# Nonexistence and invalid formulas in modal logic

In first-order logic, I can essentially just ignore issues related to nonexistence and invalid formulas, without losing much. There is also free logic, in case I'm not happy with simply ignoring these issues.

While trying to make sense of a description and discussion of the Barcan formula, I started to wonder whether ignoring these issues also for modal logic could "work". Of course, I would want to have a predicate which tells me whether an object is nonexistent in a given world, but such a predicate in itself doesn't create much inconvenience. Ignoring also seems to imply that formulas which are only invalid in some worlds must be forbidden, hence I can't declare formulas which contain a nonexistent object in an "inconvenient" place as invalid. However, this would be a small price to pay if it would "work".

I have not yet read a systematic introduction to modal logic, but only SEP and wikipedia articles on modal logic (and Kripke semantics). So my question is just whether ignoring issues related to nonexistence and invalid formulas is a reasonable option for (quantified) modal logic, in the same sense as I typically ignore these issues for first-order logic.

• It might be a while before I can post a complete answer here. Check out p. 61 of these lecture notes by Kevin Klement. The Barcan forumla is falsified if domains increase from one accessible world to another, the Converse Barcan formula is falsified if domains shrink from one accessible world to another. I suppose in one sense you could think of this as things coming in and out of existence. – Dennis Jun 27 '13 at 17:43
• it took me a while to understand the question, presumably question of existence & non-existence become more problematic in modal logic as one is discussing possibilities? – Mozibur Ullah Jun 27 '13 at 20:56

"Existence" can mean very different things, but there's a tradition to include everything (sometimes even inconceviable things) in (any) logic's domain.

It is then natural to create predicates for various meanings of "exists", and this predicates would verify different "things" at different worlds (i.e. Pink(pink_unicorn), []Pink(pink_unicorn), <>Pink(pink_unicorn), but ~PhysicallyExists(pink_unicorn) and <>PhysicallyExists(pink_unicorn))

• My experiences from dealing with the peculiarities of NaNs (not a number) in C++ tell me that non-existent values can be quite tricky and create a whole lot of inconveniences. The question notes that classical first order logic managed to avoid all these problems by doing exactly what you call "there's a tradition to include everything (sometimes even inconceviable things) in (any) logic's domain". The question is whether this (or a similar) trick will also safe the day for modal logic. Are you telling me that it will work, and can handle simultaneously multiple definitions of "existence"? – Thomas Klimpel Aug 7 '13 at 1:02
• Well, there should be no problems. If you set "Blue" to contain {desk, chair} at world1, introducing e.g. blue_unicorn won't change anything here - if you express things in the form of Ax(PhysicallyExists(x) -> ...) or in similiar fashion. This way, every world has it's own universe, a set of things it accepts (but the things are still shared in the sense that you can talk about the same thing in two different worlds). – Luka Mikec Aug 7 '13 at 1:34
• Regarding C++, if you first check for restrictions (e.g. you are expecting int from 0 to 10 in a function that displays game score), you can't get into trouble (well, you can, but hopefully there are no dangling pointers and such in modal logic!) – Luka Mikec Aug 7 '13 at 1:35
• If I always have to write "ExistsModal(x)->..." instead of "...", this is inconvenient. And what do you do with formulas that don't start with "ExistsModal(x)->"? To go back to the C++ example, if I write a "sort" algorithm, it is inconvenient if I always have to write "if (validNumber(x)) ..." before I can write anything actually useful. If I deal with an individual number, it is perfectly fine to check if it is a valid number, but if I write a "generic" algorithm, this is just noise and overhead. And what can I actually do with a number which is "not a number"? "Maybe x" (haskell) is cleaner – Thomas Klimpel Aug 7 '13 at 13:28
• Yes, but unrestricted quantification is, after all, unrestricted, and (depending on universe, but often) includes those things that exist in a different way than normal objects. So saying things about non-existent objects is undefined, but that's expected behaviour, if your (hypothetical) class of predicates {PhysicallyExist, ICanThinkOf, ...} doesn't recognize a specific object, than that object probably shouldn't be mentioned at that world. – Luka Mikec Aug 7 '13 at 15:08

The fact that you have to deal with issues of (non-)existence and (in-)validity might actually be an important feature of modal logic. For any discussion of "existence" in the context of classical single-sorted first-order logic somehow seems to fail to capture the meaning of the word "existence" as it applies to (what we perceive as) reality. The structure of reality is better captured by one actual world from which we talk about this actual world and the worlds reachable from this actual world, instead of talking about the single static single-sorted universe of classical logic.

