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.

So you would say that a reductio ad absurdum is not an inference in the proper sense? — Leontiskos

It is an inference in the syntactical sense of implication, but not in the semantical sense of implication as ordinarily used by scientists and legal practitioners who are in the business of inferring facts as opposed to uninterpretable sentences.

In a consistent deductive system , If the sign "Not A" is either taken to be an axiom, or is inferred as a theorem, then it means that the sign "A" is non-referring and hence meaningless in that it fails to denote any element of any possible world among any set of possible worlds that constitutes a model of the axioms. By symmetry, the same could be said of the sign "Not A" being meaningless if A is taken as an axiom, but by model-theoretic traditional the sign A is said to not denote anything in a model if ~A is provable.

For instance, let the sign "A" denote the proposition that the weather is wet in some possible world. If "A" is deductively assumed or proved, then A is a tautology, meaning that the logical interpretation of "A" is stronger than being a mere possibility and denotes the weather being wet in all possible worlds. On the other hand, if "~A" is provable, then no possible world is wet, in which case the sign "A" fails to refer.

In conclusion, A and ~A can only both be meaningful if they both stand for possible but unnecessary states of affairs, in which case neither are provable. So the OP's problem isn't a problem, because the signs of the implication A --> ~A aren't simultaneously meaningful.

If I uttered: "If it is raining then it is not raining." ... If formal logic is "mappable" onto ordinary language, then you should be able to infer "oh okay, it's not raining." But no one speaks like that and no one would make such an inference. At least, no one would consider such an "argument" "valid." That being so, while I would prefer there not to be equivocal definitions of validity, it appears that there are, one formal, the other informal. — NotAristotle

I disagree with regards to ordinary language, because we ordinarily reject contradictory premises for sake of avoiding contradiction; we naturally reject A whenever A implies (B And Not B) for any proposition B.

One isn't inferring Not A in such cases, rather one is establishing a consistent set of premises for subsequent inferencing. This is reflected by the fact that the case you find to be problematic, is actually an alternative axiom used in the definition of negation in intuitionistic logic.

I believe you should review the definition of Dedekind cuts. First, they can't be open sets, since (as Tones pointed out) L and R are sets of rationals. — fishfry

Yes, and that's what i meant. To explain myself clearer, I meant L and R to refer to open sets of rationals together with the entire set of rationals representing +Inf and the empty set of rationals representing -Inf. I'm not sure why people might have jumped to a different conclusion.

It is right and necessary to point out as I think you are meaning to imply, that traditionally Dedekind cuts are understood as being objects derived from sets of rationals, in which the rationals are understood to be constructed, or simply to exist, prior to the creation of open sets of rationals, which are then used to define the cuts called "irrational numbers". That approach to understanding the reals is very "bottom up", and possibly in contradiction with Dedekind's own understanding of his cuts, which i suspect might have been "top down" (see the SEP for more discussion on his thoughts about the continuum in relation to actual infinity).

In my case, i am stressing the benefits of a "top down" approach, in which one uses lattice theory to define a lattice of abstract elements that is isomorphic to the open sets of the rationals extended by end points. The open sets of the rationals are only intended to serve as a model of this lattice, which is free to not assume the existence of points and other closed sets.

I agree with you, but i probably didn't make myself clear enough. I'm saying that if L| R is a Dedekind cut consisting of two open sets (as is the case when the cut defines an irrational number that isn't already contained in R), then the union of L, L|R and R is a disjoint partition of the continuum, which is semantically problematic in being disconnected (even if not "disconnected" according to the narrow topological definition of connectedness in terms of open sets only). The closed interval [r,r] is what I meant by writing [r].

Instead of points one works with lattices of open sets. I don't see this as improving the intuitive understanding of continua. Continuity in elementary topological spaces rests upon the idea of connectedness. The topology of the reals is fairly well established, so maybe start by studying this. — jgill

I'm tempted to think of Dedekind cuts as a mathematics joke, in the sense that when open sets L and R are used to define a Dedekind cut L|R for an irrational number r, the generated closed set [r] is disjoint from both L and R, and yet their union equated with the continuum. As I understand it, this disjointed representation of the continuum is in semantic conflict with the continuum's connected topology , which is ultimately the cause of the continuum being empirically uninterpretable and practically useless in real life without abuse of notation.

I think the interesting thing about the open-sets of the extended continuum (with -Inf and +Inf introduced as end points), is that they can be interpreted as representing propositions, due to the fact that they form a distributive lattice with a top element (-Inf,+Inf), whose join operation is set union representing logical disjunction, and whose meet operation is set intersection representing logical disjunction, in spite of this lattice lacking a bottom element (since the empty set isn't an open set).

