• Nicholas Ferreira
    69
    I'm studying modal logic and I'm reading Kripke's paper "Semantical analysis of modal logic I - Normal modal propositional calculi" (1963).
    When he talks about the accessibility relation R (p. 4), he says that a world H2 is accessible to H1 if every proposition A that is true in H2 is possible in H1. But then he defines a proposition A is possible in the world H1 iff there is a world H2, accessible to H1, in which A is true. But isn't this circular? In order to understand what the accessibility relation means, I need to know what it means a proposition be possible, but to understand what means a proposition be possible, I need to know what the accessibility relation means.
    I've checked in some modal logic books and they seems to use the same definitions, but seems like this definitions are empty. What do you think?
  • boethius
    318
    Based on just what you report here, my guess would be you have conflated accessible and possible.

    Accessible I would assume means expressible; the statement can be understood with the system of rules and postulates in question.

    Possible is not the same as accessible. A statement can be accessible but always false.

    To be possible means there is a world where that statement is actually true (where world means a system of rules and postulates, not the actual world).

    The main theme in this sort of framework is that "modal necessity" is equivalent to the proposition being true in all possible worlds accessible to the system in question, so in this framework you can hop around systems and relate them to each other and try to find counterexamples or then prove no counterexamples can exist to investigate and conceptualize necessity, as well as investigate what possibility means by building a concrete example of an accessible world where the proposition is true.

    In a sort of colloquial "everyday" modal logic, you can understand this process as explaining the concept of "it might rain tomorrow" by describing that possible world of tomorrow where it's raining (which is different to explaining what is true about the world today which makes rain tomorrow possible; in this colloquial everyday sense, Kripke is building what relations need to hold between what's true today and the imagined world of tomorrow to imply possibility or impossibility; we can not only imagine it but it is a world accessible from what we believe to be true today).

    Likewise, explaining the concept that "the sun will necessarily rise tomorrow" is the process of explaining how every possible world of tomorrow includes the sun rising. (of course this isn't rigorous as someone with powerful rockets could stop the spin of the earth, or the entire sun could potentially quantum tunnel to the edge of the galaxy or any number of other ways the sun wouldn't rise, but we implicitly ignore these possibilities as so low as to be irrelevant by treating these things in a modal way; a sort of "modal lite" that is the quickest way to reason for a wide range of cases: I necessarily need to eat to live, poison will necessarily kill me, I necessarily am unable to fly on the earth's surface without technology, going to work is not necessary but various possible things may happen as a consequence, various things are contingent on various other things, time travel backwards is necessarily not going to happen etc. The purpose of this "modal lite" way of reasoning is to narrow down the scope of factors that have an ambiguous range to consider by first finding relevant details that are close enough to 100% or 0% to be treated in a necessary or necessarily-not way, and then applying our intuition to the possibilities that remain, as we can't explicitly calculate probabilities for most situations; the necessary and necessarily-not game allows us to build a trunk of necessary things and then at least cut off entire branches that no longer need to be considered if they are necessarily-not, and so better make use of our intuitive capacity on the branching possibilities that remain).
  • Terrapin Station
    13.8k
    When he talks about the accessibility relation R (p. 4), he says that a world H2 is accessible to H1 if every proposition A that is true in H2 is possible in H1. But then he defines a proposition A is possible in the world H1 iff there is a world H2, accessible to H1, in which A is true. But isn't this circular? In order to understand what the accessibility relation means, I need to know what it means a proposition be possible, but to understand what means a proposition be possible, I need to know what the accessibility relation means.Nicholas Ferreira

    He's not really defining either "possible" or "accessible." He's rather defining what he's calling the accessibility relation. He's saying that it obtains when there are the relations he's describing between H1 and H2 with respect to true and possible propositions. So it's just one definition and not two (where the two could be circular). One needs to have a grasp beforehand re what "possible," "accessible," and "true" refer to.
  • Nicholas Ferreira
    69
    Well, I don't think I've conflated accessible and possible, for me it's very clear the difference, and I agree with everything you said. But I still cannot understand exactly how this two terms are to be formally expressed without requiring one another.

    Hmm, it might not be exactly a definition, but anyway he expresses the accessibility relation in terms of possibility (H2 is accessible to H1 if every proposition true in H2 is possible in H1, that is, (h1Rh2 ↔ ∀p(V(p, h2)=T ⊃ V(◊p, h1)=T)), and expresses possibility in terms of accessibility (A is possible in h1 iff there is a world h2 accessible to h1 in which A is true, that is, (V(◊A, h1)=T ↔ ∃w(h1Rw∧V(A, w)=T))) (Here, V(x, y) is de valuation function of the formula x in the world y). I cannot understand exactly what he formally means with one term without understand another.
    He previously (p. 2) defines (and here he actually uses the term 'define') □B as follows: V(□B, H)=T iff for every H' such that HRH', V(B, H')=T. He then says that a formula A is true in a model associated to the world G if V(A, G)=T.

