Stochastic mechanics shows mathematically that entanglement follows from a non-dissipative / conservative diffusion. — Apustimelogist

Well not according to your source "On the Stochastic Mechanics Foundation of Quantum Mechanics ".

There is no mention let alone explanation of entanglement anywhere in that paper, although there is a mention of the Bohm potential, indicating that the authors are perhaps imagining their stochastic mechanics supplemented with some other foundational interpretation, perhaps to account for the non-locality of their background hypothesis. As it stands, it is a metaphysical interpretation of the Schrodinger equation that reproduces a fragment of the least problematic parts of Quantum Mechanics with deafening silence on the most critical aspects of QM that the interpretation either fails to address, or helps itself to by appealing to unstated non-local premises.

As it stands, I view that paper, which I have admittedly scantly read, is a non-earth shattering exercise in using stochastic differential equations to simulate whatever one wishes.

This is absolutely false and A.I. do not reliably give you information. — Apustimelogist

Neither do publicity seeking authors advertising grandoise and unproven claims.

For instance, I can tell that the following phrase came from an A.I. :

that implement non-local aspects of the time-symmetric transactional interpretation of QM.
— sime — Apustimelogist

They were actually my own opinion in my own words, prompted by my understanding that the authors of the other paper you mention were reconstructing quantum diffusion out of time symmetric diffusions that is reminiscent of the symmetric casuality inherent in the transactional interpretation of QM. Personally I think that more modest paper is much more informative.

I cannot access the Levy & Krener (1996) paper, so instead I asked Grok for an account of reciprocal stochastic processes, of which I am unfamiliar which gave an interesting reply.

If the papers referred to are "Dynamics and kinematics of reciprocal diffusions" and "Stochastic mechanics of reciprocal diffusions", then I can see how these papers are of relevance to foundational questions of QM, in the sense of attempting to reconstruct properties of quantum diffusion in terms of time-symmetric but otherwise classical stochastic processes that implement non-local aspects of the time-symmetric transactional interpretation of QM.

However, Grok's conclusion at the end is along the lines of my initial thoughts, namely that such processes fail to account for quantum entanglement as should be expected by the Kohen Specker Theorem.


" Reciprocal stochastic processes are compelling because they challenge the necessity of quantum mechanics’ formalism for describing quantum diffusion. If a classical-like stochastic model can replicate quantum behavior, it suggests that some quantum phenomena might be emergent from underlying probabilistic structures, aligning with interpretations like stochastic mechanics or hidden-variable theories. However, their inability to fully capture quantum non-locality (e.g., entanglement) reinforces the uniqueness of quantum mechanics, prompting deeper inquiry into what makes quantum systems distinct.
Specific foundational questions they address:
Can quantum non-locality be reduced to time-symmetric stochastic correlations?

Is the wave function a physical entity, or can it be replaced by a stochastic process with equivalent predictive power?

How does the time-symmetric nature of quantum mechanics relate to causality and the arrow of time?

Can the quantum-classical transition be fully understood as a shift from reciprocal to Markovian stochastic processes?

Conclusion
Reciprocal stochastic processes can reproduce many aspects of quantum diffusion, such as probability density evolution, interference-like patterns, and non-local effects, making them a powerful tool for modeling quantum dynamics in a probabilistic framework. Their time-symmetric and non-Markovian nature makes them particularly relevant to foundational questions about quantum non-locality, the quantum-classical transition, and the ontology of the wave function. While they do not fully explain quantum non-locality (e.g., entanglement), they offer a semi-classical perspective that challenges quantum mechanics’ uniqueness and invites exploration of alternative formulations, such as stochastic mechanics or time-symmetric interpretations."

So i stand partially corrected.

If someone wants to call it a variation of Bohmian mechanics, I don't really see an inherent issue but you don't explicitly need Bohmian kind of non-locality for the theory to produce Bell violations. There is at least one version by Levy & Krener (1996) which is does not have Bohmian non-locality, produces all the correct predictons, and even explains that non-locality only comes when an artificially idealized assumption is used for constructing the theory. — Apustimelogist

I have a suspicion that the authors you mention aren't intending to address foundational questions of QM ,and are instead focusing on the technicalities of constructing laws and diffusion models that cohere with the Schrodinger equation, with potential relevance to the subject of modelling quantum decoherence, by which classical diffusion can emerge in the limit of quantum diffusion, but without relevance as to the question of the nature and ontology of quantum states and quantum measurements.

A quantum stochastic process that in principle can model non-local correlations, i.e. a random vector field of a complex Hilbert Space that is interpretable as the evolution of a quantum state vector, cannot be explicated in terms of the local interactions of a regular stochastic process such as Brownian motion. The former can serve to explicate the latter, but not conversely unless one supplements entanglement relations.

It is an interpretation in the sense of Bohmian mechanics, in fact their underlying mathematical structure is very similar. — Apustimelogist

That is only true if the so-called background hypothesis, which is typically assumed to consist of a random field locally perturbing the motions of the particles, is assumed to have supplementary non-local Bohmian character as necessary to explain the statistics of quantum entanglement ... in which case your preferred interpretation becomes a variant of the Bohmian interpretation.

The stochastic interpretation provides a conception of wave-particle duality with an uncertainty principle, but without additional Bohmian mechanics it cannot explain Bells Theorem, for there is no getting around the fact that classical particle-field interactions that communicate slower than the speed of light cannot explain the 'action at a distance' of Bells inequalities; either the medium denoted by the background hypothesis is local, in which case we do not have quantum semantics, else the medium has non-local effects, in which case we have Bohmian mechanics.

I think the stochastic interpretation is pedagogically useful for providing a common-sense physical explanation for potentially classical aspects of complementarity that are often mistaken for inherently quantum phenomena, analogous to how Spekken's toy model of quantum mechanics is useful for providing common-sense epistemic intuition for understanding complementarity without assuming a physical account of the Schrodinger equation. But in neither case is there either a physical or epistemic explanation for entanglement.

Bohmian mechanics also uses classical particles but it effectively just takes the quantum wavefunction and puts deterministic trajectories on top - it doesn't explain anything about why quantum behavior occurs. In contrast, stochastic mechanics starts with a classical description of particles being pushed about like the pollen in a glass of water, and shows that under specific conditions related to energy conservation, as I previously described, all quantum behavior occurs for regular classical particles. — Apustimelogist

The stochastic interpretation of QM isn't an interpretation of QM in the sense that Bohmian Mechanics is, i.e. in the sense of being a non-local or anti-realist account of quantum entanglement,, rather the stochastic interpretation amounts to a phenomenological interpretation of quantum statistics that doesn't explain entanglement and the origin of Bells inequalities.

