# Defining logical possibility without circularity

In explaining logical possibility (broad), I tend to say that a thing is logically possible if it exists in some possible world, some possible description of reality, but in what sense is such a world/description of reality qualified as "possible"? In a broadly logical sense, of course. "Possible world"/"possible description of reality" could be replaced with "a way the world could have been," but there's still a tautologically-defined notion of possibility implicit in "could." Similarly for "a way the world might have been."

Does this semantic tautology actually show itself in modal logic discourse, and if not, what is the proper way to avoid it?

• While I sympathize with the question, aren't all definitions circular in basically the same way as your suggestion with "possible" <-> "could be"? Aug 17, 2016 at 0:35
• Hmm, I'm not sure I see that. If we define a giant as a "large human," the concept a giant isn't then required to define "large human." Rather, we can understand "large human" by understanding the meaning of the words "large" and "human," even if we don't know in advance what a giant is. By contrast, it seems difficult to understand "could" without a notion of possibility. Aug 17, 2016 at 0:50
• There is a small semantic circle at the foundation of any discipline: lines, points, contain, pass through, etc. in geometry; sets, elements, belongs to, etc. in set theory; analyticity, synonymy, inference, meaning in semantics itself, etc. This is inevitable, definitional chains have to end somewhere. Foundational notions are "interdefined" by postulating how they interrelate, and clarified by pointing to prototypical examples. Modal logic arguably has particularly obscure and controversial prototypes, but the overall structure of discourse is not particularly different. Aug 17, 2016 at 0:54
• @user20658 I somewhat see your point but agree with Conifold that sure we can define "higher-level" things that way (unicorn is non-circularly horse + horn), but the more basic the concept, the less room to do that. Try defining "red" using words without saying things that just mean red (wavelengths of light btw would be a different kind of definition). Also try defining "human" without saying things that mean the same thing. Aug 17, 2016 at 1:31
• Well, a human could be defined as a rational animal, as Aristotle suggested. I see the point though; logical possibility may just be a foundational concept. Aug 17, 2016 at 2:07

Something logically possible is something that is not logically impossible, and logical impossibility need not be defined circularly. Much of our modern ways of thinking about modal logic can be traced back to Leibnizian thought, and Leibniz associated impossibility with contradiction. For him, contradictions were impossible combinations such that, for any ontology, the only necessarily false statements were contradictory statements. To be more specific, necessitatem absolutum, in Leibnizian philosophy, entails truth in all possible worlds by virtue of principium contradictionis. “Les vérités nécessaires sont fondées sur le principe de contradiction” (Leibniz 1686). Likewise, impossibility entails falsity in every possible world, which justifies principium exclusi tertii sive medii inter duo contradictori. “In like manner as […] an assertion cannot be both true and false, so […] an assertion must be either true or false” (Mill 1843).

[Note: Necessitatem absolutum ought not to be confused with necessitatem ex hypothesi. “Necessity […] consists either in the constant conjunction of like objects, or in the inference of the understating from one object to another” (Hume 1748). Necessitatem ex hypothesi is the truth/falsity of an apodosis as being contingently necessary for the truth/falsity of any hypothesis to which that apodosis belongs (irrespective of any protasis in particular). On the other hand, necessitatem absolutum may invoke a Parmenideanistic mundus intelligibilis (perhaps evocative of Platonic-Pythagorean εἶδοη). “Indépendamment de la preuve qu'on appelle apodictique [il y a donc] une certitude que nous avons souvent [...] qualifier de philosophique ou de rationelle, parce qu'elle résulte d'un jugement de la raison” (Cournot 1851).]

(As a neo-intuitionist) I would suggest the only place for logic to ever go for grounding is psychology. And (having a fondness for Lacan) I would suggest that 'possible', as used in logic, is really a generalization of 'fully conceivable', meaning something that some human should be able to hold in his mind in a way robust enough to explore.

We cannot ever generate 'possible' worlds as test cases, only 'fully conceivable' ones, because we are us. So it is pretentious for us to imagine the less restrictive word 'possible' has any more content than the more confining psychological notion of imagination.

I would go so far as to contend that only the latter term has any definition, and the one we use is an inappropriate attempt to dodge human identity, pushing possibility up into the mind of God, or out into the universe under the assumption that the universe is comprehensible, but ultimately coming down to human psychology.

What we are really after when we use this modality is acknowledging the human capacity for imagination and internal modeling, and estimating bounds on the flexibility of that process.

• Surely it is conceivable that some non-contingent entity (i.e., the number seven) objectively exists, and also conceivable that it does not objectively exist. Both are possible in an epistemic sense, but only one is possible in a logical sense. My question was about logical possibility. Aug 17, 2016 at 20:51
• How exactly is a world with no 7 clearly conceivable, in the sense given here that you should be able to hold it in mind? You would immediately reach a contradiction. You can only hold a pointer to it in mind, if you try to actually develop the concept, it vanishes. (If you don't like the word conceivable, and would like to reserve it for the next lower mode, for wishes or unresolvable references, you can surely still get the meaning here.)
– user9166
Aug 17, 2016 at 21:36
• I have added an adjective and extended the definition for clarity.
– user9166
Aug 17, 2016 at 21:41
• Plenty of people believe that numbers are actual objects, which necessarily exist in a mind-independent sense. It's an epistemic possibility that they are correct, as we have no knock-down proof against the view. However, if in fact such non-contingent objects as numbers do not exist, then it is not logically possible that they do exist. It's just two different types of possibility. Jan 15, 2017 at 3:14
• @user20658 It is logically possible that black unicorns live on the other side of the moon. I am pretty sure they would immediately explode or suffocate, so it is not actually possible. So actual facts do not necessarily entail logical ones. Logical facts entail actual ones. There several layers of meaning for 'possible' but they are not independent, they are nested.
– user9166
Jan 17, 2017 at 19:13