Yep. Yet the limit is not something the sequence is chasing, but a property of the sequence as a whole...? — Banno

It depends on whether you are referring to a recursive sequence or to a choice-sequence.

A recursive-sequence is an algorithm for generating a sequence prefix of any finite length, where a limit refers to a convergence property of the algorithm, as opposed to referring to a property of any prefix that is generated by using the algorithm.

On the other hand, a choice-sequence S is an unfinishable sequence of choices that is both

- Dedekind finite - meaning we don't have an injection N --> S.
- Of unbounded length - meaning we don't have an injection S --> {0,1,2,..n} for any finite n.

Such potentially infinite sequences do not possess a limit unless the choices are made in accordance with an epsilon-delta strategy that obeys the definition of "limit". So in this case, we can speak of approaching a limit, because Eloise and Abelard are endlessly cooperating to produce a strategy for continuing a live sequence that literally approaches their desired limit, as opposed to the previous case of Eloise having a one-move winning-strategy when competing against Abelard for proving a convergence property of a dead algorithm.

Unfortunately, ZFC grounded classical mathematics cannot formally recognize potentially infinite (live) sequences due to the axiom of Choice that "finishes" them. Hence there is a clash between common-sense mathematical intuition (i.e. intuitionism) on the one hand, that correctly thinks of infinite sequences as referring to either unfinishable processes or algorithms, versus the formal straight-jacket imposed by the timeless world of ZFC, that cannot express the notion of a live process approaching a limit.

By default, classical mathematics is implied when talking about calculus, and even though ZFC isn't explictly assumed in textbook discussions of calculus, the logic they appeal to when discussing logical concepts such as limits, is classical in which calculus proofs are inductive proofs, which aren't applicable when reasoning about choice sequences, whose proof-theory is coinductive.

(I've never read a textbook definition of a limit as a two-player game - but they nevertheless informally appeal to such games when encouraging students to rote learn - a short term pedegogical payoff leading to long-term confusion after the students forget the game-theoretic reasoning behind the proofs)

A series limit isn't a literal sum of an infinite series, unless the number of summed terms that are non-zero is finite. E.g.

The infinite sum of (1,1,1,1,0,0,0,0....) = 4, as is its limit.

The infinite sum of the geometric series (1,0.5,0.25,...) is technically undefined, for in this case, every partial sum S(n) is non zero, since S(n) = 2 - 0.5^(n-1).

Sure, we can call 2 the "infinite sum", but it isn't an infinite sum in any literal sense of the word, rather 2 is the limit of the geometric series.

A limit isn't defined as a position on a sequence, but is defined as a finite winning strategy in a finite game, that involves cutting off the tail of an infinite sequence at a position δ, such that the height of the tail is within a prespecified distance ε to a prespecified value called "the limit", as per Cauchy's (ε, δ) definition of a limit.

E.g we have a sequence game S. Eloise first chooses a value v, then Abelard chooses a positive rational ε, then Eloise chooses a natural number δ in response. Abelard is now tasked with choosing an n greater than δ such that |S(n) - v| >= ε, otherwise Abelard loses. If Eloise's choice of v can guarantee her victory over Abelard, then we say that the limit of S is v.

There is no approaching the limit when proving a limit, for a proof of a winning strategy for Eloise doesn't involve multiple rounds of Abelard choosing ε1 then Eloise choosing δ1, then Abelard backtracking to choose ε2 < ε1 then Alice choosing δ2 > δ1 etc. Rather, a proof of a limit is just an inductively defined function ε -> δ established in two steps, for mapping Abelard's possible choice of ε to Eloise's choice of δ.

Since theories are usually ambiguous and interpreted within a specific context, I would say:

A theory's interpretation is unfalsifiable if the interpretation does not imply a means for potentially refuting the theory under that intepretation.

For example, we can probably all agree that "All Swans are White" is a falsifiable proposition; I say "probably" because I am assuming we can all agree that a "swan" isn't white de dicto but de re, and that we can all agree that whiteness is a publicly observable and testable empirical category.

On the other hand, if either of those two assumptions fail, such that the theory is no longer interpreted as implying a means of potential refutation, then the interpretation of that theory is unfalsifiable.

We should first forget all about AI and focus instead on the meaning of "other minds".

If Alice judges her human friend Bob to be sentient, then does her judgement concern properties that are intrinsic to Bob, or does her judgement merely express her relationship to Bob?

It all hinges upon whether the cartesian notion of belief is admissible. According to naturalized epistemology, beliefs are stimulus-response dispositions that are conditioned by a community to approximately reproduce some aspect of a shared semantic network, as when training an LLM to respond "correctly".

Ask ChatGPT what its hinges are. Even if you agree with its answer, is it in a position to know what it says, given that its output is a deterministic consequence a sequence of transformer blocks applied to a query? Where precisely do hinges fit in the machine learning pipeline?

We have two semantically separate layers:

1) The stimulus-response conditioning of a particular individual, which fully explains that particular individual's verbal behaviour; he is indeed referring to a beetle in a box that only he can see.

