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Impossible Worlds and Content Inclusion
In this paper, I will argue that the relevant logicianIn general, a relevant logician is one which has requirements on the material conditional. They aim to eliminate paradoxes of the material conditional found in classical logic. The primary requirement is that the antecedent and consequent share some propositional parameter or variable. Then, for an interpretation v, vw(A B) = 1 iff for all x, y W such that vx(A) = 1, vy(B) = 1. This evaluation requires a ternary relation, R. This is a necessary condition for relevance logics. For more, see Priest (1992, 2008). is required to accept philosophically and metaphysically intractable positions in order to preserve the Law of Excluded Middle (LEM). I will do this by first explaining the interpretations of a relevant logic in terms of the ternary relation R, the worlds W, the Routley Star, and content inclusion. I will consider two possible understandings of worlds, one being in the sense used by classical modal logic, and another in terms of information states from authors like Restall. I will explain what the semantics of R in a relevant logic mean with respect to each of these concepts of worlds, and what the Routley Star and its characterization of star worlds offers to the relevant logician in each of these cases. Then, I will discuss what content inclusion means for both understandings of worlds and how it is required for LEM to be valid. All of these things come together in the interpretations I of a relevant logic, with these interpretations being some structure , with W being the set of all worlds and N the set of all normal worlds, R the ternary relation, ∗ the Routley Star operation on worlds in W, ⊑ being the binary operation on worlds called content inclusion, and ν as the interpretation function. Something is valid in a relevant logicI say “a relevant logic” here not because I am confused, but because there are no fewer than 10 relevant logics to be considered. The primary targets here are those like BX, DWX, and TWX, which accept LEM (see Priest (2008) page 203 for more information on their differences). with this structure just in case truth is preserved in normal worlds. That is to say, it is impossible for the premises to be true and the conclusion false in all normal worlds of our interpretations. Thus, in introducing the binary relation of content inclusion to keep LEM as a logical truth, the relevant logician is required to accept the claim that logically impossible worlds inform our understanding of logical truths in a normal world. That is to say, for LEM to be valid, the truths of logically impossible worlds
I will use “worlds” to include both senses of worlds outlined above, but will distinguish the two when necessary.
must be appealed to. The most distinguishing characteristic of a logic is its interpretation. Demonstrations of validity for the relevance logician will rely heavily on the relation in the interpretation. Thus, one of the most important aspects of a relevant logic is its ternary relation. There are two primary things that this relation can be between; that of worlds, or that of information states. The case in which the relation is between worlds is rather simple and is what one would usually expect. However, the relation changes slightly to accommodate additions to the interpretation like ∗ or ⊑. The ternary relation primarily changes the evaluation of the material conditional, and the change from a binary relation to that of a ternary one is mainly motivated by the requirement of relevance between the antecedent and the consequent in a material conditional (see footnote 1, above). Thus, R is impacted on how it bears out in evaluating the material conditional, and thus only matters in such cases with respect to normal worlds, and otherwise behaves as we would usually expect it to.
See Priest, (2008), page 190-1.
However, there is an alternative view to what this relation could be between. One such position is that of possible information states.
See Restall, (1999) for a more interesting discussion of states.
For instance, we can have what are called “states”, or ways in which the world could be. These states are information in parts of worlds, and behave as we may expect states of information to behave. Thus, the way in which a relevant logician could read the ternary relation R is as follows. Rxyz will hold when the information given by x (in a particular state), when considered with respect to another information state y, also holds in z. As Restall puts it, “you are to think of the information supported by x as ‘data’, and that supported by y as information to be applied to the data. If the results are all supported by z, then Rxyz holds”.
Restall, (1999) page 7.
This is the best that I can do for an intuitive reading on what R is. Although R resembles the modal relation due to Kripke and others in modal semantics, it is by no means the same and the two should not be confused.
See Berto (2007) for a more detailed discussion on R’s intuitive readings.
This discussion on R will be important later in discussing worlds, most notably for impossible ones. Let us now turn to W, the set of worlds in our interpretations. We have both normal and nonnormal worlds in this interpretation.
recall that our interpretation includes N, the set of all normal worlds, a subset of W. Nonnormal worlds are the worlds in W – N
Normal worlds are the worlds in which validity is determined and the conditional may be evaluated.
Using something called the ‘normality condition’ we can have R handle conditionals at all worlds, not just normal ones. See Priest (2008) page 189.
