contained in @, which will be inconsistent. So will be verified by the -state distinct from @, which must be . Then any falsifier of , and any part of any incompletion of a verifier of , and therefore any verifier of , is a part of @. So is verified solely by @.

Positivity
Our informal reasoning about the behaviour of the completion operation has been strongly informed by the idea that goes beyond in excluding the obtaining of any positive state not contained in . Let us consider the relation between the notions in a bit more detail. It turns out that there is a plausible way to define a notion of a state's being positive in terms of and the mereology of the state-space. First of all, it is plausible that -Identity(2) may be strengthened in the following way. Say that states and are -equivalent iff . Now let be a set ofequivalent states. Then -Identity(2) implies that if is finite, then for all . But surely the principle is just as plausible in the case of an infinite set . So suppose it holds in general. Then for any state , the state will be the smallest state -equivalent to . We may call that state the -core of and denote it as ; note that always . Now in terms of the notion of a -core, we can plausibly define the notion of the positive part of a world . For it seems clear that this will always simply thecore of the world in question, so that . From this definition, we can then straightforwardly derive the two assumptions about @ we used in order to establish ( S): that @ @ and that @ whenever @. Defining the monadic notion of a wholly positive state is a little more tricky. Call a state -minimal iff it is its own -core, i.e. . Roughly, a -minimal state contains no parts that are rendundant in the sense that if one were to remove them from , they would be added back in when we form the completion of the result. At first glance, it is tempting to suppose that all and only the wholly positive states will be -minimal. It is clear, firstly, that a wholly positive state must be -minimal. This is because if a state is not -minimal, it must have a proper part with , and so some part of must be added to in forming its completion . But since nothing positive may be added to a state in forming its completion, it follows that is not wholly positive.
But the converse is not plausible. For given some wholly positive state , we may plausibly extend it by a negative state that excludes some parts of . Call the resulting state . Then should be -minimal. For suppose is a proper part of . If lacks some positive part of , then clearly and are -inequivalent. And if only lacks some negative part of , then that part will exclude part of , and therefore not be added back in upon forming the completion . So again and must be -inequivalent.
So -minimality is not sufficient for positivity. However, it is plausible that any state that is not wholly positive, although possibly -minimal, will have a part that is not -minimal. For instance, in the case of , the result of removing from any positive part that is excluded by will not be -minimal. A wholly positive state, on the other hand, will not only itself be -minimal, but will also have exclusively -minimal parts. So we may plausibly maintain that: (Df. -Positivity) A state is wholly positive iff for all It follows from this definition that any part of a wholly positive state is itself wholly positive, as one would expect. 22