2) An inter-subjective protocol for coordinating verbal behavior, where the correctness criteria defining the protocol is invariant with respect to every speaker's stimulus-conditioning, and so has no conception of beetles in boxes.

Hence if we say "In Michael's opinion, the water is cold", it would be a grave but popular philosophical misconception to think of this "opinion" as referring to a belief-state of Michael's in relation to a universally accessible truth; for every speaker can only accesses their private truth conditions and nobody elses, as determined by their personal and bespoke mental conditioning.

So instead, we should think of the above sentence as distinguishing Michael's stimulus-response conditioning from the rules of the protocol.

However, we might in accordance with our use of the protocol, decide "Michael is wrong, for the water is actually hot". In which case, we aren't talking about an epistemic error on behalf of Michael in the absolute sense of Michael having a false belief state in relation to a universal truth, rather we are simply referring to Michael's remarks being in violation of the rules of our protocol:

For if we accept that Michael's verbal behaviour is the causal expression of Michael's stimulus-response conditioning, then Michael cannot be literally intepreted as having a false belief in relation to a universal truth. All that we can allege when alleging epistemic errors, is that a person's verbal behaviour was in violation of our lovely communication protocol.

Or, we can represent motion as discontinuous, which is the way that quantum physics seems to demonstrate is the real way. The particle has a position, then it has another position, without traversing the intermediary. I believe, that what happens in between cannot completely be represented as "a smooth and differentiable continuous topology". Issues with the wavefunction demonstrate that this is not quite right. So what happens in between ought to be represented as truly unknown, though it is actually represented in a not very accurate way, as a continuous topology of superpositions. — Metaphysician Undercover

Yes, there is no traversing anything unless a particle is in a smooth motion-state as a result of applying a motion operator to it, which cannot be the case if the particle is in a spiky position-state as a resulting of applying a position operator to it. The question is, at what level of explanation should this incompatibility be situated? at the physical level, as physics usually assumes, or at the level of the rules of mathematics?

I think we should consider the fact that Newton and Leibniz didn't invent calculus for the purpose of solving Zeno's paradox, but for describing trajectories under gravity. Hence the mathematical definition of differentiation that we inherited from them and use today, isn't defined as a resource-transforming operation that takes a mutable function and mutates it into its derivative; rather our classical differentiation is merely defined as a mapping between two stateless and immutable functions.

But if Zeno's paradox is to be exorcised from calculus, such that calculus has a dynamical model, then I can't see an alternative than to treat abstract functions like pieces of plasticine, that can be sliced into bits or rolled into a smooth curve, but not at the same time.

Zeno pointed out the impossibility of enumerating, in order, the dense order of rationals.

E.g, starting from 0, what is the next rational number to count? since this doesn't have an answer (when counting in order), this means that the topology of the rational numbers cannot represent dynamics. Sure, the rationals can represent displacement, including an infinite sum of displacements, but not the process of displacement, namely motion.

To represent motion in a way that avoids the paradox, requires a smooth and differentiable continuous topology that doesn't contain points that are in need of traversal, but only open sets that can finitely intersect to create spots, but not infinitely often so as to create points. Yet on the other hand, to represent positions requires a discrete point-based topology of infinitely thin spikes that doesn't blur position information. Hence motion and position require incompatible topologies.

Fourier Transforms are the best way to visually understand the solution to zeno's paradox, but on the proviso that the FT is understood as creating an output topology (e.g motion) from a different input topology (e.g position).

Again, there seems to me to be a bunch of errors in what you have said here. The core one seems to be equating P(N) with the decidable sets.

The statement “We can construct an injection P(N)→N via Turing machine encoding of decidable sets”
would mean every subset of N can be uniquely encoded by a natural number. But that is equivalent to saying ∣P(N)∣≤∣N∣, which directly contradicts Cantor’s theorem. So if the statement were true, Cantor’s theorem would already be false.

There are undecidable subsets of N. We cannot construct an injection P(N)→N via Turing machine encoding of decidable sets

I'll stop there. I can't see that your account works. — Banno

I'm not saying that the powerset of N is defined as only referring to the decidable sets (apologies if that is how it looked). Rather, I was overloading the notation of P(N) to refer exclusively to the decidable subsets of the natural numbers (i.e. to what is sometimes written Pdec(N))), in order to inspect what CSB implies in that special case, because the results are illuminating.

A decidable set A is a set whose members can be enumerated, and whose complement ~A can also be enumerated. Applying diagonalization to A, as per Cantor's theorem, must produce a novel decidable set, at least if we assume that an injection N --> Pdec(N) represents an effective procedure (a point that I ought to have stressed earlier). Thus to improve upon the above, diagonalization shows that:

if N --> Pdec(N) is a computable injection, then N --> Pdec(N) cannot be a computable surjection. Hence in this case, diagonalisation is a proof of the undecidability of the halting problem, rather than a proof of "more numbers".

Furthemore, Pdec(N) --> N exists as a computable injection. Hence according to CSB, N --> Pdec(N) must necessarily exist as a surjection, which is false if by surjection we mean a computable surjection.