In my understanding, a physical language per se is purely a communication protocol for coordinating human actions, that is to say physical languages per-se do not transmit information about the world from the mind of the speaker to the mind of the listener. Physical languages are to my mind, analogous to software written in a high level computer programming language, which per se is not meaningful until compiled via additional rules that are external to the language, onto a particular CPU or GPU. Physical languages, like high level programming languages, are useful as universal protocols because they do not specify their "compilation rules". And this is what it means to say that physical language isn't a phemonological language.

Physical languages are de-dicto not phenomenological; otherwise their meaning would become relativized to the thoughts and judgements of a particular speaker which would hinder their ability to function as universal protocols. Hence the semantics of physical languages are realist and this semantic characteristic of physical languages is often mistaken for a metaphysical assumption, or worse a "hard problem" of consciousness, for the phemonenological independence of physical language is in fact a hard feature of any useful communication protocol.

On the opposite extreme, a purely phenomenological language is analogous to a bespoke machine code language that is only recognized by a unique CPU with an unknown architecture. In both cases such private languages might be reverse engineered into the common parlance of physical language or a high level programming language respectively, but it should be born in mind that the assertability conditions of such private languages are not publicly known in advance, nor publicly controlled - unless such languages and the hardware they are executed on are publicly conditioned. This situation is of course the case for designed machine code running on a manufactured CPU, but much less true of the psychological dialects that people think in, since psychological conditioning is only crudely and sparsely conditioned by public input.

Ordinary "common" language is in fact an aggregate of semi-conditioned but generally divergent psychological dialects; compare the "optical redness" of physical language - whose assertibility conditions are strictly public and non-phenomenological, to ordinary "redness" whose assertibility conditions are an open-ended mixture of public and privately decided rules that vary somewhat from speaker to speaker.

(I think the realist can be one without being dogmatic!) Not sure what seems un-entertainable about that idea. Could you expand? — J

Suppose a realist insists "Metaphysical realism is true". If we understand the realist's beliefs as having a causal explanation in terms of the realist's psychological conditioning and sensory input, then we cannot interpret the realist's assertion "metaphysical realism is true" as representing metaphysical realism. Instead we must interpret his assertion as meaning what we might prefer to express by saying "metaphysical realism is false".

First we have to consider the meta-metaphysics of "mind-independence"; should mind-independence be understood to be an existential claim that the world literally exists independently of the senses? Or should mind-independence be understood as merely a semantic proposal that physical concepts are definitionally not reducible to the senses?

And even if an apparently dogmatic realist insists upon the former interpretation, should we nevertheless interpret him to be a semantic realist? For can we really entertain the idea that the realist is conceiving the world as existing independently of his senses?

But, on the other hand, our experience seems to be fundamentally temporal. A process. Not something static. If time stops, can we really speak of beatitude or torment? — boundless

The impermanence of emotions and sensations isn't necessarily in conflict with the thought that an emotion or sensation is temporally unbounded. Consider for instance the mood of grief. On the one hand the mood is all absorbing and the grieving cannot comprehend an end to their grief and locate it on a timeline, yet on the other hand the emotions of grief do in fact come to an end, in spite of the inconceivability of the end when in the state of grief.

I'm not a theologist, but I think that eternity should be distinguished from unending procession. Notably, the present can be regarded as 'eternal' in that the meaning of "now" isn't defined in relation to a time series. In this sense, your present emotions, as in the mood you have now, can be regarded as 'eternal' even though your moods are not permanent. Moods can also feel timeless in that those feelings do not involve temporal cognition. Also, the seven deadly sins seem to refer to moods rather than to actions; so I would guess that biblical references to eternal heaven or to eternal punishment should probably be interpreted in the presentist's sense of timelessness, rather than in the sense of unbounded duration.

I suspect that hinges refer to what Frege called "taking as true".

For example, I often "take it as true" that my colour judgements are synonymous with the optical colours, due to learning the colors by ostensive definition; in spite of the fact that the definition of the optical colours makes no mention of my color judgements.

If this is the case, then hinges represent an extension of thought from Wittgenstein's earlier remarks in relation to private language, and possible represent a footnote to, or even an attack on, Frege's anti-psychologism that sought to clearly delineate truth from "taking as true".

Can you explain a bit more thoroughly what you mean by "resource-conscious"? — Metaphysician Undercover

Resource conscious logics such as Linear Logic don't automatically assume that the premise of a conditional can be used more than once. They are extensions or refinements of relevance logic. The best article relating resource-sensitivity to the principles of quantum mechanics is probably nlabs description of linear logic

https://ncatlab.org/nlab/show/linear+logic

As for uncertainty principles:

Recall that classical logic has the propositional distributive law, that for all A, B and C

A ∧ ( B ∨ C) = (A ∧ B) ∨ ( A ∧ C)

Here, the meaning of "and" is modelled as the Set cartesian product, and the meaning of "or" by set disjunction, neither of which are resource conscious - therefore one always has the same cartesian product, even after taking an element from one of its sets. The negation of this principle is more or less a definition of the uncertainty principle and characterizes the most remarkable aspect of quantum logic, which is in fact a common-sense principle that is used extensively in ordinary life.

The connectives of Linear logic cannot be interpreted in terms of the cartesian product and set disjunction. Instead it has the tautology

A ⊗ ( B ⊕ C) ≡ ( A ⊗ B ) ⊕ ( A ⊗ C )

If this formula is interpreted to be a true conclusion that needs to be proven with respect to unknown premises , then it has the interpretation "Assume that we are sent an A i.e. an element (a : A), and that we are also sent either (b : B) or (c : C) at our opponent's discretion, neither of which consume the (a : A) (that is to say B and C are independent of A). Then we end up with either (a : A) and (b : B), or (a : A) and a (c : C)".

Likewise, our opponent's side of this interaction is then described by the tautology

¬A ⅋ ( ¬B & ¬C) ≡ ( ¬A ⅋ ¬B ) & ( ¬A ⅋ ¬C)

"If our sending of (a : A) also implies our sending of either (b : B) or (c : C), where B and C are independent of A , then we either send both (a : A) and (b : B), or we send both (a : A) and (c: C).