The Logics of Totality
In our choice of language, we largely follow the lead of [8] and extend a standard propositional language by a one-place sentential operator 23 to express the target concept of totality. More precisely, we take as given a set of (non-logical) sentence letters or atoms. The formulas of our language of totality are: the members of , a logical constant for triviality , as well as , , , and whenever and are formulas. We abbreviate by . The point of including a special purpose triviality constant among the formulas is to enable us to express within that the world is empty. For will be interpreted as verified only by the nullstate (and falsified only by the fullstate), so will be verified only by , the one wholly negative world (if consistent). 24 Definition 11 A model is a tuple @ where is a C-space, @ is a world, and maps each member of to an exclusive and exhaustive bilateral proposition.
Given a model , we define extensions of to all formulas of , one for the conjunctive and one for the disjunctive conception of , written with a or a as subscript. For an atom and and any formula of , and and 22 It is also very plausible to hold that the fusion of some wholly positive states must be wholly positive as well. As it stands, this is not guaranteed by our definition and our constraints on C-spaces. As we shall see, requiring C-spaces to satisfy this additional condition makes no difference to the logic, so I have opted against it. 23 Leuenberger, naturally enough, uses . I prefer because of the relationship to the state-completion function (for which would have been unfortunate, since etc. are standardly used as variables ranging over states). 24 It would not serve our purposes to introduce as an abbreviation of an arbitrary tautology, since these, although necessarily true, are not in general verified by the nullstate. To specify adequate deductive systems we proceed in two steps. First, we specify systems for establishing when two formulas and will be interchangeable in the scope of due to their logical form. For disjunctive , we can use a version of the system shown in [7] to prove exactly when and have the same truthmakers in every model, given that neither closure nor convexity is assumed. 25
For conjunctive , we have to slightly strengthen this system. Recall that under the conjunctive conception, is true iff every verifier of is between @ and @. This condition is satisfied by the set of verifiers of iff it is satisfied by that set's closure under non-empty fusions. As a result, and are interchangeable in 25 Krämer [7] uses a language that does not include the logical constant , so the logic does not have the axioms Collapse( ) and Collapse( ). The soundness of these axioms is readily verified. The completeness proof in [7] is by disjunctive normal forms, and may be adapted to the present setting by adjusting the relevant notion of normal form, and simplifying conjunctions of literals to if is among them, and removing all occurrences of as a literal otherwise.
the context of conjunctive . So for that context, we replace ECollapse( ) by the stronger Collapse( ) We write iff and are formulas of for which can be derived using the resulting set of axioms and rules.
Next, we turn to derivations within . For the logic of conjunctive , we extend any standard axiomatization of classical propositional logic by the instances of 26 and write iff is derivable from in the resulting system. We prove in the appendix that this system is sound and complete with respect to , so that iff . The final two axioms require some explanation. As we noted, says that the actual world of the model is empty, so that @ . functions as a strengthening of this; it holds iff the actual world is empty, and the only -state aside from @ is . So -NE1 says that the negation of a -claim can only form part of a weakly complete truth in this sort of model, and -NE2 says that in such a model, is a weakly complete truth iff is. Note that by instantiating -NE2 with and using the fact that , we can derive . For the logic of disjunctive , we extend a classical propositional logic by the instances of -EquivD whenever -Fact -PFix -DisjD -AbsT -NFix and write iff is derivable from in the resulting system. We prove in the appendix that this system is sound and complete with respect to , so that iff . 27 26 Strictly speaking, our language does not include the material conditional or bi-conditional, but we can regard them as meta-linguistic abbreviations in the usual way. Note that while the various logically equivalent candidates for formulas to abbreviate using and may differ with regard to their truthmakers, since the conditionals do not appear within the scope of in our axioms, these hyperintensional differences do not lead to a difference in the logic. 27 If one wished to endorse Collapse( ) even in the context of the logic for disjunctive , one would simply need to replace -EquivD by -EquivC and modify -DisjD to read . Disjunction would then need to be given the inclusive interpretation described above, and we would impose a general constraint to the effect that every unilateral proposition by closed under non-empty fusions.

Comparison
We now turn to the comparison between our hyperintensional logics of totality and Leuenberger's intensional one. As one might expect, there is significant overlap, but there are also significant differences, resulting from our hyperintensional orientation and, relatedly, our conjunctively or disjunctively distributive interpretation of with respect to disjunctive arguments.
Following the suggestion in [1], Leuenberger takes to be true at a world iff is true at , but at no world outstripped by . In other words, satisfies iff is a minimal -world. The conditions under which is true at a world under our two semantics can also be formulated partially in terms of outstripping and minimality, where a world is taken to outstrip another iff . We can bring out the relationship between the accounts as follows, ordering them by increasing strength of the satisfaction condition for .
1. Under the disjunctive interpretation, satisfies iff for some state verifying , is a minimal -world.