The biggest failure of CSB in this context, is its insistence that if A --> B is a surjection, then a surjection B --> A must also exist. This is constructively false as discussed above, and the reason for why cardinal arithmetic is pointless, misleading, and false from a computational perspective.


Formally, it is might be said that CSB isn't refuted by the above, due to an "apples versus oranges" argument: Classical set theory makes no reference to decidability, meaning that N --> Pdec(N) is allowed to exist in ZFC 'rent free' as a non-computable surjection in the strictly syntactical sense of a quantiified predicate that cannot be converted into data.

In the constructive setting however, CSB is usually said to be false rather than inapplicable.

Personally, I am of the opinion that CSB along with infinite cardinal analysis, should be called "correct" in relation to the language of ZFC, but false when no background set theory is specified, due to the fact 1) that exclusively classical theorems aren't relatable to reality, and 2) they are often appealed to without anyone remembering that their correctness is relative to a classical axiomatization of set theory.

Are you suggesting that is a reason for rejecting his conclusions? Either way, I would suggest that we leave Cantor's theology as a matter between Cantor and his God. — Ludwig V

I'm saying that in the presence of an inconsistency between ZFC and computable notions of mathematics, coupled with the obvious uselessness of of non-constructive cardinal analysis, the theological origin of ZFC becomes conspicuous.

i don't consider this to be a solution, because the result is the uncertainty principle. What you indicate is two distinct concepts of space which are incompatible, "position space", and "momentum space". — Metaphysician Undercover

yes, there are two incompatible bases for describing dynamics, and in line with your proposal, what looks like a hypertask iwhen measured in one basis, is a mere task in the other (where measurement is understood as destructive interference).

I presently suspect that the structure of the uncertainty principle, that concerns non-commutative measurements, is a logical principle derivable from Zeno's arguments, without needing to appeal to Physics.

"Infinite" means limitless, boundless. The natural numbers are defined as infinite, endless. limitless. All measurement is base on boundaries. To say a specific parameter is infinite, means that it cannot be measured. — Metaphysician Undercover

As a slogan, that looks almost right.

To say that a parameter is infinite, means that it cannot be measured relative to a given basis of description. Hence the distinction between an ordinary task and a hypertask depends on how the task is described, and this distinction can be regarded as the logically correct solution to Zeno's Dichotomy paradox.

(The finitude of an object's exact position in position space, becomes infinite when described in momentum space, and vice versa. Zeno's paradox is dissolved by giving up the assumption that either position space or momentum space is primal)

I've tried to follow what you are doing here, but scattered inaccuracies and errors make it very difficult. I gather you want to Cantor’s argument into a constructive or even computational lens. It’s valid in that framework, yet you seem to think it can be taken as refuting classical results about cardinality. — Banno

Yes absolutely, if we interpret "refuting cardinal analysis" as ditching ZF/ZFC for being computationally inadmissible due to the infinite hierarchy of cardinals being computationally meaningless and poorly motivated, given the fact that ZF/ZFC are set theories that are descendents of Cantor's theological prejudices that aren't true by correspondence to anything of relevance to science and engineering.

More specifically, the Cantor-Schröder-Bernstein Theorem that is the foundation of infinite cardinal analysis, is abjectly false in any constructive intepretation of the diagonalization argument that is conscious of undecidability.

In reverse mathematics, where we start by analysing a theorem without first assuming a particular axiomatic foundation, then the CSB theorem becomes the assumption that if f : A --> B is an injection (written |A| <= |B|) and g : B --> A is an injection (written |B| <= |A|) , then there must exist a bijection between A and B (written |A| = |B|).

So let's take P(N) to be the decidable subsets of the natural numbers. Then is CSB true or false?

1. We know that we can construct an injection P(N) --> N via Turing machine encoding of decidable sets. (|P(N)| <= N)

2. We can build 'any old' injection f : N -> P(N) to show that |N| <= |P(N)|.

3. Hence according to CSB, the set of decidable sets of P(N) has the same size as N.

And yet f cannot be a surjection: For diagonalising over f must produce a new member of P(N), but this isn't possible if f is surjective. Hence f cannot be a surjection, and this is the reason why diagonalization can produce new members of P(N), without P(N) ever being greater than N.

This particular case turns CSB against it's originator Cantor, for CSB insists that P(N) and N must necessarily have the size, in spite of the fact a surjection N --> P(N) cannot exist.


So this is where Cantor specifically went wrong: he should have interpreted diagonalization as showing that a surjection cannot always exist between countable sets. But instead, Cantor started with the premise that a surjection A --> B must always exist when A and B are countable, which forces the conclusion that diagonalisation implies "even bigger" uncountable sets, which is a conclusion that Cantor accepted because it resonated with his theology.

As the popularity of this post shows, we do need clarity on the mathematical object called infinity.