But there isn't the theorem

A ⊗ ( B & C) ≡ ( A ⊗ B ) & ( A ⊗ C )

The inability to derive this theorem is the common-sense uncertainty principle of linear logic: getting an A together with a choice of B or C for which this act of choosing is independent of the existence of A, isn't equivalent to the outcome of the choice being independent of the existence of A.

(Imagine winning a bag of sugar together with a choice between winning either ordinary ice cream or diet ice cream. It might be that the awarders of the prizes use the awarded bag of sugar to produce the chosen ice-cream.)

By analogy, by using a resource-conscious logic as the foundation of an alternative calculus, smoothness and pointedness can be reconciled by defining them to be opposite and incompatible extremes of the state of a mutable function that is affected by the operations that are applied to it. This is also computationally realistic.

In my view, Zeno's arguments pointed towards position and motion being incompatible properties, but the continuum which presumes both to coexist doesn't permit this semantic interpretation.

Is this in any way motivated by the uncertainty principle? — Moliere

If you mean the Heisenberg uncertainty principle no - although I'm tempted to think that Zeno was close to discovering a logical precursor to the Heisenberg Uncertainty Principle on the basis of a priori arguments.

The semantic problems of calculus with regards to Zeno's arguments stem from the fact that calculus isn't resource conscious. Sir Isaac Newton and Leibniz had no reason in 17th century to formulate calculus that way, given the use cases of calculus that they had in mind.

A notable feature of resource-conscious logics is how they naturally have "quantum-like" properties, due to the fact their semantic models are state spaces of decisions that are generally irreversible, thereby prohibiting the reuse of resources; indeed, the assumption that resources can be reused, is generally a cause of erroneous counterfactual reasoning, such as when arguing that a moving object must have a position because it might have been stopped.

So in the case of a resource-conscious calculus that avoids mathematical interpretations of Zeno's paradoxes (as in a function having a gradient but also consisting of points), a function must be treated as a mutable object whose topology undergoes a change of state whenever the function is projected onto a basis of functions that "measure" the function's properties -- Thus the uncertainty principle of Fourier analysis has to be part of the foundations of a resource-conscious calculus rather than a theorem derived from real-analysis of the continuum that is the cause of the semantic unsoundness of calculus with respect to the real world.

An obvious candidate for contributing to the foundations of such an alternative calculus is some variant of differential linear logic, which incidentally has many uses in quantum computing applications.

Is your point that Zeno treats motion as a series of steps, while both physics and maths treat it as continuous?

I'll go along with that. — Banno

More or less in the case of Zeno. Mathematics is often said to resolve the paradox in terms of the topological continuity of the continuum, by treating the open sets of the real line as solid lines and by forgetting the fact that continuum has points, meaning that the paradox resurfaces when the continuum is deconstructed in terms of points.

In my view, Zeno's arguments pointed towards position and motion being incompatible properties, but the continuum which presumes both to coexist doesn't permit this semantic interpretation.



Mathematical limits are proved in two steps using mathematical induction - which obviously does not involve a literal traversal of each and every rational number in order, which leads nowhere. (The proof of a limit is intensional, whereas the empirical concept of motion is extensional).

The mathematical interpretation of Zeno's paradox seems straightforward to me. Evaluating limits makes the so-called paradox disappear. What is illogical about that? And what does this have to do with calculus. Representing a continuum as an infinite series of infinitesimals seems like a good model of how the universe works, simple and intuitive. — T Clark

Zeno's dichotomy paradox corresponds to the mathematical fact that every pair of rational numbers is separated by a countably infinite number of other rational numbers. Because of this, a limit in mathematics stating that f(x) tends to L as x tends to p, cannot be interpreted in terms of the variable x assuming the value of each and every point in turn between its current position and p. Hence calculus does not say that f(x) moves towards L as x moves towards p.

Not something with which I am familiar. But in intuitionistic type theory, isn't a theorem synthetic if its truth depends on constructive proof rather than mere definitions? That is, not all synthetic theorems contain existential quantifiers. Consider "Every red bead appears before every blue bead on the string", which is not analytic, which must be determined by inspecting the arrangement of beads, and which uses universal quantification only. I may be misunderstanding your point, but being synthetic is not dependent on existential quantification only. However if your point is just that theorems containing an existential quantification are always synthetic because they require constructive proof, then yep. — Banno

Apologies for any misleading. To clarify, in type theory synthetic judgments can be identified with existential quantification due to the fact all propositions are types: having a proof that proposition A is true is equivalent to constructing a term a of type A, written a : A.

When referring to existential quantification, Lof wasn't referring to an existential quantifier within a proposition, but to an existential quantifier over terms representing a proof of a proposition type. Furthermore, the terms of a proposition type are definitionally equal by fiat, i.e a proposition type is the equivalence class of all proofs of that proposition.

My example referring to the swans was potentially misleading for conflating the two sorts of existential quantification, but nevertheless valid. A term cannot be constructed for the proposition type "All swans are white", indeed for any proposition containing a universal quantifier over an infinite domain, unless the proposition is interpreted intuitionistically such that the proposition can be proved by mathematical induction.

Perhaps a better example is the proposition "Nothing can accelerate beyond the speed of light". In relativity, a proof of that proposition implies contradiction. Hence presumably, the negation of the proposition is analytic in the theory special relativity, meaning that the proposition doesn't imply the physical impossibility of faster than light travel.

So we need necessity in order to do physics; but we must debar it from logic. A difficult path to tread. — Banno

Compare Quine to Martin Lof, the inventor of intuitionistic type theory. According to Lof, analytic sentences, at least in the context of intuitionistic type theory, are de-dicto definitions that are regarded to consist of perfect information, as in a complete table.

So in terms of your beads example, Lof would regard your proposed function mapping numbers to colors as analytic. But it is important to note the utility of calling this function definition "analytic" is only in relation to existentially quantified propositions about the analytic definition, which Lof classifies as "synthetic". E.g the theorem "there exists three red beads" is synthetic for Lof in relation to your bead function definition, because to determine the truth of the theorem requires checking.

In general, Lof regards a theorem in relation to intuitionistic logic to be 'synthetic' if the theorem contains an existential quantifier whose existence requires a proof in relation to the analytic definitions provided. Lof regards a synthetic theorem to be 'a priori' if the theorem can be proved de dicto via a process of deduction using the supplied analytic definitions that makes no recourse to facts about the external world. This is of course the case with intuitionistic logic, since its deductive system is constructive, i.e. de dicto. Hence for Lof, most of the theorems of intuitionistic mathematics are synthetic a priori (with the exception being postulated mathematical axioms). Generally, synthetic a priori propositions are undecidable.

