Objectivist Modal Logic

Metaphysics and Epistemology
1

So some Objectivists with whom I’ve spoken reject the concept of possibility except where human will is involved. All non-human events are then necessary, and matters of choice are possible. I wonder what axioms we would take for modality, then.

We would probably have to reject the idea that if something is actual then it’s possible, and that if something is necessary then it’s possible–since we may want to restrict possibility to things that derive from humans. And that would probably imply that ~□~ ≠ ◊. So you actually wouldn’t have a K system.

So you would probably just re-interpret your operators to mean “subject to human will” (◊), and “not subject to human will” (□). So □ = ~◊. You would also have ~(◊A → ◊◊A), since there are at least some instances where a thing is subject to your will and yet it’s being subject to your will is not subject to your will. In fact, that seems to be true all the time, so ◊A → □◊A would be an axiom, which is ◊A → ~◊◊A.

Any others?

2

If you’re talking about just the syntax, the idea of “Objectivist modal logic” is meaningless. Anyone can posit whatever formal postulates they want. If you’re talking about semantics, you’ll need to do some serious recasting, since there are no “possible worlds” in the Kripkean sense. “Possible” is the same for human agents and others: it is a well-formed concept, and you can equally say “It is possible that it will rain” of “It is possible that he will arrive on time” – you don’t know. “Necessary” is applicable to humans and non-humans alike – “He must pay the fine”, “The dog must go out”.

All propositions, and therefore all modal statements, exist within the scope of human will (a human formulates and evaluates the proposition). All modal statements assert a judgment (by the human). But the proposition can be about anything. See OPAR ch. 5 for the meaning of “possible”.

3

If you’re talking about just the syntax, the idea of “Objectivist modal logic” is meaningless. Anyone can posit whatever formal postulates they want. If you’re talking about semantics, you’ll need to do some serious recasting, since there are no “possible worlds” in the Kripkean sense.

Huh? Why would Objectivism reject possible world semantics? It’s not like just using possible world semantics entails the truth of modal realism.

4

Well, I’m not sure what you mean by “just the syntax”. Para-consistent logicians are just about the only people I guess are working on pure syntax with no semantics. I mean a syntax with semantic interpretation. As for possible worlds, I don’t really see the problem with them since–as far as I know, in the Kripkean sense–they’re just sets (or groups) of propositions. But we can do away with possible world semantics and talk about propositions about possible choices.

As for doxastic, deontic, and epistemic modal systems, that’s a dead horse and I don’t have the energy for beating it. Smarter people than me have thought about those things.

Another axiom for this system: ◊A → ◊¬A.

Now that the system has a couple of axioms under its belt, I think it deserves a name. Volitionist modal logic?

I think, once I’m happy with my set of axioms, I’ll prove soundness and completeness as a fun exercise.

5

Oh, and ◊A → □~◊A, and ~◊A → □~◊A, and ~◊A → □◊A. I’m betting at least some of these axioms entail some others.

6

So some Objectivists with whom I’ve spoken reject the concept of possibility except where human will is involved.

  • What are some examples of the concept of “possibility” without human will?
  • What does it mean for “will” to be involved with a concept as opposed to “will” being uninvolved with a concept?
  • What do you mean by “possible”?
  • What do you mean by “impossible”?
7

Another axiom for this system:

Now that the system has a couple of axioms

I think, once I’m happy with my set of axioms, I’ll prove soundness and completeness as a fun exercise.

*What do you mean by “axiom”?
*What do you mean by “proof”?
*How do you believe “proof” is given, i.e., how does one “prove” a statement to you?

8
  • What are some examples of the concept of “possibility” without human will?
  • What does it mean for “will” to be involved with a concept as opposed to “will” being uninvolved with a concept?
  • What do you mean by “possible”?
  • What do you mean by “impossible”?

Concepts of possibility: epistemic, metaphysic, and deontic (associated with the term “must”).