In my view the question comes down to simply just what does it really mean when Cantor showed us that the natural numbers cannot be put into 1-to-1 correspondence with the reals. The standard answer, that the infinity is simply larger, and thus we have larger infinities etc. doesn't really answer everything. It simply lacks the rigorous logic that is so ever present in mathematics. The problem with the Continuum Hypothesis shouldn't come as a surprise. — ssu

Yes, the purely constructive meaning of the diagonal argument, is that any constructable injection from the naturals to the Reals defines the construction of a new real. And all that this constructively implies, is that it is impossible to define a surjection from the natural numbers to the reals. Hence it says nothing about whether or not an injection exists from the reals to the natural numbers, and hence the diagonal argument does not rule out the possibility that the real numbers might in fact be a subset of the natural numbers.

As far as the computable Reals are concerned, all that the diagonalization argument implies is that the computable reals are a subset of the Naturals that cannot be "detached" from the rest of the Naturals by an algorithm. For we know by definition that there aren't more computable reals than naturals, since a computable real refers to a computer program of some sort that has a godel number.

The hypothesis that every real number can be listed by an algorithm, is equivalent to knowing the limiting behaviour of every computer program. So what Cantor actually showed, is an indirect proof that the halting problem cannot be solved, and not that there are "more" real numbers than natural numbers.

I suppose that while transfinite numbers are not much used in physics, continuum cardinality and so on are present as background commitments. So if the space-time manifold in General Relativity is continuous, then I suppose transfinite cardinals are included by default in that formalisation; or so I believe. — Banno

the existence and meaningfulness of transfinite cardinals rests upon the Axiom of Choice, but that principle also implies unfettered resource duplication via the Banach Tarski Paradox, which goes way what is needed to define mathematical continuity, and well beyond the physical requirements and assumptions of general relativity, not to mentioning violating energy conservation.

By definition, the computational content of physics (i.e. physical inferential semantics) cannot rest upon choice axioms, because they represent what cannot be computed. In practice, physical continuity doesn't refer to an idealised continuum in the sky, but only to the ability to construct or measure vanishingly small changes in output in response to vanishingly small changes in input, for which multiple alternative languages are available, that don't carry the metaphysical baggage.

Moreover, the standard dogma of the transfinite cardinals, harms theoretical physics, by denying the ability speak of potentially infinite sets that naturally describe the content of a physical process better than Dedekind-infinite completed set balony.

Once AOC is relinquished, the unreal "beauty" of the ideal cardinals is replaced with the ugly and uncertain truth of equivalence classes of set bijections that are generally undecidable, and only potentially infinite, such they cannot hide their complexity behind a veil of cardinal representatives obeying a simple cardinal arithmetic.

In addition, it cannot even be proven, without begging the question, that transfinite induction up to ε0 is sound, since ε0 might not be well-founded. Hence a theoretical physics claim cannot rest upon transfinite induction. In practice, a "proof" that PA is consistent essentially amounts to beating the skeptic into submission with pseudo-religious dogma about the metaphysical "truth" of transfinite induction as decided by classical mathematicians who don't ultimately care about the physical truth and use-value of such theories.

Only the boring transfinite ordinals up to ε0 are empirically and computationally meaningful.

Even though the java programming language can be compiled to run on any computer, it is an additional fact of the world that which specific computer it actually runs on. It is convenient to ignore this fact in order to "avoid inconsistent semantics", but that ignorance is wrong nevertheless, when we talk about the world in its totality. — bizso09

Suppose Alice the realist talks to Bob the irrealist, about an Alice-independent world that she believes to subsume the world of Bob the irrealist. Then should Bob think that Alice is wrong to assert this and correct her for being ignorant of irrealism, or should Bob simply nod on his irrealist understanding that no matter what Alice expresses, she must be expressing facts about her Alice-dependent world?

What truth-criteria should Bob use when interpreting Alice's claims about an Alice-independent world? As an irrealist, Bob presumably would not claim to understand Alice's claims about Alice's world, but presuambly Bob would also not claim that Alice has cognitive access to Bob's world; in which case Bob cannot interpret her as making claims about his world after all, and so can only nod and smile when Alice speaks about an absolute reality.

You've hit upon the reason why Frege distinguished ideas from sense and reference. The intersubjective meaning of language must be invariant to the perspective-dependent realities of each individual in order to avoid inconsistent semantics, in spite of the fact that from the perspective of each individual, only their own ideas exist and intersubjective truth is dependent upon their perspective.

It is analogous to the platform-independent definitions of programming languages. The semantics of a programming language, e.g Java, and source code written in java, is oblivious/invariant to the fact that it will be compiled/interpreted and executed in terms of different hardware instructions running on different machines. So on the one hand, the meaning of a java program is invariant to which piece of hardware it will be interpreted and executed on, and yet the truth of an executed java program reduces to the hardware operations of a specific computer.

Magnus is right in spirit, but isn't referring to natural numbers, but to "lawless choice sequences" that are infinite yet Dedekind-finite, meaning that the sequences are of finite but growing length.

By contrast, the naturals are "lawful" choice sequences, which by construction are essentially dedekind-infinite functions that don't represent sequences in the flesh, and are what a type-theorist would say are purely intensional sequences that shouldn't be confused with actual sequences.