Likewise, the closed-sets of the extended continuum can be interpreted as representing negated propositions, due to the fact that the intersection of an open set (-Inf, x) with a closed-set [x,y] is the empty set representing falsity. More specifically, any point [x] represents a false proposition under this interpretation, i.e. p[x] := NOT { p(-Inf,x) OR p(x,+Inf)}, where p denotes a predicate that maps open sets to propositions of some sort. This interpretation refrains from asserting the existence of a point x for which p is true, but it doesn't deny the existence of such points either. (To deny the existence of such points is to go from a pointfree topology to a pointless topology).

In short, the open sets of the extended rational numbers can represent propositions derived by coinduction with respect to a 'top' proposition that is continuous in the sense that it isn't isomorphic to any union of propositions whose domains are disjoint. This top proposition is empirically meaningful. For example, we generally don't consider a priceless Ming vase to be the same after smashing it and gluing the pieces back together. By going point-free with our continuous topology, at least initially, allows us to consider "points" as being defects that are introduced when damaging a continuum to produce a non-continuum, such as in the destructive testing of a smooth object.

Consider the operational meaning of "infinity" that refers to circular control flow that lacks a termination condition. That is what the lemniscate symbol represents. So there is at least a pictorial, operational and geometric meaning of infinity. Such flows generate infinite processes that often produce observable data on each iteration, so there is also empirical meaning with regards to the execution of an infinite process. And the controllers of the execution of the process get to decide when to terminate it once it has served its purpose, and so the ability to control such a process, as well as it's forced termination by the user, gives controlled infinite processes empirical meaning.

Karl Popper's principle of falsification refers to the case in which an observer hypothesizes that an infinite process that the observer does not control, is responsible for producing a stream of observations.

For example, suppose that an investigator conjectures that all swans are white and that he will terminate his investigation upon observing the first non-white swan. Here, the only empirical meaning that the investigator can be ascribe to his conjecture are the conditions under which his infinite process conjecture is refuted by observation of a counter-example. For Popper, such potential refutation is enough for the investigators hypothesis to be considered as scientifically meaningful. However, suppose that the investigator considered his infinite process hypothesis to be true. As a true hypothesis, it would have no empirical implications, since if it were true then the investigator will never exhaust his stream of observations so as to know that it is true. So I interpret Popper's falsification criteria as implying that infinite process hypotheses are empirically and scientifically meaningful in the sense of the criteria that falsify them, but that such hypotheses cannot be interpreted as true hypotheses, since such interpretations are meaningless.

I am however puzzled how all the members of the natural number set are finite yet it has aleph_0 members. — MoK

Semantic puzzlement at the fact that one cannot finitely bound the natural numbers is understandable, even when having no syntactical puzzlement with regards to a formal definition of the naturals.

Firstly, there is presumably no semantic puzzlement about the situation in which one constructs the naturals one by one by counting upwards from zero, for in that case one assumes full control of the number generation process that one never finishes, for which at any time one has only constructed naturals that are a constructively finite distance from zero.

So i think that semantic puzzlement is in relation to arbitrarily large natural numbers that one assumes to exist but which aren't constructively defined.

For example, consider a guessing game between Alice and Bob, in which Alice privately thinks of a natural number which Bob then has to guess. Suppose that Bob is given an unlimited number of chances for guessing Alice's number. If Alice doesn't cheat by changing her number in response to Bob's guesses, then Bob has a winning strategy that will eventually terminate, such as guessing each natural number in turn by counting upwards from zero. But then suppose Alice does cheat to ensure that Bob will lose. How could Bob ever know it? Alice could for example secretly define her number to be one greater than Bob's largest guess. In which case her natural number isn't constructively finite in relation to Bob's strategy, in spite of being constructively finite in relation to Alice's strategy in the situation where Bob eventually gives up.

Semantic contextualism needs to be distinguished from truth relativism. According to the former position, differences of opinion are not interpreted as reflecting differences in truth assessment with respect to the same set of facts, but as reflecting differences in the contextual meaning of what each opinion is asserting.

Semantic contexualism when pushed to the extreme as a dogma, interprets all assertions as being necessarily true when contextually understood and trivialism ensues, which raises the question as to whether all of the problems of epistemology reduce to the trivialities of semantics.

Formally, the classical continuum "exists" in the sense that that it is possible to axiomatically define connected and compact sets of dimensionless points that possesses a model that is unique up to isomorphism thanks to the categoricity of second order logic.