I don’t know what it means for will to be involved with a concept, but I know what it is for something to be subject to will. For instance, getting up or reading, or thinking. I mean by “possible” that which might be.

*What do you mean by “axiom”?
*What do you mean by “proof”?
*How do you believe “proof” is given, i.e., how does one “prove” a statement to you?

An axiom is a fundamental truth, also, a theorem from which all other theorems in a system are derived. A proof of a theorem is a set of theorems which are either axioms or results of applying an inference rule to an axiom or inference, which ends in the theorem proved. This is a formal notion of proof. I believe other things may also be proved, by appeal to consistency and axiomatic knowledge.

9

Well, I’m not sure what you mean by “just the syntax”.

I’m sure you understand the distinction between syntax and semantics. Objectivism has no position on or use for meaningless expressions. No “axioms” of modal logic are true axioms, and anything resembling K or T are theorems. The notions of epistemic, metaphysic, and deontic modality (as distinct) are confused – they reduce to one fundamental thing (the epistemic) and separate elaborations (to specify the metaphysical versus deontic). This does not mean that as an exercise in symbol-manipulation using 9d8f6d53ad070771a6910933810023c4.png and 6a2084de787fb2d1c3e408d926425904.png you can’t engage in formal modal logic, just that it has no connection to Objectivism.

The semantics is primary, so start there.

As for possible worlds, I don’t really see the problem with them since–as far as I know, in the Kripkean sense–they’re just sets (or groups) of propositions.

I didn’t think you would see. The sets don’t exist in any sense of “exist”. A “world” is not actually a world, it is an integrated human knowledge context. Saying that a proposition p is true purely by dint of being in some w is wrong, and it’s impossible to cover the empirical subject matter of modal logic without including errors (e.g. you could not talk about counterfactuals; common law reasoning cannot be discussed).

I think, once I’m happy with my set of axioms, I’ll prove soundness and completeness as a fun exercise.

As a meaningless exercise in systems of symbol manipulation, that’s a possible action. However, it has no relation to logic, meaning the art of non-contradictory identification. Identification requires a real-world subject matter, which is lacking in formal logic. You might be able to relate your formal system to real-world semantics and eventually come up with a systematic mapping between natural language semantics and your formal system. I feel confident that that is not your interest.

10

I’m sure you understand the distinction between syntax and semantics.

I’m sure I do, too, which is why I wasn’t sure I understood your question.

Objectivism has no position on or use for meaningless expressions. No “axioms” of modal logic are true axioms, and anything resembling K or T are theorems.

I figured you’d say as much, which is partially why I’m talking about a system that is not K or T. This system, however, is meant as a representation of axioms that an Objectivist would assent to. I’m supposing that an Objectivist should assent to the idea that one cannot will to be unable to will that which he can will. That is, given some n different possible choices, one cannot will that one hasn’t those choices. (One might will one of the n, and that particular choice might eliminate the possibility of the others, but that’s after choosing one of the n.) I’m also supposing that, if you can will A, then you can will ~A. If an Objectivist would not assent to this, then I’ll consider an appropriately different system.

The semantics is primary, so start there.

It should be clear that’s what I’m doing. Note, I’m investigating what an Objectivist believes about willing and possibility. It seems your later-expressed confidence in my disinterest toward real-world language and semantics is a bit misplaced.

I didn’t think you would see. The sets don’t exist in any sense of “exist”.

You don’t think people express propositions? Or are you a nihilist about groupings (note: we don’t have to talk about sets, if you’re anti-set theory).

A “world” is not actually a world, it is an integrated human knowledge context.

What does that mean? A proposition about the context of an individual who knows certain propositions about his context? Perhaps this will help: How is this different from the concept of a set of propositions?

Saying that a proposition p is true purely by dint of being in some w is wrong, and it’s impossible to cover the empirical subject matter of modal logic without including errors (e.g. you could not talk about counterfactuals; common law reasoning cannot be discussed).