To rectify an earlier confusion, the computer-science meaning of "extension" refers to explicit data. According to this definition, the identity function on the naturals ( \lamda (n : N) => n ) is an extension in the sense of a function, whereas the graph of that function, namely the set { (n,n) | n is a Natural number} isn't an extension. But confusingly for philsophy that graph is considered an extension according to the Fregean notion of extension, since Frege defined an extension as referring to the arguments of a predicate that make it true.

In effect, Frege conflated the notion of data-at-hand with the notion of functions that can produce data on demand, as a result of thinking that functions exist independently of their domains and ranges. For Frege, and unlike the computer scientist, a function isn't a causal operation that transforms input into output, but a transcendental relation that relates a static domain to a static range. Hence Frege interpreted predicates (which he called "concepts") as being non-destructive testers of their domains, which naturally implies that concepts and Fregean extensions exist independently and in one-to-one correspondence, leading to Russell's Paradox and also led to the failure of formalists like Hilbert to predict incompleteness.

The dispute concerns the notion of Dedekind Infinity.

Dedekind Infinity, referring to the "fact" that the set of natural numbers N is equinumerous to a proper subset of itself, is an intensional concept referring only to injections of the type N --> N . This isn't the same asserting that 1,2,3,... is extensionally of the same length as 2,4,6,... since the dots "..." don't have an extensional interpretation.

A sequence S := 1,2,3,.. that is understood to be unfinished rather than complete, refers to the notion of a Dedekind-finite infinite set. This means that

1) There doesn't exist a bijection between S and and a finite set, meaning that S is unfinished.
2) Any injection S --> S is necessarily a bijection, meaning that S isn't Dedekind Infinite.
3) Any function N --> S isn't an injection, meaning that S isn't countably infinite or larger.

Unfortunately, this indispensible common-sense notion of the potentially infinite set, cannot be formulated in ZFC, because it isn't compatible with the axiom of countable choice which insists upon completing every set.

it is right for amateur philosophers to object to Dedekind Infinity being misued as an extensional concept by the media and the general public (including some physicists who ought to know better). This misuse is due to mainstream mathematics being grounded in adhoc 20th century Hilbertian foundations that assumes the existence of a completed "set" of natural numbers, to the detriment of common-sense, as well as to science and engineering.

Recall that Hilbert believed that finitary proof methods could be used to ground the notion of absolute infinity that he considered to be indispensible for mathematics, due to being under the spell of Cantor, and which the incompleteness theorems conclusively debunked - a negative result that should have been obvious from the outset - namely that an infinite amount of information is obviously not finitely compressible into finite axiom schema.

This example was to draw attention to what Kripke says in N&N:

"If there were a substance, even actually, which had a completely different atomic structure from that of water, but resembled water in these respects, would we say that some water wasn't H2O? I think not. We would say instead that just as there is fool's gold there could be a fool's water."

I was told that possible world semantics/rigid designation as a formal language does not perfect our language but can guide it. So, I wanted to know if a community that treated "fool's gold"(FeS2) the same as "regular gold" would we need to correct them when they say "some gold wasn't Au." I did not get an answer, but I would say "no" we do not correct them because we can clearly understand how they are using "some" and what is "essential" to them is not some microscopic atomic structure but similar macroscopic properties they find valuable. Another good example is things we call "diamonds." What has science discovered here? That a diamond is C. But wait I thought science also discovered that graphite is C. I am confused about what is happening in that possible world where they both exist. It reminds me of a favorite passage in Quine's paper from "On what there is": — Richard B

Possible world semantics as presently understood by computer science, is just denotational semantics for dependently typed languages, as formally expressed using functors in the language of category theory. This purely formal and de dicto understanding of possible world semantics in terms of a kripke frame or categorical equivalent, needs to be distinguished from Kripke's de re metaphysical thesis of rigid designation that refers to a causal theory of reference, by which referents are 'baptised' with a speaker's use of name via ostensive definition. Whilst Kripke frames are formally useful, the same cannot be said of Kripke's metaphysics.

On a purely technical level, your examples illustrate the ontological decisions that must be decided when formally specifying domain knowledge; such decisions are formally expressible in a dependently typed language such as Lean 4 or Coq. The question is, are your examples counterexamples to Kripke's metaphysical concept of rigid designation?

I would think that Kripke would likely argue that his concept of rigid designation refers to the first external cause of a linguistic community's use of a name, that Kripke would call an "initial baptism", involving an ostensive definition that is usually unknown to successive generations of the lingusitic community, who continue to use the name as a result of tradition for many years later. On this understanding, Kripke's hypothesis of rigid designation isn't a semantic guide for understanding the intensional meaning of a language in the sense you questioned above and that I proposed a solution to earlier, rather Kripke's rigid designation is a metaphysical hypothesis about causality and its relationship to language, particulary with respect to natural kinds - a hypothesis that cannot be appraised by simply asking a community about their usage of language and intended meaning.