Under the intensional interpretation, satisfies
iff is minimal among the worlds containing some verifier of . 3. Under the conjunctive interpretation, satisfies iff for every state verifying , is a minimal -world.
This difference is reflected in the logical behaviour of with respect to disjunction. As we observed, under the disjunctive interpretation, we have , and under the conjunctive interpretation we have . The intensional interpretation is strictly in between, we have , but not the converse, and we have , but not the converse. Indeed, it is clear that the converses are unacceptable under any intensional approach. For suppose . Then firstly, , so if we can infer from this, as under our disjunctive interpretation, that , then by intensionality it would follow that . And secondly, by intensionality we can infer from that , from which we had better not be able to infer , as we could under the conjunctive interpretation. What we can infer, under Leuenberger's account, from , is and . So under this account, every true disjunct of a total truth is itself total (where is called a total truth iff holds). It is interesting also to consider under what conditions may be inferred from . Under the conjunctive interpretation, we may do so only if . Under the disjunctive interpretation, we may always do so. Under the intensional account, we may do so if, and only if, for every -minimal world , no world outstripped by makes true , i.e. if cannot be made true with strictly less material, as it were, than is available in . This will hold when requires more than is available in , but also when just requires something else than is available in . For example, suppose that is a physicalist world, and that states all the physical facts in , so is true at . Now let say that there are angels. Then will also be true at , since any -world will contain angels, and no such world is outstripped by .
I do not think that these observations constitute a compelling objection to Leuenberger's account. All three accounts constitute, from their respective semantic points of view, natural ways to handle disjunction in the scope of . Something of an advantage may perhaps be claimed for the hyperintensional approaches in that the disjunctive reading does seem to align with a natural use of 'that's it' in some ordinary contexts, while the conjunctive one fits a reading of 'that's it' as expressing weak completeness. As the angels-example just presented suggests, it is not clear that there is any similarly intuitive reading that is tracked by the specific conditions under which, on the intensional account, we may infer from . But since it is not Leuenberger's aim to track the ordinary usage of 'that's it', or to characterize an operator expressing weak completeness, that is not by itself an objection to his account.
Let us consider the overall logic obtained under the intensional account. Leuenberger first presents a base system , which he proves sound and complete relative to the class of all totality frames, which consist of a set of worlds and some binary relation of outstripping , with being true at a world iff is true at and at no world outstripped by . The system consists of the propositional tautologies, the rule of modus ponens, as well as the following axioms and rules distinctively concerned with : The intended relation of outstripping, much like the relation of proper part, is clearly a strict partial order, i.e. transitive and asymmetric. Leuenberger shows that is not complete with respect to the class of partial order frames, but that a sound and complete axiomatization of the class of the partial order frames, and indeed the wider class of transitive frames, is obtained by adding to the somewhat hard to interpret axiom schema (A3) He calls the resulting system 3. (A1) and (A2) are of course valid in both our logics. I shall set aside the 'transitivity axiom' (A3). As Leuenberger highlights, its connection to the transitivity of outstripping is not exactly transparent, as a result of the fact that is related to outstripping in a tight but somewhat complicated way. Moreover, to make the connection to transitivity, (A3) exploits the distinctive behaviour of Leuenberger's with respect to disjunctive arguments, and for this reason it does not say in our contexts quite what it says in the intensional context. In particular, it does not appear to be tied in any clear way to the transitivity of outstripping, which is guaranteed under the proposed definition of outstripping as . (RIM)-the Rule of Inverse Monotonicity-is valid in neither. Nor should it be; it is unacceptable given our general hyperintensional orientation regarding . For (RIM) says that any truth that logically entails a total truth is itself a total truth, which means that in constructing a total truth, logical consequences come for free, contrary to our hyperintensionalism.
In addition to the stated axioms and rules, Leuenberger considers, but does not endorse, some additional axioms corresponding to the existence of unrestrictedly minimal worlds and related constraints. They are: In effect, within the context of Leuenberger's semantics, (A4) says that no world is minimal, (A5) says that every world outstrips a world which itself outstrips a world, and (A6) says that every non-minimal world outstrips a world that outstrips a world.
Since we are giving a non-disjunctive interpretation, (A4)'s status is the same under both our accounts, and it says something close to what it says in the intensional setting: that the actual world is not null. Since any world in a C-space is eligible to be the actual world in a model, the validity of (A4) comes down to the condition on C-spaces that . We might impose such a constraint, but, like Leuenberger, I see no clear reason to exclude the empty world. (Such an exclusion seems especially questionable when we consider applications in which worlds merely represent complete ways for things to stand with respect to the possibly restricted subject matter of the language under consideration.) It is worth noting that we can use (A4) to exclude the empty world, without thereby prohibiting minimal worlds. Let be a set of atoms and consider the canonical Cspace based on in which exactly the atoms are regarded as positive literals. We may exclude the empty world by regarding the set of all negative literals as inconsistent, but then have many minimal worlds, namely the worlds with exactly one atom as member. So this is one way in which our framework allows us to express somewhat more fine-grained distinctions than the intensional one regarding the structure of the space of worlds in terms of outstripping and positivity.
(A5) and (A6) concern the potential of negated -claims to be total truths, so both their status and what they say about minimal worlds differs strongly between our two approaches. Under the conjunctive interpretation, (A5) is invalid, but valid in the class of models with more than two -states, and hence in the class of models in which some world is non-empty. (A6) is valid, and registers the fact that the negation of a -claim can only be a weakly complete truth if the actual world is empty.
Under the disjunctive interpretation, (A5) and (A6) are invalid, and indeed holds whenever . This is another instance in which the different orientation towards disjunction makes comparison between the accounts difficult. For us, the fact that @ is an exact falsifier of any given false statement means that must then be true. Now, as we have observed, will then not normally be the complete truth, but we can even give an example of a C-space without minimal worlds and a selection of a world as actual so that a proposition of the form is a complete truth. 28 28 Let be the set of natural numbers plus , and let be , so is the nullstate, and there is no minimal non-null state. Now modify the simple C-space as defined above simply by letting be inconsistent, so that while there is a minimal -state, there is no minimal world. Let @ 0 and let P be verified by all and only the proper parts of @, and falsified by @ and only @. Then @ is the only -state not verifying P, and therefore verifying P, and so @ will be the sole verifier of P.
Having discussed how the principle Leuenberger considers fare within our system, let us briefly turn to the converse question of how our axioms fare within Leuenberger's logics. We have already seen that the disjunction-rules -DisjC and -DisjD and the negated--rules -N and -NFix are not valid and why. The remaining rules are readily seen to hold within Leuenberger's base system .
And that's it.