However, even this "Kripke structure" of modal logic described above doesn't fully capture the structure of reality, because we often also talk about the worlds from which the actual world is reachable, and about the "abstract" truths valid in any world. Note however the disclaimer in the question that I have not yet read a systematic introduction to modal logic, so some of my statements above might be inappropriate as a description of the current practice and interpretation of modal logic. However, I thought it would be appropriate to write down these thoughts as an answer to my own question, because they look at the problem from a significantly different perspective than I had initially in mind when I asked this question.

I just asked, what I believe is, a similar question recently about what you call "inconvenient formulas" in possibilist modal logic. That is, formulas that attribute properties to an object that doesn't actually exist at the world the formula is being evaluated at, but to avoid truth value gaps, must have an interpretation. Like you, I believe that logic should be metaphysically neutral (or at least that's what I think you are saying). With quantified modal logic this seems especially difficult since there are at least two different approaches to semantics, possibilism and actualism. Possibilism (in modal logic) is roughly the view that there is one domain across all possible worlds, and all objects in this domain exist in some sense in all worlds. The referent of a name is fixed across all worlds, as the possible object which it picks out in this one domain. Quantified statements are statements about all objects in this single domain. Actualism (in modal logic) is roughly the view that there is a domain for each world which can vary, so an object which is in the domain of one world may not be in the domain of another. Quantification is world-relative, only about the objects that exist in the domain for that world. Depending on which approach you take, certain formulas that are valid on one approach (Barcan Formula, Converse Barcan Formula, and the Necessary Existence Formula in possibilism) are not valid on the other. So, it seems the choice of semantics here does make a difference to the truth of some formulas, and to valid deduction.

On the other hand, some philosophers argue that possibilism is the simpler and more general approach to the formal semantics, since actualist reasoning might be adequately captured by possibilist semantics. In "In Defense of the Simplest Quantified Modal Logic", Linsky and Zalta argue that there are readings of "inconvenient" possibilist formulas that should satisfy actualists, as long as they are willing to regard the troublesome objects as non-concrete objects, instead of possible objects ("non-concrete" means something like "abstract", but only contingently so). In this way, "the possible red pen on my desk" is read as "the non-concrete possibly red pen on my desk". It doesn't even have to have the property of being red (though it is possibly red), since it's non-concrete, and non-concrete objects don't possess properties such as being red, having spatial or temporal locations, and mass, just like how abstract objects can't have these properties either. Or, maybe they can (if your metaphysics permits), and so the non-concrete red pen on my desk might actually be red. The choice is yours, and the formal semantics doesn't dictate the answer one way or the other.

I don't really know the rules of this site too well, so I'm not sure if this paragraph is too subjective. But personally, I've sort of just accepted possibilism and the simplest quantified modal logic as a "default" logic for reasoning about metaphysics, and I just think of it as the logic of possibilities and possible objects. As such, we need to be able to refer to possible objects in some way, just like how in probability we need to refer to alternative outcomes of an experiment (ie. elements of a sample space). I think this approach to possibility is kind of like doing mathematics without settling questions about the philosophy of mathematics. Whatever ontology of numbers, outcomes, etc., you believe in, you still talk about numbers, outcomes, etc., "existing" in first-order logic formulas ("there exists a natural number x, such that..."). But whether such a number really, metaphysically, exists is a different question that we can hope to ignore for as long as possible while still doing math. So, even though we might talk like a possibilist regarding the existence of possible objects, metaphysically we might not believe in them. Or, à la Linsky and Zalta, believe in their existence as non-concrete entities. Or as something else.

All this flexibility might be a good thing as it gives us a lot of choice in how to metaphysically interpret modal logic formulas, but doesn't seem to commit us forcefully to any one position.