If we interpret Kripke's axiom in terms of his theory of causal reference, then his x=y → □x=y axiom can be interpreted as saying that for any world w' that is accessible from our actual world w, w' shares the same history as w by definition, including having the same initial baptisms for the names used by the speakers of w and w'. So if by definition, the referents of names reduce to their initial baptisms, then it follows that if x and y are names that have identical referents in w, then x and y must also have identical referents in w'. Naturally, Kripke's examples of rigid designation center upon 'static' types in the C++ sense: natural kinds, astronomically large and unchanging objects, and so on, upon which we build theories of causation.

But since Kripke's rigid designation cannot be used to make predictions about future discoveries or predict theory change, and since there is no way of inferring whether or not two names identically refer, his axiom is useless, whilst also reinforcing arguably outdated metaphysical conceptions of historical time, such as the block universe which conflict with, or fail to be useful for, the purposes of Quantum Mechanics.

Let's

I can conceive some gold is Au if fool’s gold gets $4500 oz. Do I need guidance from possible world semantics to clarify that my use of “some” needs correction? — Richard B

Semantics in general, can be understood as situationally defining types, e.g the natural kind gold, in terms of a set of tests, where a failure to pass any of the tests implies the negation of the type, e.g. if this nugget fails to pass the tests of heaviness, malleability and yellowness, then by definition this nugget is not "Gold", i.e. it is of type "Not-Gold". So we have a de dicto component, namely the set of tests that we use to situationally define the type "Not-Gold", and a de re component, namely whether or not our nugget passes or fails ours tests.

Strictly speaking, since natural kinds, indeed any physical kind, are not exhaustively defineable on the one hand, and yet we can only specify finite sets of tests and make finite observations on the other, we should only conclude that something is of type "Not-X" in line with Popper's principle of falsification. E.g we should define "Gold" to be "Not-Not-Gold" whose meaning is situational and in relation to a finite set of tests that we using situationally to define and test for "Not-Gold" (softness, brittleness, etc).

This points towards local definitions for natural kinds that lose their transcendental significance; for different sets of tests will be used in different contexts. e.g although gold has a unique spectral "fingerprint" that can be observed using techniques like X-ray Fluorescence, in most situations such techniques aren't available and hence are not included in the tests that are used to situationally define the presence of "gold". But even when such techniques are available, perhaps the spectrometer is broken, fraud takes place, new scientific laws pop into existence etc. So tests themselves require higher-order tests (i.e. "necessity" is contigent and loses epistemological significance).

We can now think of rigid-designation as a harmless and comedic truism which says that "if" we could non-situationally define gold, by exhaustively specifying what "gold" is in terms of an infinite number of infallible tests covering all contexts and use cases, and "if" our purported sample of 'gold' passed every one of those tests, then we would necessarily "have gold" de dicto, in accordance with our definitions of "gold" and first-order "necessity".

This brings us to the "possible world" variety of semantics: we can think of the execution of a test as updating the state of our world, which is modelled in possible world semantics as a transition to another world. Tests reinforce the idea that the accessibility relation used in in possible world semantics expresses a non-deterministic version of causal implication in relation to a set of non-deterministic causal assumptions. When possible world semantics goes wrong or is under-constrained, it is because the underlying causal assumptions are wrong or under-constrained.

Another thing to bear in mind, is the relationship between Kripke's axiom x=y → □x=y in relation to his "Naming and Necessity" lectures that he gave in 1970 on the one hand, versus Kripke's resolution of his sceptical paradox on the other, that he discussed in lectures in the late seventies that led to the 1982 publication "Wittgenstein on Rules and Private Language".

Notably, Kripke's positions on both positions seem on the surface to come into collision, that might suggest an "early Kripke" with potentially radically different views from an emerging "later Kripke", that parallels the distinction between the early/late Wittgenstein.

Recall that it only hit Kripke later on in his career, that the truth of a mathematical formula might refer to intersubjective social and environmental assertibility conditions that exist independently of the speaker's subjective mental states and personal use of the formula. So the problem for the later Kripke is how to still make sense of x=y → □x=y in light of the skeptical rule following paradox that he later readily acknowledges, and whether it forces him to abandon this axiom.

Actually, in this regard the equivalent contrapositive form ¬□(x = y) → x ≠ y shines brighter in the face of semantic skepticism, in saying something like "If x isn't necessarily asserted to be equal to y in the future, then I cannot in good conscience declare x to be definitionally equal to y today". And since it is impossible to know in advance what terms will necessarily be intersubstitutable after the next theory change, then one can deduce from the contrapositive of Kripke's axiom that equality 'in good conscience' is purely reflexive: x=y → x=x. In which case, Kripke's axiom has an anti-metaphysical reading on the understanding that no object is necessarily identical to any other object, in which case Kripke's axiom becomes structurally synonymous with propositional equality in intensional type theories,which is precisely of the form x=y → x=x, indicating that terms x and y are only equal if they reduce to the same term after term rewriting.

Also recall that the early Kripke proposed the category of a priori contigent propositions. This category is much more resilient to theory change and skeptical paradoxes than the necessary a posteriori category, since they represent the fact that our axioms are perpetually subject to reinterpretation and even revision in light of new information. In light of semantic skepticism, the expression ¬□(x = y) → x ≠ y very much looks like a metalogical theorem stating that equality is a priori contigent - an interpretation that flips Kripke's rigid desgination on its head by interpreting it as implying that identity is non-rigid via a denial of the consequent.