Appendix A: Soundness
Theorem 4 Let and . Then implies .
Proof The soundness of classical propositional logic is easily established using the fact that @ is a world, and that any formula of is assigned an exclusive and exhaustive proposition. It remains to show that our additional axioms are true in every model.
The soundness of -EquivC follows from the soundness for for the interpretation of as true iff the sets of verifiers of and have the same closure under (non-empty) fusions and the fact that the truth-value of and depends only on the closure under fusion of the verifier-sets of and .
For -Fact and -PFix, assume is true in a model, so some part of @ is the sole verifier of . By the clause for , some -state is part of . But the only -state contained in @ is @, so @ is the sole verifier of . By the clause for again, has a verifier and every verifier of is part of @, so is true in the model, and @ is also the sole verifier of , so is also true in the model. For -DisjC, note that is true in a model iff @ is the sole verifier of , which holds iff @ is the fusion of all where verifies . This is so just in case @ for all verifying and hence iff @ for all verifying and for all verifying , so iff @ is the sole verifier of both and , and hence of their conjunction, so iff is true in the model. For -AbsP, suppose is true in a model and hence verified solely by @. Then the same holds for both and . It follows that every verifier of , and every verifier of has @ as its completion, and so that every verifier of is part of @. But then using -Identity(1), we may infer that every verifier of has @ as its completion, and thus that is verified by @ and hence true in our model.
For -NE1, suppose is true in a model. Then is verified only by parts of @. Since it is verified by any incompletion of any verifier of , @ must be the only incompletion of a verifier of . So every -state other than @ must be the completion of every verifier of , so there can only be two -states. Since is astate distinct from @, it follows that @ and are the only -states. By -Identity(1), cannot be , and so must be @. Then is falsified by the one incompletion of , which will be @, and by , the falsifier of . Both have @ as their completion, so is verified by @, and hence true in the model. 29 For -NE2, suppose is true in a model. By the same reasoning as before, the model can only have two -states, with @ . Now suppose is true in the model, so every falsifier of is part of @. Then every verifier of has as its completion, so @ is the only incompletion of any verifier of . It follows that is verified only by parts of @, and thus by states that have @ as their completion, so is true in the model. Conversely, suppose is true in the model. Then all verifiers of , and hence all falsifiers of , must be parts of @. But then they all have @ as their completion, so is true in the model. is true, and so verified by some part of @. Since falsifiers of are all -states, @. Since @ is a world, @ @, and so @ verifies . 30 For -AbsT, suppose part of @ verifies . Then for some and verifying and , respectively. Since @ is consistent, is consistent, and so is consistent, and therefore a world. And since @, it follows that @, und @. Then also @ and hence @. So @ , and hence @.

Appendix B: Normal Forms
We establish completeness via normal forms. For each logic, we prove a reduction theorem to the effect that any formula of is provably equivalent to a formula in a kind of disjunctive normal form.
For these purposes, we count as atoms the members of , , as well assomewhat unusually-the formulas and . Any atom and any negation of an atom is a literal; the members of and their negations are the non-logical literals and their set is denoted by . We use the variables as well as variously adorned versions of them to range over the literals of . The converse of a 29 Note that will still be true if we countenance more falsifiers of -statements, as long as these are all parts of an incompletion of a verifier of the argument. For in the present case, this will mean that these added falsifiers are still parts of @, and hence have @ as their completion. 30 Note that we do not require the assumption that only -states falsify to obtain the result; it is enough that some state with @ falsifies . So as long as @, or at least its positive part, is included among the falsifiers of any given false , -NFix will be sound.
literal is if is an atom, and if . Formulas of the form 1 and their negations will be called -literals. A is any disjunction of conjunctions in which each conjunct is either a literal or a -literal.
We will show that under either logic, every formula of is provably equivalent to a -DNF. To do this, we classify formulas of in accordance with their maximum nesting depth of . For each , we define a set of -atoms, -literals, and -formulas. The 0-atoms are the atoms, the 0-literals are the 0-atoms and their negations, and the 0-formulas are all the formulas built up from 0-atoms without the use of . The 1 -atoms are the -atoms as well as all expressions of the form , where is an -formula. The 1 -literals are the 1-atoms and their negations, and the 1 -formulas are all the formulas built up from 1 -atoms without the use of . A conjunction of -literals will be called an -CF, and a disjunction of -CFs will be called an -DNF. The degree of a formula is the smallest for which is an -formula.