Our collection of worlds also include a curious set of counterparts to the worlds we usually have. These worlds are star (∗) worlds, and the ∗ operation is a function between worlds. This function produces a new understanding of negation. With this new operation, vw(¬p) = 1 if vw*(p) = 0 and is otherwise zero. Thus, for every world w in W, there is a star world (or mate world) called w*. These star worlds are of serious import to the relevant logician. For instance, to validate contraposition, the relation R is such that if Rww1w2, then Rww2*w1*. In addition, it is important that w** = w. This way, double negation is also preserved for the relevant logician. Here now we turn to the primary concern: what are these star worlds supposed to be? It’s certainly nice that they allow us to verify particular logical validities that we appreciate having in classical logic
Note that, classically, w* = w for all worlds.
, but we presumably decided that classical logic was getting it wrong at some point; the worry may be that these star worlds are too problematic if we want to be sure that relevant logic is getting it right in the ways classical logic wasn’t.
These star worlds offer an interesting turn on negation. Indeed, negation now becomes an intensional operator, and behaves much differently from how we expect it to behave as it does in the classical case. It does not on the face of it seem obvious as to why one might require star worlds in their semantics if they have nonnormal worlds. The primary issue one runs into with only including normal worlds in a relevant logic is that with negation. If negation is defined intrinsically for the world itself – that is to say, evaluated at the world in question – then the purpose of a relevant logic is ignored. For instance, if negation is evaluable at a world as being the complement of a statement’s truth value, A (B ∨ ¬B) is valid at that world.
This is pointed out by Priest (1992) regarding Routley. The important point here is that a relevant logic requires something important about implication, and negation will overwrite this importance if not treated carefully.
In essence, if the interpretation only considers negation intrinsically, the entire point of introducing nonnormal worlds is lost. That is to say, LEM could become a logical truth at even nonnormal worlds, but the entire purpose of nonnormal worlds was to introduce possibilities where laws of logic may fail. Graham Priest offers an explanation of the ∗ operator and why it belongs. He says that “∗ is, in fact, a device for ensuring that there are non-normal worlds that are inconsistent and in-complete, and so where certain “laws of logic” may fail”.
Priest (1992), page 298-9.
Thus, the ∗ operator provides a look into precisely the kinds of impossible worlds we wish to include. While possible worlds are in general not a problematic topic, these nonnormal and logically impossible worlds are of a strikingly distinct character than the usual worlds. What we are left with is the question of what to make of these nonnormal worlds, and determine how precisely they might differ from the actual world and other possible worlds. There are several different kinds of impossible worlds that have been proposed, each deserving its own short treatment and explanation.
The first kind of impossible world
Many thanks are due to Berto (2009) for explicitly underlining these various definitions of impossible worlds across different articles.
are those which are most analogous to possible worlds. Take possible worlds to be representations of sentences or propositions, or as states of affairs that could have been but are not. For instance, it could perhaps be physically possible that I am six inches taller. Thus, there is some world different from the actual world we inhabit in which I am such a height and not much else is different. Likewise, for impossible worlds, it might be that there are some things that simply cannot be the case. Impossible worlds would be those where states of affairs that cannot happen, happen. These things that can’t happen are impossibilities and, as such, populate the set of impossible worlds; these worlds are things which we would expect to be exactly not possible. The other three options for what an impossible world is are relative to a logic or a logical law instead of being an analogy to traditional possible worlds. For instance, an impossible world may be one in which the laws of logic are simply different than they are in possible worlds. That is to say, relative to some logic, an impossible world would be one in which the collection of truths in that world would not be true under that logic’s interpretations. For example, there may be some world at which A and A → C are true. However, at such an impossible world, it might not follow that C is true simply because modus ponens fails.
This way of thinking is perhaps similar to how other logics operate. For instance, the paraconsistent logician already knows what it is for modus ponens to fail. Why we should necessarily believe impossible worlds to be paraconsistent worlds is perhaps a matter of personal preference on the truth of paraconsistent logics.
Another case may be one in which impossible worlds are regarded as simply being worlds in which classical interpretations are different. This then would be a special case of the previous kind of impossible world, privileging the classical perspective in classifying these worlds. Finally, impossible worlds may just be worlds in which actual contradictions exist. For instance, violations of the law of noncontradiction may be common-place and accepted. These definitions are not necessarily exclusive nor exhaustive ones and many can be taken simultaneously. The real question might be why we should allow impossible worlds at all in the first place. Some authors
See Priest (2008) sec. 9.7 for an elucidation on this point.
argue that there is no good reason to rule out impossible worlds if one accepts possible worlds. If one is comfortable with the idea of physically impossible worlds where things travel faster than the speed of light, one should also consider those worlds in which logical laws are different.
See Priest (1992) for a similar and more detailed description of these impossible worlds.
These raise the question, why must the whole world itself be impossible? Why not just have parts of a world be impossible? This runs along the idea of our worlds not being real, existing things but instead states of information. As Restall puts it, “we can consider ways that parts of our world could be, and ways that parts of our world couldn’t be. All of these entities will be called ''states''” (emphasis his).
Restall (1999), page 4.