Well, I don’t agree, but it’s not essential since I’m not clinging to possible world semantics. Also, though, this is not itself essential to possible world semantics. (Note KD to KD45 modal logic systems, that attempt to model epistemic modality.)

However, it has no relation to logic, meaning the art of non-contradictory identification.

It’s every bit as much “the art of non-contradictory identification” as mathematical logic (which is currently a field being led by an Objectivist, no less, who just gave a presentation on Objectivism at the NYU Philosophy of Math conference!).

11

This system, however, is meant as a representation of axioms that an Objectivist would assent to.

That would be good; but I have not seen any examples here of such an axiom. I have seen some mentions of possible theorems that could be derived from axioms.

I’m supposing that an Objectivist should assent to the idea that one cannot will to be unable to will that which he can will.

Why should we assent to that idea? Can you reduce that to “embracing a contradiction” (and prove that this is a correct reduction)? You would need an articulated theory of the words “will”, “can”, “able”. Maybe you have that, I don’t know. Besides, modal logic is not exclusively about “willing”. Are you specifically focusing on deontic meanings?

That is, given some n different possible choices, one cannot will that one hasn’t those choices.

If you were to reduce modal logic to “knowledge context”, as I have said you should, then what you are talking about might be a theorem. Go for it.

It should be clear that’s what I’m doing.

Yet another is-ought problem. You can solve the problem, which is of your own making.

Note, I’m investigating what an Objectivist believes about willing and possibility. It seems your later-expressed confidence in my disinterest toward real-world language and semantics is a bit misplaced.

If you are actually interested in semantics, then set aside your interest in “willing” and “possibility”, and focus on the actual semantics of that which is primary. Is that “will” and “may”? I’m not sure. I think a more useful first cut would be “may” and “must”. But whatever. Can you specify the semantics of “will” and “may”?

You don’t think people express propositions? Or are you a nihilist about groupings (note: we don’t have to talk about sets, if you’re anti-set theory).

Then you presumably will not henceforth talk about “possible worlds” – that will be useful.

Perhaps this will help: How is this different from the concept of a set of propositions?

Let’s put it this way: for you to make any headway, you have to be speaking of knowledge contexts.

Since we don’t have to buy into the power set of power set view of “possible worlds” to pursue the actual content of modal logic, I imagine you could start by specifying the content of “will” and “may”.

12

That would be good; but I have not seen any examples here of such an axiom. I have seen some mentions of possible theorems that could be derived from axioms.Why should we assent to that idea? Can you reduce that to “embracing a contradiction” (and prove that this is a correct reduction)? You would need an articulated theory of the words “will”, “can”, “able”. Maybe you have that, I don’t know. Besides, modal logic is not exclusively about “willing”. Are you specifically focusing on deontic meanings?If you were to reduce modal logic to “knowledge context”, as I have said you should, then what you are talking about might be a theorem. Go for it.

So what would be some axioms? How do you derive from them the theorems I have listed? Naturally, adding theorems as axioms doesn’t hurt the theory–just makes it more cumbersome. So at the very least, these could be investigated. And, just as naturally, I assume standard predicate logic built on Aristotelian classical logic.

I take the meaning of “willing” to apply only to cases where “willing the negation” is equally valid, all from the axiom that man is volitional. To deny it would be to assert that one might will A, but it be impossible to do otherwise, in which case A is not willed but merely done. Likewise for the other axioms. Here I am not talking about any deontic notions, like “should” or “good”, so this is conceptually divorced from other notions of modality.

Yet another is-ought problem. You can solve the problem, which is of your own making.

I disagree with the last sentence clause.

Then you presumably will not henceforth talk about “possible worlds” – that will be useful.Let’s put it this way: for you to make any headway, you have to be speaking of knowledge contexts.

I haven’t been speaking about possible worlds–you brought that up. You also brought up knowledge contexts (of some sort), but haven’t explained what they are or how they’re relevant to this “volitionist modal logic”.

13

So what would be some axioms?