B.1 Conjunctive
Throughout this subsection, reference to provability is to be understood in terms of , and reference to provable exact equivalence in terms of . We first establish some useful facts in preparation of the reduction theorem.

Lemma 4
1. Any -formula is provably exactly equivalent to an -DNF.

For an -formula,
is provably equivalent to a formula , where is an -DNF.
is provably equivalent to . 5. If is an -CF of degree 0, then is provably equivalent to someformula. 6. For 0, if is a formula of degree , then is provably equivalent to some -formula.
(3): By -Fact, -PFix, and -AbsP. (4): Left-to-right: By -NE1, from we infer , from which we obtain, using -NE1, . Using and -AbsP, we can derive every instance of which allows us to infer both and from . Using -NE2 again, we may infer . Right-to-left: Using -NE2, we may infer from and . Using -AbsP and , we obtain . (5): Suppose is an -CF of degree 0. So is a conjunction of -literals 1 , at least one of which is of degree . Let be the number of degree literals among 1 . We may then use an 'extraction' procedure to obtain a formula provably equivalent to , which is a conjunction of an -formula and a sentence with an -CF with 1 degree literals among its conjuncts. So by repeated application of that procedure, we obtain a sentence provably equivalent to which is a conjunction of -formulas, and hence itself an -formula. The extraction procedure is as follows. First, we pick one of the degree literals among 1 , call it 1 , and turn into a provably exactly equivalent -CF of degree 1 . Then is provably equivalent to . 1  . If it is of degree , we do nothing. If it is of degree , then by (3) it may be replaced by a provably equivalent -formula. In this way we obtain an -formula provably equivalent to , as desired.

Theorem 6 Any formula is provably equivalent within to a -DNF.
Proof The case in which is of degree 0 is trivial. Suppose that is degree 1, and consider any degree 1 atom in . This will be of the form , with a 0-formula. By Lemma 6 (2), there is a 0-DNF with provably equivalent to . By -DisjC, in turn is provably equivalent to a conjunction of -literals. So we may replace any degree 1 atom in by a provably equivalent conjunction of -literals. The result is provably equivalent to , and, by propositional logic, provably equivalent to a -DNF. Now suppose is degree 1 with 0, and suppose the claim holds for all formulas up to degree (IH). Consider any degree 1 atom in , which will be of the form , with a degree -formula. Then by Lemma 6 (6), is provably equivalent to some -formula. We may thus replace each degree 1 atom in by a provably equivalent -formula. The result will itself be an -formula provably equivalent to . By IH, that formula, and therefore , is provably equivalent to a -DNF.

B.2 Disjunctive
Throughout this subsection, reference to provability is to be understood in terms of , and reference to provable exact equivalence in terms of .

Lemma 5
1. Any -formula is provably exactly equivalent to an -DNF.

2.
For an -formula, is provably equivalent to a formula , where is an -DNF.

3.
is provably equivalent to .

4.
is provably equivalent to . 5. If is an -CF of degree 0, then is provably equivalent to . 6. For 0, if is a formula of degree , then is provably equivalent to some -formula.
Proof (1) and (2) are as before. (3) and (4) may be established using -Fact, -AbsT, and -PFix or -NFix, respectively. (5): has at least one conjunct of degree 0, i.e. an -literal of the form or . In the former case, is provably exactly equivalent to some conjunction of the form . So by -EquivD, is provably equivalent to , which by (3) is provably equivalent to , which is provably equivalent to . In the latter case, is provably exactly equivalent to some conjunction of the form . So by -EquivD, is provably equivalent to , which by (4) is provably equivalent to , which is provably equivalent to . (6): By (2), is provably equivalent to for some -DNF 1 . Then by -DisjD, is provably equivalent to 1 . Now consider each disjunct . If it is of degree , we do nothing. If it is of degree , then we replace it by . Since is an -CF, by (5), is provably equivalent to . So through these replacements we obtain an -formula provably equivalent to , as desired.