Thus, our impossible worlds can be thought of as states in which our information is incomplete or perhaps inconsistent. These states are ways in which worlds could be, and again there is perhaps no good reason to suspect that this could not be the case.
As Priest might say, while there are no states of information at which logical laws may not be the case, these are precisely the kinds of states where we might expect them to be false; namely, impossible ones. For a better understanding of impossible worlds, see Priest (1992).
It is at this point that we turn to the Law of Excluded Middle. LEM is not valid for any interpretations I that are structurally similar to the one I have explained thus far.
For reference, all interpretations I that are similar will be such that I =
What follows is the tableau for LEM:
A ∨ ¬A, -0 (1)
A, -0 (2)
¬A, -0 (3)
r00#0# (4)
A, +0# (5)
As you can see, we end up stuck at (5), being unable to infer anything about the truth of A in 0.
For those confused, (4) is required by the normality condition (as 0 is normal) and (5) follows from (3) and (4).
The only move we can make between worlds and their mate worlds is with negation itself. Without something like ¬A, -0# (which would yield A, +0 and close our branch), LEM is not a validity. The question thus becomes, what is missing from our interpretation to keep LEM valid? Here is where we introduce the so far overlooked relation of content inclusion. Content inclusion is a reflexive and transitive binary relation on worlds in W
These facts are not necessary for the demonstration of LEM. They are just useful to know.
and is a rather intuitive relation. We say that if w ⊑ w’, then everything that is true at w is true at w’.
Due to Priest (2008), we see that ⊑ satisfies the following constraints:
# If vw(p) = 1 then vw’(p) = 1 # w’* ⊑ w* # if Rw’w1w2 then (w N and w1 ⊑ w2) or (w N and Rww1w2)
Note that the identity relation is one which satisfies these conditions, and so our previous interpretations which were lacking content inclusion simply need identity and we have a similar structure.
As a result of this introduction, we have a new set of tableaux rules and ways of evaluating validity in our language.
See Priest (2008) and Read (1988) for an introduction and an in-depth discussion of the intuitive reading.
The relation of content inclusion allows us to make the crucial inference we needed before to demonstrate LEM’s validity. For instance, we can introduce the following constraint: if a is a normal world, then whatever is true in a’s star world is also true in a itself.
If a N, a* ⊑ a.
A ∨ ¬A, -0 (6)
A, -0 (7)
¬A, -0 (8)
r00#0# (9)
A, +0# (10)
0# ≼
0 (11)
A, +0 (12)
The crucial line for the branch to close is now available in (11) and follows from the constraint imposed above. While this preserves the validity of LEM and the tableaux rules make intuitive sense, we find ourselves appealing to the notion of inclusion between normal worlds and their counterparts. We must evaluate LEM as being valid by appealing to impossible worlds. Is this a satisfying fact? I argue that this development is unsettling and that accepting these costs for LEM is too large a price. Content inclusion seems like an intuitive kind of relation. Indeed, it might make a lot of intuitive sense that a world contains the truths of the worlds that are included in it. If we consider possible worlds, for instance, it seems that, at least in principle, we would be able to organize these worlds in terms of proximity to the actual world, where the nearest of possible worlds are the most like the actual world, such that for almost every fact p both the actual world and the possible worlds agree. It makes even more intuitive sense in the case where our worlds are simply states of information or ways the world can be. Per Restall, “there is a relationship of ''involvement'' x  y between states. To say that x  y is just to say that being y includes being x, or that y involves x”.
Restall (1999) page 5. He offers a far more feature rich account of inclusion than Priest (2008) and fleshes out the notion of “states” much more clearly than I can do justice in the space I have.
Another way of phrasing this is that if we have a certain amount of information in one state, we do not lose that information in a later one.
This discussion is similar to how an intuitionist logician may read their account of validity. For the intuitionist account, see Priest (2008). Involvement behaves similarly to the heredity rule in intuitionist logic. The only strong reason I have for the intuitionist logician not being a target here as well is related to the fact that the intuitionist does not require nonnormal worlds, whereas the relevant logician (in at least this case) demands them.
From this position we may now see where content inclusion and impossible worlds fit together. Indeed, Edwin Mares says that “we must distinguish between the assertion of ¬A and the denial of A… If ¬A obtains in [a situation] s, then it obtains in all ⊑–extensions of s”.
Mares (2008). Mares has an excellent discussion on situations and states and goes into great detail on their import to the semantics of relevant logic.
Looking at the constraint we introduce for the content inclusion operation
a N, a* ⊑ a, N the set of normal worlds.