Of course one might question whether the rules of the deductive system are correctly applied or whether one's analytic definitions are correct, in which case one's definitions are treated as being truth apt synthetic propositions in relation to some other underlying analytic definitions. So the analytic-synthetic distinction Lof intended is pragmatic without implying an absolute metaphysical distinction.

I think that Lof's reasoning is very much in line with Quine, whose notion of "physical necessity" I understand to be synthetic a posteriori, being in relation to the external world, but nevertheless also in relation to an analytic definition of physical terminology that undergoes constant revision on the basis of a posteriori evidence.

For example I imagine that Quine would consider the theorem "All swans are white" to be an analytic definition in the sense that Lof referred to, namely that the theorem doesn't contain a non-negated existential quantifier and so cannot be regarded as "true" except in the de dicto sense. This of course doesn't imply that the theorem's negation is analytic, which consists of a non-negated existential quantifier that answers to de re evidence. To me, such examples suggest that when counter-examples cause theory change, the falsified older theory is often not even wrong, in that the older theory cannot express the counter-example that it is wrong about.

It is my understanding that the appropriate mathematics didn’t exist in Zeno’s time. — T Clark

I'm not aware of a mathematical definition of an alternative continuum that resolves all of the logical puzzles posed by Zeno.

Zeno's paradoxes when interpreted mathematically, pose fundamental questions concerning the relationship between mathematics and logic, and in particular the question as to the logical foundation of calculus. The existence and utility of the classical continuum is also called into question.

I consider the most important and radical implication of Wittgenstein’s later work to be his critique of Frege’s theory of sense as reference. — Joshs

A critique of Frege's theory of sense and reference by Wittgenstein isn't possible, because Frege never provided an explicit theory or definition of sense. Frege only demonstrated his semantic category of sense (i.e. modes of presentation) through examples. And he was at pains to point out that sense referred to communicable information that leads from proposition to referent - information that is therefore neither subjective nor psychological. Therefore Fregean sense does not refer to private language - a concept that Frege was first to implicitly refer to and reject - but to sharable linguistic representations that can be used.

The later Wittgenstein's concept of language games, together with his commentary on private language, helps to 'earth' the notion of Fregean sense and to elucidate the mechanics of a generalized version of the concept, as well as to provide hints as to how Frege's conception of sense was unduly limited by the state of logic and formal methods during the time at which Frege wrote.

Frege - the first ordinary language philosopher? ;-)

Frege remained mired in a formalistic metaphysics centered on logic, without ever grasping f Wittgenstein’s distinction between the epistemic and the grammatical. — Joshs

Definitely not, for that makes it sounds like Frege was a dogmatic contrarian as opposed to the innovative and respectable founder of analytic philosophy - apparently the only thinker for whom Wittgenstein expressed admiration. As previously mentioned, Frege had already distinguished the epistemic from the grammatical when he introduced the turnstile. He knew the maxim "garbage in, garbage out".

Yet Frege's perception of propositions having eternal truth suggests that Frege might have been dogmatically wedded to classical logic that has no ability to represent truth dynamics. Indeed, I suspect that the later Wittgenstein's anti-theoretical stance was not a reaction against logic and system-building per-se, but a reaction against the inability of propositional calculus and first-order logic to capture the notion of dynamic truth and intersubjective agreement - a task that requires modern resource sensitive logics such as linear logic, as well as an ability to define intersubjective truth or "winning conditions", as exemplified by Girard's Ludics that breaks free from Tarskian semantics.

I agree. Intuition isn’t really what I was after. Wittgenstein said it better. — Joshs

Many of Wittgenstein's contemporaries said it better than Wittgenstein by formally distinguishing assertions from propositions. In particular, Frege introduced turnstile notation to make the distinction between propositions on the one hand, and assertions about propositions that he called judgements on the other.

If P denotes a proposition, then ⊢ P expresses a judgement that P holds true. Judgements can also be conditioned on the hypothetical existence of other judgements, written Q ⊢ P, where Q expresses a hypothetical judgement.

Notably, turnstile expressions don't denote truth values but rather practical or epistemic commitments, and the logical closure of such implications forms bedrocks of reasoning referred to as syntactic consequence. Of course, this does not preclude the possibility of such a collection of judgements from being treated as an object language, thereby allowing such judgements to be analysed, derived or explicated in terms of the finer-grained meta-judgements of a meta-language.

I presume the later Wittgenstein's remarks were not directed towards Frege or Russell - who essentially robbed the turnstile of philosophical significance by automating it, but at his earlier self who argued in the Tractatus that the turnstyle of logical assertion is redundant, due to thinking of propositions as unambiguous pictures of reality whose sense automatically conveyed their truth. But if this Tractatarian notion of the proposition is rejected, thereby leaving a semantic gap between what a proposition asserts and its truth value, then what does the gap signify and how must it be filled?

Evidently Frege was content to leave the gap unfilled and to signify it with a turnstile, and every logician since Russell has been content to build mathematics upon the turnstile by restricting the role of deduction to mapping judgements to judgements.

Logicians generally aren't bothered by the implication of infinite regress when explicating the judgements of object languages in terms of the meta-judgements of meta-languages, as aren't software engineers who often don't rely upon any meta-logical regression (with occasionally horrific consequences). but it apparently took Wittgenstein more time to feel comfortable with the turnstile and to reach a similar pragmatic conclusion.

A suggested computational analogy:

Non-rigid designators: Reassignable Pointers. Namely, mutable variables that range over the address space of other variables of a particular type. E.g, a pointer implementing the primary key of a relational database.

A rigid designator: A pointer that cannot be reassigned, representing a specific row of a table.

An indexical: A non-rigid designator used as a foreign key, so as to interpret its meaning as context sensitive and subject to change.

I can appreciate the distinction you are pointing out between stipulation and observation. Indeed, classical probability theory explicitly accommodates that distinction, by enabling analytic truths to be identified with an a priori choice of a sample space together with propositions that describe the a priori decided properties of the possible worlds in terms of measurable functions that map worlds to values. By contrast, statistical knowledge referring to observations of the sample space is encoded post-hoc through a choice of probability measure. I think this to be the most natural interpretation of classical probability theory, so I am tempted to think of probability theory as modal logic + statistics.