Theorem 7 Any formula is provably equivalent within
to a -DNF. Proof The reasoning is as for Theorem 7, using -DisjD in place of -DisjC and Lemma 8 (6) in place of Lemma 6 (6).

Definition 13 Let and a 0-CF in . Then
is complete iff is minimally complete iff is complete, and the result of replacing any conjunct of by is not complete is the set of non-logical literals occurring as a conjunct in is actual iff is pure iff is a conjunct in some complete 0-CF if is actual: is positive iff either there is no complete 0-CF, or is a conjunct in some minimally complete 0-CF; is negative otherwise if is non-actual: is positive iff its actual converse is negative; is negative otherwise is the negative member of We write @ for the set of actual non-logical literals and @ for the set of positive members of @. Note that @ is empty just in case , in which case there is a complete 0-CF, but no minimally complete 0-CF. We denote the set of all (nonlogical) 31 literals by (since we now regard them as proto-states). For , we say that is minimally complete iff some conjunction of all its members is, and we let be if no subset of is minimally complete, and @ otherwise. Letthe set of states-be . (The point of excluding from the set of states those sets of literals which have a minimally complete subset but do not include @ is that this will make it easy to ensure that all minimally complete 0-CFs are verified by @ .)

Definition 14 The canonical state-space for is with
In preparation of the proof that is indeed a state-space, we note some useful facts about and .
: @ or has no minimally complete subset 6.
is closed under intersection 7. If are minimally complete and has at least two members, then is not closed under union 8. Proof (1) and (2) are obvious from the definition of .
(3) and (4) are immediate from the fact that extends at most by members of @ , and @ @. (5): It is immediate from the definition of that always includes @ or has no minimally complete subset, and that is equal to and hence a member of whenever includes @ or has no minimally complete subset. (6): Let . Either some member of has no minimally complete subsets, in which case the same is true of , or all members of have @ as a subset, in which case again the same is true of . (7): Let . Then and have no minimally complete subset, but their union does. Since is also minimally complete and distinct from , @ . So and are members of but their union is not. . Suppose now that at least one has a minimally complete subset. Then @ @ . But then also has a minimally complete subset, so also @ .

Lemma 7
The canonical state-space for , , is a state-space.
Proof From the fact that the subset-order on any set is a partial order together with Lemma 10 (9).
Next, say that a member of is consistent iff there is no literal such that and let is consistent . Say that decides a given literal iff either or is a member of , and let and does not decide .

Lemma 8 is a C-space
Proof It is clear that is non-empty and closed under part. Next, we show that is a world if is consistent. So suppose is consistent. Then clearly is also consistent. Now let be any state not contained in . Let be a literal which is in but not in . Suppose for reductio that . Then does not decide , and so . But is either or . Contradiction. So and hence is incompatible with . It follows that is a world. Moreover, -Containment is immediate from the definition of . So we may conclude that every consistent state is part of a world-and hence the conditions on a W-space are satisfied-, and that -Completeness holds. It is straightforward to verify that @ is a world in this C-space, that @ iff @ @ and that @ is the positive part of @ under the definition proposed in Section 4.5. Moreover, a state is wholly positive under the definition from that section just in case it does not contain for any , so the set of wholly positive states is closed under fusion, as one would expect.
We now define the interpretation function for the atoms; it is straightforward that it satisfies the conditions of exclusivity and exhaustivity. . Then is complete, so every non-logical literal occurring as a conjunct in is pure, and therefore is verified only by . By Lemma 10 (10) and the clause for conjunction, the sole verifier of the conjunction of non-logical conjuncts of is . Note that since , the only logical literals that can occur in are and . Suppose first that , so @ , and is verified only by @. Then . By -Fact and conjunction elimination, every literal in is in , so @. Since is verified only by and only by @, it follows that the sole verifier of is part of @, and hence is verified by @. Suppose now that . Then also does not occur as a conjunct in , and so the sole verifier of is . Since is complete but is not complete, some subset of is minimally complete, so @ . By -Fact and conjunction elimination, every literal in is in , so @. It follows that @ @, and since is the sole verifier of , that @ verifies . So .
. By the falsifier clause for , @ will be a verifier of provided that @ is not the sole verifier of . So suppose for contradiction that @ is the sole verifier of , and hence that every verifier of is between @ and @. It follows that no literal that occurs as a conjunct in has a non-actual verifier. So every non-logical literal is pure, and hence occurs in some complete 0-CF , so that . Any such is then verified only by , so by Lemma 10 (10), the sole verifier of their conjunction is . Moreover, the only logical literals possibly occurring in are and , and the latter can occur in only if it is in , since it has non-actual verifiers otherwise.
So suppose first that . Then whenever is a conjunction solely of and . So must include some non-logical literals, which must then be pure. But then using the for all non-logical literals in , using -AbsP and the assumption , we can derive , contrary to the fact that is consistent and . Suppose now that . Then is the sole verifier of , and hence @ , so some subset of is minimally complete. Then for some conjunction with , . By -Equiv and -AbsP, from and for each non-logical we can again derive , contrary to the consistency of . So @ is not the sole verifier of and hence @ verifies , and so .
We now construct a very simple, 'empty' model for the case in which does include . Let 0 1 and 0 , @ 0, 0 0 and 1 1, and let be the partial order on in which 0 1. It is readily verified that is a C-space. For , we let 0 if is pure, 1 if , and 0 1 otherwise, and 0 if is pure, 1 if , and 0 1 otherwise. Again, exclusivity and exhaustivity are straightforward.