The perceptual is the axiomatic.

I take the meaning of “willing” to apply only to cases where “willing the negation” is equally valid, all from the axiom that man is volitional.

Yet we can say “The tree will fall”, and trees have no consciousness. To talk about volitional predicates in modal logic, you need to first get at the broadest concept “will”.

You may, if you want, want to focus just on one use of “will”, but I don’t know exactly what that use would be. You could clarify by exemplification, without using the word “will” (so as to avoid circularity). I am just trying to figure out what thing in reality you are concerned with.

14

A is the perceptual. Where making A the case is subject to your will, we notate it ◊A and take it from there as I’ve done able. Yay.

Clearly, no semantics where “will” is a verb and not a noun will fit, so the non-standard model you were trying to give doesn’t fit.

15

Are you saying that you just aren’t interested or aren’t going to try? Or is it that you do not know how to express that which you are interested in with any greater degree of clarify? If your only interest is understanding choice, then you needn’t be concerned with “making it happen” because that involves things besides simply choice (e.g. the laws of physics, the choices of other people). But then I still don’t know if your concern is over “creating the proposition” (recognition of a hypothetical state) versus “evaluating alternatives” (making the choice).

16

Concepts of possibility: epistemic, metaphysic, and deontic (associated with the term “must”).

I don’t know what it means for will to be involved with a concept, but I know what it is for something to be subject to will. For instance, getting up or reading, or thinking. I mean by “possible” that which might be.

An axiom is a fundamental truth, also, a theorem from which all other theorems in a system are derived. A proof of a theorem is a set of theorems which are either axioms or results of applying an inference rule to an axiom or inference, which ends in the theorem proved. This is a formal notion of proof. I believe other things may also be proved, by appeal to consistency and axiomatic knowledge.

Thanks for the clarifications, they help me understand your positions a little better.

?What is the ultimate purpose or goal of making the distinction of “modal logic” as opposed to just logic?

I.e., what is one trying to accomplish?

17

Are you saying that you just aren’t interested or aren’t going to try? Or is it that you do not know how to express that which you are interested in with any greater degree of clarify? If your only interest is understanding choice, then you needn’t be concerned with “making it happen” because that involves things besides simply choice (e.g. the laws of physics, the choices of other people). But then I still don’t know if your concern is over “creating the proposition” (recognition of a hypothetical state) versus “evaluating alternatives” (making the choice).

I just met your demands is all.

I don’t know what you are talking about with any of the phrases you put in quotes, or parentheses. I’m not talking about hypothetical states in any straight-forward sense, or actually making choices..

Thanks for the clarifications, they help me understand your positions a little better.

?What is the ultimate purpose or goal of making the distinction of “modal logic” as opposed to just logic?

I.e., what is one trying to accomplish?

Well, logic is a long-debated topic, and it’s often clarified by formalizing it. (For instance, we now have a clear distinction between epistemic, deontic, and metaphysical modal claims due to formalized modal logic. We see that deontic systems are helped greatly by eliminating the axiom □A → A, because what holds in the deontic modal claims [about what should be] are free from truth-values of statements about the actual world, which verifies our intuitions. On the other hand, results like the failure of ◊( A → B ) → ( A → B ) to be a theorem is a bit of a surprise, which corrects and refines our intuitions.) So I’m proposing a formalization of the concept of possibility that’s relevant to human will for the same purpose. Already, if my formalization is in the right ballpark, we see that the notion does not admit of the standard modal equation: ~□~ = ◊. It’s also interesting that modal operators don’t stack in any normal way: Usually □A is equivalent to □□A, and ◊A is equivalent to ◊◊A. But for volition, where □ means “necessary, i.e., not subject to will” and ◊ means “possible, i.e., subject to will”, □A expresses completely different content from □□A. One is a statement about the nature of A, and the other a statement about unequivocal metaphysical reality. The two expressions can have different truth-conditions. You can have that ~□A and yet have □□A.