In particular, we can define a proposition p to be analytically true in relation to a possible world w if p is "True" for every path that includes w (or 'pathlet' if transitivity fails), in an analogous fashion to the definition of modal necessity for a Kripkean frame. (But here, I am suggesting that we say p is analytically true at w rather than necessarily true at w, due to the assumption that the sample space was decided in advance, prior to making observations).

By contrast, we can define p to be necessarily true at w if the set of paths including w for which p is true is assigned a probability equal to one. Thus a proposition can be necessarily true without being analytically true, by there existing a set of paths through w that has probability zero for which p is false.

That's what I thought. "One simple space" - so the step-wise structure disappears? That would presumably be the case if we implemented S5 in this way. — Banno

I'm not quite sure what you meant there, but to clarify, a sample space S can fully and faithfully represent any relation that is defined over a countable number of nodes, in terms of a set of infinite paths over those nodes.

However, speaking of probability theory in the same breath as modal logic seems to be uncommon, in spite of the fact that modal logic and probability theory have practically the same models in terms of Boolean algebras with minor changes or small additional structure that has no bearing with respect to the toy examples that are used to demonstrate the meaning of the theories.

Notably, the logical quantifiers of any decidable theory that has a countable number of formulas can be eliminated from the theory by simply introducing additional n-ary predicate symbols. And since modal logic refers only to fragments of first order logic, then unless the modalities/quantifiers are used with respect to undecidable or uncountable sets of propositions, then they have no theoretical significance and one might as well just stick to propositional logic. To me this raises a philosophical paradox, in that the only propositions that give the quantifiers/modalities philosophical significance are the very propositions that the quantifiers/modalities cannot decide.

In considering this I have been struck by how accessibility in modal logic resembles a Markov process, with states resembling possible worlds and transition probabilities resembling Accessibility relations. A directed graph resembles a Kripke frame... but Markov processes are not binary, unlike modal logic. Would that I had a stronger background in the maths involved. — Banno

Your suggestion is essentially equivalent to what I suggested in my last post, and indeed the likely tool for constructing the sample space i was referring to.

A Markov Kernel on a measurable space (S,B) onto itself, i.e. (S,B) --> (S,B), is a direct way of defining a state-transition probability matrix on a generally infinite set S. But as you indicate, what is needed is a binary valued state-transition matrix rather than a probability matrix. This just means swapping the state-transition probability measure B x S --> [0,1] for an unnormalized binary valued measure B x S ---> {0,1}. By iterating this 'markov process', one obtains a trip on S. The construction I suggested earlier that directly identified trips with events, has one sample space that consists of the product of n copies of S:

S1 x S2 x .... Sn.

in which the sigma algebra of possible trips obeys the accessibility relation.

But surely, ignorance is directly related to probabilities. If an event has a probability of 1, you can predict it perfectly; if all the probabilities are equal, then its like maximal unpredictability. — Apustimelogist

The distribution of an unknown random number generator could equal anything. If an analyst knows that he doesn't know the rng, then why should he represent his credence with a uniform distribution? And why should the ignorance of the analyst be of interest when the important thing is determining the function of the unknown distribution?

The probability that some hypothesis was the cause of your observation; and even if your prior is wrong, probability theory is the only logical way of changing probabilities when you see the evidence if you know the likelihood afaik. — Apustimelogist

Ever heard of imprecise probability?

K. In probability theory possible worlds are elements in a sample space, which consists in all possible outcomes of some experiment. These possible worlds are fixed by the definition of the probability space, they are mutually exclusive in that only one world can be the outcome of any one experiment. They are not hypothetical, but points in a mathematical space.

Wearers possible worlds in modal logic are stipulated, are not mutually exclusive and sit within a structure R which determines what worlds are accessible, one form the other. — Banno

Yes, you're right to challenge my previous post, as I realize that I wasn't quite correct in my interpretation of possible worlds in probability theory. But I still see no fundamental incompatibility.

Ultimately, i think the question we're addressing is "Can a set of possible worlds be adequately modelled in terms of a sigma algebra defined over a sample space?"

I think the key is to think of an element of the sample space as a trip through possible worlds that obeys the accessibility relation. This is essentially how finance uses probability theory when modelling movements of a stock price, where an element of the sample space is a sequence of binary values representing a sequence of price directions. Following this approach,

- An event is a possible trip through possible worlds.
- The sigma algebra defined on the sample space represents the possible history of the trip at each stage.
-A stochastic process represents possible histories of observations as the trip proceeds.
- An additional element can be added to the sample space to represent termination of the trip.

in probability theory the possible worlds are the outcome of a stochastic process, a coin flip or whatever. But in Modal Logic possible worlds are stipulated, hypothetical stats of affairs. They are not the same sort of thing. Care is needed in order to not be misled by the analogy. — Banno

No, it is the same in probability theory. There, the "set of possible worlds" refers to the sample space, where a "possible world" is normally referred to as an event or element of the sample space. A coin flip or stochastic process refers to a random variable, namely a function whose domain is the sample space and whose codomain is another set, usually the reals or the naturals.

So the input to a stochastic process is a particular possible world, of which the output is a set of observations of that possible world.

Any accessibility relation defined on a set of possible worlds can be interpreted as placing restrictions on the probability measure defined on (a sigma algebra of) sets of the possible worlds.

(post recently edited due to a mistake when describing the codomain of random variables)

Then how are you supposed to update your ignorance when you encounter new evidence? — Apustimelogist

Knowledge is represented in terms of

1) A deductive system, that apart from the logical connectives is comprised only of constants, sets, types and functions, e.g such as a model of a road network.

2) Statistics that report how the deductive system is used, e.g traffic statistics.

It makes no sense to represent ignorance. To me that's a contradiction in terms.

Structural Equation Models are another reasonable example, provided one steers clear of non-informative priors and sticks to making deductions rather than making inductive inferences; Personally, I think Bayes rule should only be used when inferring a conditional distribution of a known multivariate distribution, for what does it mean to say that " Hypothesis A is inductively twice as probable as Hypothesis B when conditioning on an observation"?

It doesn't, at least not in the Principle of Indifference as described by Leplace, Keynes, etc. It's the simplest non-informative prior. Obviously, it cannot be applied to all cases, rather a special set of them. But the general reasoning used here tends to be at work in more complex non-informative priors. — Count Timothy von Icarus

The Principle of Indifference is supposed to be a normative principle for assigning probabilities on the basis of ignorance. As soon as a non-informative prior is used, posterior probabilities are epistemically meaningless in general, even if their distributions are useful for convergent machine learning.