Lemma 12
Let be the canonical empty model, any literal, and any 0-CF. If , then if .
Proof (1): Let . If is a non-logical literal, then by definition of , is verified by 0 @, so . If is a logical literal then is one of , , and . It is readily verified that in either case, is again verified by 0, and indeed only by 0. So again, . . Then since , it follows that some literal in is not in or not pure. Either way, it is verified by 1, and hence so is . Since 0 is an incompletion of 1, it follows that is verified by 0, and hence .

If
, then for all .
Proof Suppose . By Theorem 7, is provably equivalent to some -DNF . Since is maximal consistent, some disjunct of is also in . Since is a -DNF, is a conjunction each conjunct of which is a literal or a -literal and also a member of . By Lemmas 15 and 16, every such conjunct, and hence their conjunction , and thus also , is true in the relevant model. By soundness, it follows that so is .
From this lemma, completeness follows straightforwardly.

Theorem 8 Let and
. Then implies .

C.2 Disjunctive
Let be a subset of that is maximal consistent with respect to . Again, we begin by defining some important notions in terms of .

Definition 17
Let and a 0-CF in . Then is complete iff is minimally complete iff is complete, and the result of replacing any conjunct of by is not complete is the number of times occurs as a conjunct in is max is minimally complete With each (non-logical) literal , we associate countably many indexed literals (short: i-literals) ( ), and with each indexed literal we associate a unique shadow . Roughly, the various will correspond to different ways for to be true, and acts as a negation of . We countenance an infinite number of ways for each literal to be true in order to prevent the sets of verifiers and falsifiers from being closed under fusion. The need for negations of (some of) these will become clear later on.
Definition 18 Let and . Then is the sentence letter on which is based is if and otherwise is the unique member of is actual iff and either 0 or We let and be: 0 if ; 0 if 0 and 0 for some ; 0 otherwise is negative iff and positive otherwise Intuitively, we may think of as the true, or obtaining state, and of as the negative state w.r.t. . We are taking 0 to be negative iff the actual world is empty according to , or some non-logical literals occur in minimally complete 0-CFs, but is not one of them. (Note that the actual world may be non-empty while no nonlogical literals occur in minimally complete 0-CFs, because it would take an infinite 0-CF to exhaust the positive part of the world.) Then is non-actual if the truth w.r.t. is positive, and actual otherwise. We denote the set of actual indexed literals by @, and the set of actual and positive (short: a-positive) ones by @ . Note that @ is empty iff . Next, with any 0-CF including only non-logical conjuncts, we associate a unique matching set of indexed literals . Note that iff and correspond to the same multi-set of literals, i.e. and contain the same conjuncts, and they contain them the same number of times, though possibly in a different order. Moreover, say that a 0-CF is included in a 0-CF iff each literal occurring in as a conjunct occurs at least as many times as a conjunct in . Then implies that is included in . We call a set of i-literals minimally complete iff for some minimally complete conjunction of non-logical literals . Now suppose is such a conjunction. Suppose , so occurs at least 1 times in . Then , so @ . So @ whenever is a minimally complete conjunction of non-logical conjuncts.
We construct the set of states similarly as before, but using as the set of protostates not the set of literals, but the set of the i-literals together with the shadows of the a-positive i-literals. Then for , let if no subset of is minimally complete, and @ otherwise, and let be .