The Philsophical Investigations doesn't sanction the rigid designation axiom x=y implies □x=y, because PI didn't sanction the sort a priori metaphysical, formal, or semantic speculation that Kripke was clearly indulging in; for there is simply no compelling epistemic or semantic justification for this axiom.

I suspect that Kripke might have been influenced by early seventies work in Martin-Lof type theory when he proposed his axiom, because it looks suspiciously like the reflection rule of extensional type theories, namely a rule which asserts that computationally equal terms are definitionally equal de dicto.

For example, 1 + x isn't definitionally equal to x + 1 in a typical so-called "intensional" type theory for they are not the same construction, yet these formulas behave identically under evaluation, for clearly the Boolean equality 1 + x == x + 1 evaluates to True for all natural numbers x. In an extensional type theory, the reflection rule when applied to this case says that a proof of 1 + x == x + 1 for any x, de dicto implies that 1+ x = x + 1. The reflection rule isn't very popular because it implies that type-checking is generally undecidable. Hence in most programming languages, 1 + x and x + 1 aren't considered to be definitionally equal, in spite of always evaluating to the same value.

The reflection rule is structurally identical to Kripke's x=y implies □x=y, but is innocously de dicto. By contrast, Kripke proposal is meant de re, in a way that is metaphysically speculative and also debunked by the history of science theory change - indeed, the whole point of theory-change is to render the possibilities implied by the previous theory as inconsistent, as for instance when promoting the equivalence between mass and energy to analytic status or downgrading it to synthetic status.

If I recall correctly, Quine's position relates to the following observation:

Suppose that the modal operators merely refer to the quantifiers of First Order Logic (FOL), which is the case if every set of possible worlds is describable by a first-order predicate. In which case, we can eliminate the modal operators from any modal formula f to produce an equi-satisfiable formula f' without the modal operators, e.g by using skolemization. This means that although we obviously cannot represent modal truth as a formula in modal logic unless the modal logic is trivial (as per Tarski's undefinability theorem), we do at least have an automatable procedure for verifying the logical falsity of a modal formula f , i.e by substituting it for an equi-satisfiable formula f' without quantifiers, and then checking whether ~ f' is satisfiable. If it is, then f isn't satisfiable in all models, meaning that f cannot be logically true and hence isn't provable, as per Godel's completeness thorem.

On the other hand, suppose that the modal operators refer to second-order quantification over sets of possible worlds that cannot be described in terms of first-order predicates. In which case we "really have" modal operators above and beyond first-order quantification, since modal formulas are now assumed to not be reducible to FOL. But the price we pay is to lose Godelian completeness, and hence we can no longer use skolemization to determine the truth of the modal formula.

Essentially modal logic is just a game of let's pretend. Underneath the philosphical posturing one either has FOL with syntactically defined quantifiers equipped with a verifiable notion of external truth (skolemisation applied to a model), else one has second-order modal formulas without an unbiased means of deciding a truth value.

The concept of an Infinite extension can only be circurarly defined, because in that case all that we actually have is an intensional definition of a sequence in the form of a self-looping algorithm whose output is a finite extension of of random length decided by the user of the algorithm.

Possible world semantics, while used for denoting possibilia, (i.e. state relative actualia via an overspill expansion of the domain of the quantifiers), has no notion of dynamics or interaction, that is necessary for understanding the language-game of possibility.

As a static set-theoretic model. possible world semantics can describe a game tree, but not the execution path of a given game, or the prior processes of interaction by which a game tree emerges.

All I can say is that we aren't agreeing as to the semantics of the problem. Your sample space includes the counterfactual possibility (H, Tuesday), which isn't in the sample space of the experiment as explicitly defined. You appeal to "if we awoke SB on tuesday on the event of heads" might be a perfectly rational hypothetical in line with common-sense realism, but that hypothetical event isn't explicit in the problem description. Furthermore, the problem is worded as a philosophical thought experiment from the point of view of SB as a subject who cannot observe that tuesday occurred on a heads result, nor even know of her previous awakenings, in sharp contrast to an external point of view relative to which her awakenings are distinguishable and for which the existence of tuesday isn't conditional on the outcome of the event of tails.

As straw-clutching as this might sound, there are radically minded empiricists who would argue that the existence of "tuesday" for the sleeping beauty is contingent upon her being awake. For such radical empiricists the event (H,Tuesday) doesn't merely have zero probability, but is a logical contradiction from SB's perspective.

Epistemically for SB,

(h,mon) -> observable, but undiscernable.
(h,tue) - > unobservable.
(t,mon) -> observable but undiscernible.
(t,tue) -> observable but undiscernible.

So we are back to the question as to whether (h,tue) should be allowed in the sample space. This is ultimately what our dispute boils down to.