The way i interpret non-informative priors is in terms of the following analogy:

Imagine using a net to catch a fish in a lake. Using a big net that covers the entire surface of the lake is analogous to using a non-informative prior. Reeling in the net to obtain the fish is then analogous to Bayesian updating. But would you really want to say that the net represents your indifference as to where the fish is? rather, isn't the net simply part of a mechanical procedure for ensuring the fish is caught, irrespective of your state of mind?

- Perhaps a Bayesian will remark that the net represents the fisherman's credence as to where the fish is. I think my reply would be to say that the meaning of "the fisherman's credence" should be given in terms of where the net is, rather than the meaning of the net being in terms of "the fisherman's credence" which I have no prior understanding of.

Also, why choose the simplest prior? Occams Razor? what justifies the use of that?
In fact, if one isn't interested in asymptotic Bayesian convergence and has no frequency information, then why use a prior at all? Why not just stick to saying what one knows or assumes, and gamble without saying anything else?

Kolmogorov's axioms effectively define probability in terms of a collection of sets of possible worlds, together with a probability function that maps those sets of possible worlds to values in the unit interval, where the accessibility relation between worlds is implicitly represented by one's design choices. As for whether the probability function denotes logical or frequential probability, this depends on how the probability function is defined.

If the probability function is defined so as to quantify the mathematical proportion of possible worlds having a particular property, then we are dealing with logical probability, but not necessarily frequential probability. For example, if there are three possible worlds of different colours, then why should the existence of these three distinct possibilities automatically imply that each colour is equally likely or frequent? In my opinion, the fallacy that logical probability implies frequential or even epistemic probability is what gave rise to the controversial and frankly embarrassing Principle of Indifference.

On the other hand if the probability function is chosen to represent non-mathematical facts concerning observational frequencies, then we have frequentialist probability but not logical probability.

In my opinion, there is no such thing as epistemic probability or propensity probability, because I think that the belief-interpretation of probability consist of a poorly articulated muddle of logical probability, frequential probability, and unarticulated subjective bias that at best expresses the mental state of the analyst rather then the phenomena he is predicting; of course mental states and reality are sometimes correlated but not always.

The best way of expressing ignorance with regards to the likelihood of a possible outcome is simply to refrain from assigning a probability, and the best way of using Bayesian methods is to interpret them as inferring frequency information from logical information expressed in the design of the sigma algebra over the sample space, plus observational frequency information expressed in the probability measure.

I would regard the presumption that other beings are like myself as apodictic. I wouldn’t be so egotistical as to believe otherwise. And real life is not a hypothetical exercise. — Wayfarer

It depends what you mean by apodictic. Anti-realism doesn't necessarily deny the possibility of logical certainty with regards to the existence of other minds - on the contrary, if 'other' minds are considered to refer to a psychological aspect of the observer who interprets phenomena , then anti-realism could provide a more compelling account than Cartesian minded realism as to why the existence of other minds cannot be denied. On the other hand, such apodicity would be relative to the observer, perhaps with one observer insisting that a chat bot is conscious and the other insisting otherwise, without there existing an observer-transcendent matter of fact to settle the issue.

Anyway it makes perfect intuitive sense to me. Even though I don't know other people in the same way I know myself, I know they are persons like myself. 'Husserl explores this through the concept of empathy (Einfühlung). He suggests that we "appresent" or co-present the other’s mind: we perceive another body as similar to our own and, by analogy, attribute to it a consciousness like ours.' I've often opined that empathy is the natural antidote to solipsism. — Wayfarer

Certainly empathy is an antidote to psychological solipsism. But does empathy refer to other minds 'in themselves' that possess an existence that is independent of one's experiences of empathy? Didn't Husserl appreciate that methodological solipsism cannot establish the metaphysical realism of other minds?

If we consider borderline cases in the animal kingdom or in AI, the public make wildly different judgements as to the sentience that they ascribe to the entities concerned. Suppose that Alice and Bob are two equally brilliant and informed cognitive scientists who nevertheless disagree as to the sentience they each ascribe to a borderline case 'X'. Are they disagreeing about the same thing? Or is their disagreement akin to an aesthetic disagreement about X that isn't expected to have an objective answer?
According to the anti-realist, Alice and Bob's disagreement as to the sentience of X is only an objective disagreement in so far that their disagreement is the product of different scientific understandings of X. So if Bob and Alice are assumed to have a full and equal scientific understanding of X but nevertheless disagree as to its sentience, then the anti-realist considers their disagreement to be a subjective disagreement that only expresses the fact that Alice and Bob are in different psychological relationships to X. The anti-realist can consider disagreements over the sentience of X to be ethically important, without considering the disagreements to have epistemological or metaphysical significance, at least not from a public perspective.

Does Husserl believe in the existence of other minds in the literal sense of other?

I posed that question to Grok, which returned the following slop(?)

"Husserl didn’t deny the existence of other minds outright; rather, he approached the issue phenomenologically, focusing on how we experience and constitute the "other" within consciousness. He argued that we cannot directly access another person’s mind as we do our own—our own consciousness is given to us immediately through self-awareness. However, he posited that we encounter others indirectly through their bodies, which appear in our perceptual field as more than mere physical objects. This is what he called the experience of "alterity" (otherness).
In the Fifth Meditation, Husserl explores this through the concept of empathy (Einfühlung). He suggests that we "appresent" or co-present the other’s mind: we perceive another body as similar to our own and, by analogy, attribute to it a consciousness like ours. This isn’t a naive belief or a logical inference but a fundamental structure of experience. For Husserl, the other is not just a theoretical construct; the other’s existence as a minded being is given in a unique, indirect way within our intentional acts.
So, yes, Husserl did believe in the existence of other minds in a literal sense—as distinct, conscious entities separate from oneself—but he reframed this belief within the limits of phenomenological evidence. He didn’t take it as an unexamined assumption (as in everyday realism) but sought to show how this "otherness" is constituted in our subjective experience without stepping outside his transcendental method."