But the definition isn't constructive and is extensionally unintelligible for some of the reasons you pointed out in the OP. Notably, Dedekind didn't believe in the reality of cuts of the continuum at irrational numbers and only in the completeness of the uninterpreted formal definition of a cut. Furthermore, Weyl, Brouwer, Poincare and Peirce all objected to discrete conceptions of the continuum that attempted to derive continuity from discreteness. For those mathematicians and philosophers, the meaning of "continuum" cannot be represented by the modern definition that is in terms of connected and compact sets of dimensionless points. E.g, Peirce thought that there shouldn't be an upper bound on the number of points that a continuum can be said to divide into, whereas for Brouwer the continuum referred not to a set of ideal points, but to a linearly ordered set of potentially infinite but empirically meaningful choice sequences that can never be finished.

The classical continuum is unredeemable, in that weakening the definition of the reals to allow infinitesimals by removing the second-order least-upper bound principle, does not help if the underlying first-order logic remains classical, since it leads to the same paradoxes of continuity appearing at the level of infinitesimals, resulting in the need for infinitesimal infinitesimals and so on, ad infinitum.... whatever model of the axioms is chosen.
Alternatively, allowing points to have positions that are undecidable, resolves, or rather dissolves, the problem of 'gaps' existing between dimensionless points, in that it is no longer generally the case that points are either separated or not separated, meaning that most of the constructively valid cuts of the continuum occur at imprecise locations for which meta-mathematical extensional antimonies cannot be derived.
Nevertheless this constructively valid subset of the classical continuum remains extensionally uninterpretable, for when cut at any location with a decidable value, we still end up with a standard Dedekind Cut such as (-Inf,0) | [0,Inf) , in which all and only the real numbers less than 0 belong to the left fragment, and with all and only the real numbers equal or greater than 0 belonging to the right fragment, which illustrates that a decidable cut isn't located at any real valued position on the continuum. Ultimately it is this inability of the classical continuum to represent the location of a decidable cut, that is referred to when saying that the volume of a point has "Lebesgue measure zero". And so it is tempting to introduce infinitesimals so that points can have infinitesimal non-zero volume, with their associated cuts located infinitesimally close to the location of a real number.

The cheapest way to allow new locations for cuts is to axiomatize a new infinitesimal directly, that is defined to be non-zero but smaller in magnitude than every real number and whose square equals 0, as is done in smooth infinitesimal analysis, whose resulting continuum behaves much nicer than the classical continuum for purposes of analysis, even if the infinitesimal isn't extensionally meaningful. The resulting smooth continuum at least enforces that every function and its derivatives at every order is continuous, meaning that the continuum is geometrically much better behaved than the classical continuum that allows pathological functions on its domain that are discontinuous, as well as being geometrically better behaved than Brouwer's intuitionistic continuum that in any case is only supposed to be a model of temporal intuition rather than of spatial intuition, which only enforces functions to have uniform continuity.

The most straightforward way of getting an extensionally meaningful continuum such as a one dimensional line, is to define it directly in terms of a point-free topology, in an analogous manner to Dedekind's approach, but without demanding that it has enough cuts to be a model of the classical continuum. E.g, one can simply define a "line" as referring to a filter, so as to ensure that a line can never be divided an absolutely infinite number of times into lines of zero length, and conversely, one can define a collection of "points" as referring to an ideal, so as to ensure that a union of points can never be grown for an absolutely infinite amount of time into having a volume equaling that of the smallest line. This way, lines and points can be kept apart without either being definable in terms of the other, so that one never arrives at the antimonies you raised above.

Usually these sorts of discussions begin on the wrong foot by conflating communism with state capitalism under a ruling party, that is a situation resembling modern day corporate America in many respects, which is ironically reinforced by "communist" hating conservatives refusing to support progressive taxation.

I'm no Marx expert, but understand that he viewed communism descriptively as an inevitable outcome of capitalism, as much as he did as a moral imperative. With modern society's inevitable transition to universal income in the coming years, the appeal of communism seems besides the point.

Suppose y = sin(cos(x)). Which (sin or cos) would you say is inside, and which outside? — bongo fury

Quine was presumably referring to the stratification of types originally proposed by Russell, which ensures that the a map between type universes resides in a universe that is higher than both of the input and output universes. We might recall the fact that each universe contains a subclass that is isomorphic to the previous universe, as represented by the quotation marks in the liar sentence. So if we start with the highest level universe that we say contains everything we regard to be true, and use it to build in stepwise fashion an infinitely descending chain of so-called object languages that are each the meta-language of their predecessor, the liar sentence can be interpreted as stream of fluctuating truth values with respect to isomorphic, but non-identical terms of different types.

By contrast, both Sin and Cos are maps of type Real --> Real, i.e maps between terms of Reals, where the type Real --> Real resides in the same universe as the type Real, as does any function of functions of ... functions of reals; for they all reside in the universe definable in terms of second order logic. Following their example, we could alternatively interpret the liar sentence as directly referring to a stream of fluctuating values, where the stream and its values all exist in the same universe as a binary approximation to those trigonometric functions.