    Thank you for answering!
  • Terrapin Station
    13.8k


    You're not saying that you do not understand (the general modal logical sense of) "possible," are you? Again, the statements in question are both explicating what the accessibility relation is.
  • Nicholas Ferreira
    69
    No, I understand what it means for a proposition to be possible. The whole point is that Kripke explains it in terms of a concept (accessibility) which requires the notion of possibility to be understood, which seems to be circular. I mean, you know what "bachelor" means, but if I say that bachelor is a unmarried man and that an unmarried man is a bachelor, i'm not saying anything usefull, it's circular, even though you know the meaning of this terms.
  • Terrapin Station
    13.8k
    The whole point is that Kripke explains it in terms of a concept (accessibility) which requires the notion of possibility to be understood, which seems to be circular.Nicholas Ferreira

    He's not explaining possibility, though, he's explaining the accessibility relation.
  • Nicholas Ferreira
    69
    He explains both concepts, as I said before. And he uses possibility to explain the accessibility relation.
  • Terrapin Station
    13.8k


    No, that's part of explaining the accessibility relation. That's what I was telling you in my first post.
  • Nicholas Ferreira
    69
    So what it means for a world to be accessible to another?
  • Terrapin Station
    13.8k
    .

    Right. He's giving multiple characteristics is that, explaining it from different angles so to speak. That's why I said earlier that "One needs to have a grasp beforehand re what 'possible,' 'accessible,' and 'true' refer to."
  • Nicholas Ferreira
    69
    Well, ok, but I couldn't find a precise formal definition/explication of 'possible' and 'accessible' without being required to already know one of this concepts. In 'Modal Logic for Philosophers', by Garson, and in 'Basic Concepts in Modal Logic', by Zalta, the same circularity appears.
  • Terrapin Station
    13.8k


    A common formal definition of "possible" is:

    ◊A=∼□∼A
  • Nicholas Ferreira
    69
    Sure, but how would you define □A without using ◊ or accessibility? In both cases the circularity would appear again.
  • Terrapin Station
    13.8k


    You realize that all definitions are eventually circular, right? We have a finite set of symbols or terms, and within whatever system at hand, we define those symbols and terms by other symbols and terms in the system. They all sooner or later point to each other, which means that all definitions wind up being circular.
  • fdrake
    2.8k
    So the metaphysics of possibility is some other thing.

    The different modal logics specify certain conditions on the accessibility relation or certain desired axioms for them; the two are interchangeable. I think it's usually the case that specifying some property of the accessibility relation specifies some axiom of the modal logic.

    If you can fix a notion of possibility, or a notion of accessibility, then you can see what the other does through this equivalence. I think the modal logics are more of a modelling tool for different accounts of possibility or necessity (like uh... necessarily P => P makes sense for metaphysical possibility/necessity, but ought that P => P does not make much sense for deontic possibility/necessity).

    If you wanna know what possibility "is" in general, look elsewhere, if you wanna stipulate some possibility behaviour or accessibility relation behaviour and see what happens, the definitions help. You can embed lots of different metaphysical intuitions into the logics.
  • Terrapin Station
    13.8k
    Circularity only winds up being a problem when at some point in the circle, you don't have an intuitive grasp of what a term refers to, or an intuitive grasp of its connotation and denotation.

    For example, if we say "That's a snarblaff." And you go"What's a snarblaff?" And we say, "It's a grutparp." And you say, "What's a grutparp?" And we say, "It's a snarblaff." (And we can extend that any number of steps.) If you don't have an intuitive grasp (usually via experience, including ostensive references, etc.) of what either a snarblaff or a grutparp is, then it's circular in a way that's a problem.

    If we say, "A snarblaff is a toaster," then you're not going to have a problem with that, because you know what a toaster is. But if you didn't know what a toaster was, then that would refer to other words, and so on, and at some point you need to have an intuitive grasp of one of the terms, or it's going to be circular to you in a way that's problematic, even if we've gone 50 steps or whatever before completing the circle.
  • boethius
    318
    Well, I don't think I've conflated accessible and possible, for me it's very clear the difference, and I agree with everything you said. But I still cannot understand exactly how this two terms are to be formally expressed without requiring one another.Nicholas Ferreira

    If they are not the same, it's not circular, they simply depend on each other to be understood.

    There is no proof of theorem being offered that could have the problem of being circular, it is the basic concepts to provide meaning to the symbol manipulation to follow.

    You will encounter the same thing with all foundational concepts.

    For instance, logicians and mathematicians will use the word "statement". If you challenge what it means ... of course you can only get statements as explanations; if you truly don't understand what a statement is and need that understanding to understand any statement, the explanation of statement can never make sense. All such explanations will be just different versions of the same thing with various caveats and relations to each other all expressed as a series of statements. So, it's as puts it, that what's being offered is different ways of looking at the same thing, and that's all that can be offered.

    We have an intuition of what a statement is, likewise possibility. In dealing with formal reasoning systems we can relate our intuitions to some property of the system (in this case there is a relation by symbolic manipulation between the worlds in question that we can intuit as our concept of "possible"), this however is exterior meaning we give it; what exists internally to the system are the symbolic rule relations which do not require meaning. The usual analogy is long lost languages we cannot decode; we can deduce some rules between the symbols but have no idea what they mean; we can look at formal systems the same way, but we are not obliged to (if we want to make some decision based on the symbols and relations to each other, there must be some relation, some correspondence, to what we believe the actual world we live in is; and we focus on systems of rules that seem to have some innate ability to model aspects of our real world, rather than just randomly invent symbols and rules and randomly permute them without ever assigning meaning).

    It's like starting a book that begins with "the woman sat next to the tree"; what do we know about the woman, that she is next to the tree; what do we know about the tree, that it is next to the woman. Math and logic books are generally not an exception to this feature of all books and tend to start the same way too.
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