I'm not sure that I follow... In my opinion, "other" minds are a misnomer, because i consider my conception of "other" minds to logically reduce to my personal thoughts, feelings and observations. So I accept that "other minds" exist in a manner of speaking, but i am an anti-realist with respect to their existence. Therefore I am not troubled by disagreements or uncertainty with regards to whether a given machine, animal, vegetable or mineral is conscious or not, and I feel no compulsion to settle the matter one way or another. As I understand it, if Bob questions whether Alice has experiences, he is ultimately questioning the course of his experiences, for Bob doesn't possess a concept of Alice's experiences that is distinct from his own.

Is that what Husserl thought?

How is it possible for me to believe, when I am asleep, that something is real, which is completely distinct from, and inconsistent with, what I believe is real when I am awake? — Metaphysician Undercover

A simple explanation is amnesia; ordinarily, you cannot remember your waking life when dreaming. Hence the reason why wannabe lucid dreamers habitually question whether they are dreaming during their waking lives, in the hope that their habitual questioning will continue when they are dreaming.

I think an interesting philosophical question is whether lucid dreams should be regarded as being a distinct category of dreams, or whether lucid-dreams should be considered to be an oxymoron that consists of tradeoff between awareness and dreaming, or even whether lucid dreams should be regarded as ordinary dreams in which one merely dreams that one is lucid.

As Stephen LaBerge famously established, there is at least a behavioural distinction between lucid dreamers and ordinary dreamers, in that dreamers who are lucid can communicate with the outside world during REM sleep. This is coherent with the idea that lucid dreams are a trade-off between dreamfull sleep and wakefulness. Certainly my own lucid dreams are never as creative or as believable as my non-lucid dreams, and I much prefer a creative and inspiring non-lucid dream in which I have no awareness that I am dreaming, over a boring and predictable lucid dream in which I am vigilantly aware that I am dreaming. (Doesn't the "dream AI" always suck in a lucid dream in comparison to an ordinary non lucid dream?)

However, this behavioural distinction isn't available to the dreamer himself, for the dreamer doesn't have external access to his own physical body from the outside - whether asleep or awake. So in spite of the lucid/non-lucid dream distinction having objective scientific validity, this does not in itself imply that the lucid dreaming/non-lucid dreaming distinction has subjective validity. For all that is available to the dreamer is dream content. So upon waking up from a lucid dream, one is right to ask whether their lucid dream involved actual wakefulness when dreaming, or whether their lucid dream was merely a dream of wakefulness.

It isn’t necessary to use a notion of flow to address the necessity of the inclusion of past in the experience of the punctual now. Regardless of whether we attend to a discrete ‘state’ vs a flowing continuum, in either case the ‘now’ we experience includes within it the just past. — Joshs

Sure, but if the psychological past is part of a mutable mental state, then you presumably mean the "just past" in a manner of speaking, in the same way that we might say that a copy of yesterday's newspaper is about the past and Old Moore's Almanac is about the future. In both cases, we are at liberty to provide a definition as to what it means to treat an object as a 'past-referring' record or as a 'future-referring' prediction, that in the final analysis makes no mention of a B series and that reduces to observations and actions that as a matter of tautology can be said to be only of the present.

Without awareness of time there is no awareness of the continuity of the flow of experience. — Joshs

I can experience a gradual change of pitch played on a violin (portamento), but I cannot make empirical sense of a flow of "experience" unless the word "experience" is substituted for a given phenomenon, such as the portamento.

Hypothetically, I think that if I were to fully attend to the portamento, I would no longer have the impression that the portamento consisted of a sequence of particular notes. Conversely, if I were to pay full attention to the notes played, I think that I would no longer hear a portamento but a glissando consisting of a broken sequence of tones.

The intuition that a phenomenon flows is in conflict with the intuition that the phenomenon is comprised of a sequence of states, as per Zeno's Paradox. So if talk about experience deflates to talk about phenomena, and if the nature of phenomena is relative to how it is attended and phenomena doesn't always flow, then must the existence of phenomena necessitate the a priori existence of a psychological time series?

One should always start by mentioning Mctaggart on these sort of topics.

The Cartesian coordinate system represents movement, in the sense of remembered displacement spatially, in terms of a partial order on the space and time axes. Such pictures include the "Block Universe" that subsumes McTaggart's B series but does not represent any perspectival understanding of time in terms of McTaggart's A series which only makes mention of the indexicals "past" "present" and "future". This is a serious ontological limitation of pure B series reasoning, because any reasoning restricted to the B series which by assumption is an immutable series, cannot serve as a ground for present, past or future experiences, given the fact that the tenses are mutuble.

McTaggart famously argued that the A series is "unreal", on the basis of what he thought to be logical inconsistency; how can any contingent empirical proposition, say "the cat is presently on the mat", be true when said now but false when said in the past or in the future? For such propositions make no explicit reference to any underlying series. In the end McTaggart failed to find a satisfactory temporal ontology to overcome the issues he raised, but he believed that the A series when taken together with some hypothetical C series that he only partially explicated, could reconstruct the so-called B-series in a non-contradictory fashion. In my primitive understanding, his conception of the C series seems to bear similarities to what are called domains in computer science, which can be thought of as a "growing block" model of accumulated and consistent information. On that interpretation, the B series might itself reduce to some more fundamental concept of consistent and accumulative information.


In a nutshell, McTaggart meant that time was "unreal" in the Hegelian sense (i.e. still real, but in some other sense than the tenses suggest), as opposed to unreal in the Kantian sense of denying any objectivity with regards to a B series, even in the sense of rationally reconstructed noumenal object (which to many Kantians would amount to a contradiction of Kantian logic).

As for Wittgenstein, IIRC he once considered the concept of time as being factorizable in terms of a 'subjective' component he called "memory time" and an 'objective' component he called "information time". My impression of the former is that it was a weaker concept than the A series that did not include the 'eternal present' of the Tractatus, and that also did not assume that a person's memories were ordered in the asymmetric and transitive fashion assumed by McTaggart. As for Witty's conception of "information time" it also did not include the eternal present, but seemed to refer to the instrumental usage of concurrency and synchronization, as per a physicist's usage of "time".

The challenge for the presentist who prioritises the reality of phenomena to the point of denying the reality of the B series, is to reconstruct the B series 'as use', in terms of temporal cognition from the perspective of a solitary individual.

Anselm's ontological argument presents a few riddles for cognitive science, and presents a problem for Realism in general.