Definition 19
The canonical state-space for is with Since is defined from just as before, the Lemma 10 and its proof carry over unchanged.
: @ or has no minimally complete subset 6.
is closed under intersection 7. Proof From the fact that the subset-order on any set is a partial order together with Lemma 19 (9).
To extend to a canonical C-space, we need to define a set of consistent states and the -function, both of which is slightly more complicated than before.

Definition 20 Let
, and set . Then and are co-literals iff , or , or and are incompatible iff they are (a) co-literals and (b) not both actual is consistent iff (a) no two members of are incompatible i-literals, and (b) no member of is the shadow of a member of is consistent Next, if is a literal, say that a state decides iff some co-literal of is a member of . Then to form the completion of any given state, we extend it by the negative state w.r.t. any literal it does not decide, as well as the shadows of any a-positive i-literal that are not members of the state.

Definition 21 For
, let does not decide @ We take note of a useful fact about the -function used in defining :

Lemma 16 For , if then
Proof Since decides no literal not decided by , and contains no member of @ not contained in , if .

Lemma 17 Let . Then is a world in .
Proof is a state: Since is always a non-positive i-literal, and no shadow is a positive i-literal, has a minimally complete subset iff does, and consequently is a state since is.
is consistent: Firstly, no shadow of any member of is a member of . For only a-positive i-literals have shadows, and the only a-positive i-literals in are already in . Since is consistent, it contains no shadows of its a-positive members, and beyond the states in , by definition contains only shadows of i-literals not in . Secondly, no pair of i-literals in is incompatible, for only co-literals are incompatible, and by construction, any pair of co-literals in is already in , which was assumed to be consistent.
contains every state it is compatible with: Suppose that is compatible with . First, suppose is a shadow in . Since is compatible with , , and hence , so . Now suppose is an an i-literal in . Suppose first that does not decide . Then . If , then and are distinct co-literals, and since is nonpositive, they are incompatible, contrary to our assumption that is compatible with . So and hence . So suppose instead does decide , so some member of is a co-literal of . If then . If , then since and are compatible, and are both a-positive. But then since is compatible with , , so by construction of we may infer that and hence . It follows that , as desired.
Lemma 23 For all , @ iff @ @ Proof For the left-to-right direction, suppose @. Then clearly @, so it suffices to show that @ . So suppose otherwise. Then since is a state, it follows that has no minimally complete subset. So let be an a-positive literal which is not a member of . Then by construction, , contrary to the assumption that @. For the right-to-left direction, assume @ @. We show first that @. Since @ , does not decide . So it suffices to show that @ whenever does not decide . But if does not decide , then 0 is not in , and hence not in @ , so is not a-positive, so 0 . But , so 0 @, as desired. Finally, we show that @ . Suppose @. If then clearly . So suppose . It follows that is not a-positive. Since is actual, it follows that 0, and that , so 0 . Since is not a-positive, 0 . So to show that , it suffices to show that does not decide . Suppose otherwise, so contains some co-literal of . Then , since by assumption . Since @, is actual, so . By consistency of , , and so 0. But by the definition of actuality it then follows that , and hence that , contrary not being a-positive. So does not decide , and hence .

Lemma 24 is a C-space
Proof From Lemmas 20 (9) and 22 it follows that is a W-space, -Containment: Immediate from the definitions of . is -neutral @ . Since is consistent, is a world, and since is a world and , we have . -Redundancy(2): It is easily verified that decides every literal and contains every positive i-literal contained in . From that observation, it is immediate that . -Identity(1): Suppose . We show first that . By assumption, . By Lemma 21, , so , and hence . We now show that . By assumption, . So let . Then either (a) with not decided by , or (b) with an a-positive i-literal not in . Suppose (a). Note that is the only co-literal of in . If is not decided by , then is also in . If is decided by , contains a co-literal of . But and is the only co-literal of in , so and hence . Suppose (b). Then , and by also , so again . -Identity(2): Suppose . We show first that . Any member of is either (a) a member of , or (b) with not decided by , or (c) with an a-positive i-literal not in . Suppose (a). Then and hence . Suppose (b). If either or also do not decide , then clearly . So suppose both and decide . Since does not, no co-literal of is a member of both and . Let be a co-literal of that is a member of but not . Since , . Since is neither in nor a shadow, it follows that , and