, we discover a concerning fact. If our world w is a normal world, then w’s star world, w*, is content included in w. As 0, the actual world, is a normal world, then 0* ⊑ 0. As a result, those facts which obtain in 0* obtain in its ⊑-extension, 0. What this all effectively means, then, is that what is true in our logically impossible worlds has bearing on what is true in the actual world. While this works out quite nicely for a demonstration that LEM is a logical truth in the actual world, it leaves us with some concerning metaphysical considerations. Most prominently, we must answer the question of what these impossible worlds are for if content inclusion aligns with its intuitive reading
We say that if w ⊑ w’, then everything that is true at w is true at w’.
, then everything that is true at impossible worlds is true in their ⊑-extensions. Thus, it must be considered how different definitions of impossible worlds may impact the actual world. As described above, there are at least four possible kinds of impossible worlds. If impossible worlds are ways in which the world cannot be, there are seriously strong issues presented. For if a logically impossible situation obtains in w*, then that logical impossibility likewise obtains in w. But this seems paradoxical; the logical impossibility in question occurs in w* precisely because it cannot obtain in w. This seems to be the most problematic definition of an impossible world if content inclusion is part of our interpretation. The other three considerations of impossible worlds are likewise concerning. If the actual world is one in which classical logic has the correct interpretation for logical laws and our impossible worlds are ones in which classical laws do not obtain, then similarly we would not expect them to obtain in content included extensions of those impossible worlds. If an impossible world is one in which actual contradictions, all the worse for the actual world, for then those contradictions exist there as well. It appears then that the actual world is in fact some kind of impossible world, and not very normal indeed. But if the actual world is not normal, then it is not the case that the content inclusion restriction
a N, a* ⊑ a, N the set of normal worlds.
holds for our world, and LEM cannot be verified. It might seem that we are therefore left only with logically impossible worlds or information states such that some logical laws are false, but nothing violates those laws. In the case of physical laws, it may perhaps be the case that objects could travel beyond the speed of light, it just isn’t the case that any object does or ever will, perhaps due to other facts in the world. Analogously for logical laws, it perhaps is the case that LEM or modus ponens are false, but if they are not logical laws nothing is in such a way in the world to demonstrate their failure. We would therefore have no way of determining if this is the case without some sort of prior metaphysical commitment compelling us to think this is the case. As it were, there is nothing in the language of the logic itself to suggest if our logical laws are ''actually'' valid or invalid, but rather only a determination of what is possibly so. Our logic leaves it unclear what we should accept. The consideration we are forced to make is a metaphysical one: what is the nature of our worlds and are we at ease with the actual world being some kind of logical impossibility. As I have argued here, attempts by the relevant logician to save the Law of Excluded Middle entail serious metaphysical issues about the nature of this world with respect to how it relates to impossible worlds. The issues of the semantics of R and ∗, while not being very intuitive, are not necessarily the biggest issue facing particular relevant logics when it comes to intelligibility. It remains unclear what it might mean for impossible worlds to share their true content in the actual world. In order for the relevant logician to recognize LEM as valid, they need some sort of way of relating the content of a logically impossible world to the actual world. Whether these worlds are mere possibilities or existing objects, or if they are simply ways worlds can be taken as information states, the problem of content inclusion seems like it renders our logic unusable for determining validity anyways. We need some kind of deeper, more robust metaphysical or philosophical framework by which we can determine what these kinds of worlds might look like, and it must be careful to avoid being circular. Bibliography Berto, F. (2007). Is Dialetheism an Idealism? The Russellian Fallacy and the Dialetheist?s Dilemma. ''Dialectica,61''(2), 235-263. doi:10.1111/j.1746-8361.2007.01101.x Berto, F. (2009). Modal Meinongianism and Fiction: The Best of Three Worlds. ''Philosophical Studies,152''(3), 313-334. doi:10.1007/s11098-009-9479-2 Mares, E. D. (2008). General Information in Relevant Logic. ''Synthese,167''(2), 343-362. doi:10.1007/s11229-008-9412-9 Nolan, D. P. (2013). Impossible Worlds. ''Philosophy Compass,8''(4), 360-372. doi:10.1111/phc3.12027 Priest, G. (1992). What Is a Non-Normal World? ''Logique & Analyse,'' 291-302. Priest, G. (2008). ''An Introduction to Non-Classical Logic: From If to Is''. Cambridge: Cambridge University Press. Priest, G. (2016). Thinking the impossible. ''Philosophical Studies,173''(10), 2649-2662. doi:10.1007/s11098-016-0668-5 Read, S. (1988). ''Relevant Logic: A Philosophical Examination of Inference''. Oxford, OX, UK: B. Blackwell. Restall, G. (1999). Negation in Relevant Logics (How I Stopped Worrying and Learned to Love the Routley Star). ''Applied Logic Series What Is Negation?,''53-76. doi:10.1007/978-94-015-9309-0_3 ---- ________________________________________________________________________________ Dilyn Corner (C) 2020-2022