Suppose you are a self avowed Realist who is currently in America, and you want to justify making a conceptual distinction between your thoughts about Paris on the one hand, and the actual place called Paris on the other, that you like to think of as 'transcending' your personal experiences. How can you justify your a priori distinction to yourself without appealing to Anselm-like ontological arguments?

Similarly, when anxious we often like to remind ourselves that our feared imagined future isn't the same thing as the actual future. One way we might convince ourselves of this distinction is by imaging the 'actual future' more vividly and 'realistically' than our feared future. This cognitive therapy, which essentially involves replacing one delusion with another, is the same as the 'step' as in Anselm's ontological argument.

I would hazard a guess that although people neither need nor use 'reality' in the sense of an all-encompassing and absolute concept in their everyday lives, when pushed into a corner to provide a definition of 'reality' they will offer a circular definition of "reality" very much like the average philosopher, that more or less amounts to the most realistic thing they can conceive, that must 'exist' because nothing greater can be conceived.

Beliefs are curiously foundational in regard to actions. That I went to the tap to get a glass of water is explained by my belief that the tap was a suitable place to obtain water together with my desire for water. That I believe the tap a source of water is sufficient, regardless of of whether the tap works or not. While it makes sense to ask why I believe the tap a source of water, it is somehow incoherent to ask if I believe the tap to be such a source, given my actions and assertions. — Banno

In computer science, the problem of inverse reinforcement learning can be thought of as the problem of determining what an agent believes on the basis of the regularity of its actions. It is for example used by retail store websites for predicting what consumers want on the basis of their browsing behavior.

There is a chicken-and-egg problem; for any hypothesis as to what an agent believes is relative to a hypothesis as to what the agent is trying to achieve. And any hypothesis as to what an agent is trying to achieve is relative to a hypothesis as to what the agent believes. But in the end, the notions of beliefs and goal-states are only used for determining a causal model for predicting or controlling agent behavior that only employs the concepts of causation and behavioral conditioning; for once the causal model has been determined, beliefs and goals can be dispensed with entirely, along with the teleological illusion of future-directed behavior.

So at least according to the algorithmics of machine learning, beliefs and goals aren't foundational when it comes to explaining behavior, rather they are concepts concerning model-fitting strategies for determining behavioural causes and behavioural conditioning.

I understand the inscrutability of reference, and more generally the indeterminancy of translation to be more or less equivalent to contextualism as opposed to relativism, because semantic indeterminancy is a theory (for want of a better word) of meta-semantics that in effect considers the meaning of a proposition to be relative to the context of the agent who asserts the proposition, and hence the public inability to know what the speaker is referring to - as opposed to relativism that is a theory of truth that considers truth to be relative to the speaker.

To my understanding, relativism actually presupposes non-contextualism, because it must assumed by relativism that debating communicators are at least talking about the same referents if those referents are to be assigned conflicting properties or truth values by the debaters. On the other hand, if we do not assume that the debaters are referring to the same thing, then we have no basis for inferring that the debate is a disagreement about reality. In fact, I consider relativism to be self-inconsistent (for how can the truth be considered to be relative, either from an individual or collective perspective?). I think relativism is mainly motivated by a lack of appreciation for, or misunderstanding of, the logic of contextualism.

E.g suppose Bob insists that "The Earth is Flat". Then it is natural to also suppose that at the very least, there exists external physical causes and internal psychological causes for Bob's assertion, but the chances are the topology of the Earth is a negligible causal factor with regards to Bob's assertion, especially if it is assumed that the Earth is Round. So an objective semantic analysis of Bob's assertion cannot use the topology of the Earth as the referent of Bob's assertion.

Essentially, there is a conflict between

1) Interpreting a proposition as referring to a given state of affairs, and
2) Interpreting the proposition as being wrong about that state of affairs.

For this reason, I suspect that the concept of belief states is inconsistent and that beliefs don't exist in the sense of mental states, such as propositional attitudes.

"I disagree with regards to ordinary language" I'm not quite getting it, what is the disagreement you have concerning ordinary language? You think someone would make an inference from A->not-A to therefore not-A in ordinary language? — NotAristotle

The formal meaning of negation in intuitionistic logic refers to the syntactical inconsistency of the negated sign, rather than to a purported semantic counterexample denoted by the negated sign. Classical logic inherits the same meaning of negation from intuitionistic logic, except for infinitary propositions that appeal to the Law of Excluded Middle, which have no scientific or commonsensical application. So we should stick to discussing negation in intuitionistic logic, before proceeding to other formal logics such as affine linear logic, whose concept of negation is closer to ordinary use. In such cases (A --> Not A) --> Not A is not derivable, corresponding to the fact that Not A obtains the same semantic status of A.

But can we elucidate the meaning of (A --> Not A) --> Not A in the systems for which it is valid, by appealing to the mutually exclusive states of the weather? Suppose that a weather forecaster said "It is raining in Hampshire therefore it is not raining in Hampshire". Jokes about the english weather aside, wouldn't you assume that they were talking about anything apart from the weather in Hampshire? in which case your abstaining from assigning a meaning to their words would resonate with the formal meaning of negation in intuitionistic and classical logic.

As for formalities,

(A --> ~A) --> ~A is little more than the obvious identity relation ~A --> ~A, due to the fact that ~A is definitionally equal to A --> f , where f denotes absurdity. So we at least have

(A --> f) --> ~A

But the only means of obtaining f from A is via the principle of explosion (A And ~A) --> f. And so it is sufficient that A implies ~A.

(A --> ~A) --> ~A

And since the converse direction is immediately true, we could in fact define the negation of A to be the fixed point of the expression X => (A --> X) that Haskell programmers call a Reader Monad.

~A = A --> ~A
~A = (A --> (A --> (A --> ..... ) ))

which serves to highlight the meaning of Negation As Failure (NAF); A proof of ~A amounts to a finite proof that the right hand side doesn't converge, which represents an infinite failure to prove A by random search. But if we haven't managed to prove either A or ~A using our available time and resources, then we are at liberty to declare ~A by decree and reason accordingly, in which case ~A serves to nullify any hypothesized A by turning it into ~A, so as to ensure consistency with our failure to decide the issue, at least for the time being...

So in common-sense Kripkean semantics,

A --> NOT A says: all worlds that satisfy A also satisfy NOT A.

But in Kripke semantics, a world satisfies NOT A if and only if it doesn't satisfy A. So the set of worlds S that satisfy this condition is empty. A forteriori, there aren't any worlds in S satisfying A. Therefore
NOT A is true, and A refers to nothing.