The reason I keep asking for specific answers to specific questions, is that I find that nobody addresses "my sample space." Even though I keep repeating it. They change it, as you did here, to include the parts I am very intentionally trying to eliminate. — JeffJo

I think you misunderstand me. I am simply interpreting the thrux of your position in terms of an extended sample space. This isn't miscontruing your position but articulating it in terms of Bayesian probabilities. This step is methodological and not about smuggling in new premises, except those that you need to state your intuitive arguments, which do constitute additional but reasonable premises. [/quote]

There are two, not three, random elements. They are COIN and DAY. WAKE and SLEEP are not random elements, they are the consequences of certain combinations, the consequences that SB can observe. — JeffJo

Look at this way: It is certainly is the case that according to the Bayesian interpretation of probabilities, one can speak of a joint probability distribution over (Coin State, Day State, Sleep State), regardless of one's position on the topic. But in the case of the frequentist halfer, the sleep-state can be marginalised out and in effect ignored, due to their insistence upon only using the coin information and rejecting counterfactual outcomes that go over and above the stated information.

There are two sampling opportunities during the experiment, not two paths. The random experiment, as it is seen by SB's "inside" the experiment, is just one sample. It is not one day on a fixed path as seen by someone not going through the experiment, but one day only. Due to amnesia, each sample is not related, in any way SB can use, to any other. — JeffJo

You have to be careful here, because you are in danger of arguing for the halfers position on their behalf. Counterfactual intuitions, which you are appealing to below, are in effect a form of path analysis, even if you don't see it that way.

Each of the four combinations of COIN+DAY is equally likely (this is the only application of the PoI), in the prior (this means "before observation") probability distribution. Since there are four combinations, each has a prior ("before observation") probability of 1/4.

In the popular problem, SB's observation, when she is awake, is that this sample could be H+Mon, T+Mon, or T+Tue; but not H+Tue. She knows this because she is awake. One specific question I ask, is what happens if we replace SLEEP with DISNEYWORLD. Because the point that I feel derails halfers is the sleep. — JeffJo

But the Sleeping Beauty Problem per-se does not assume that the Sleeping Beauty exists on tuesday if the coin lands heads, because it does not include an outcome that measures that possibility. Hence you need an additional variable if you wish to make your counterfactual argument that SB would continue to exist on tueday in the event the coin lands heads. Otherwise you cannot formalise your argument.

Just to clarify, I'm not confusing you for a naive thirder, as I mistook you for initially, where i just assumed that you were blindly assigning a naive prior over three possible outcomes. I think your counterfactual arguments are reasonable, and I verified that they numerally check out; but they do require the introduction of a third variable to the sample space in order to express your counterfactual intuition that I called "sleep state" (which you could equally call "the time independent state of SB").

Well, I've come to the conclusion that your answer is in some sense philosophically superior to the result insisted upon by halfers like myself, even though your answer is technically false. In a nutshell, I think that although you have lost the battle, you have won the war.

As I understand it, your proposal is essentially the principle of indifference applied to a sample space that isn't the same as the stated assumptions of the SB problem, namely your sample space is based on the triple

{Coin,Day,Wakefulness}

upon which you assign the distribution Pr(Heads,Monday,Awake) = Pr(Tails,Monday,Awake) = Pr(Heads,Tuesday,Asleep) = Pr(Tails,Tuesday,Awake) = 1/4.

But the important thing isn't your appeal to indifference "on a single bell" as you put it, but the different sample space you used and the viewpoint it provides. (Any measure can be assumed on your sample space, provided that it satisfies the marginal distribution P(Heads) = 1/2 and assigns coherent condtional credences - your particular choice based on PoI is easily seen to be coherent, for the reasons you point out.

By contrast, the probability space for the classical SB problem is that of a single coinflip C = {H,T}, namely (C,{0,H,T,{H,T}},P) where P (C = H) = 0.5 . From this premise it isn't technically possible to conclude anything except for the halfers position, namely

P(C = H | Monday Or Tuesday) = P(C = H) = 1/2.

for reasons already explained many times, (and which is more rigorously proved by pushing the measure P forwards onto the different sample space of day outcomes, and then disintegrating the resulting measure and taking the inverse to obtain the conditional probability for P( C = H | Monday Or Tuesday), but this is incidental).

But what makes your argument incorrect for the SB problem à la lettre, namely the use of a non-permitted sample space that is based on commonsense counterfactual intuition that goes beyond the explicitly stated premises of the SB problem, is also what makes your argument interesting and persuasive, for your argument for the thirder's position is based on coherent counterfactual intuitions that are commonsensically valid and important to point out, even though they are inapplicable with respect to a strict interpretation of the SB problem as explicitly stated.

Essentially, if by "probability" we mean a coherence value based only on the frequential probability of the coin landing heads as explicitly assumed by the SB problem and we do not make any other assumptions no matter how intuitively plausible, then the answer can only be a half because the sample space is of one coin flip. But if we interpret "probability" more liberally to a mean a credence that includes commonsense counterfactual intuitions, then the answer can be different to a half provided that we define "probability" more precisely to permit this and extend our premises of the SB problem to include counterfactual premises that allow your chosen sample space. But then the answer isn't necessarily equal to a third, for that particular case requires the use of the Principle of Indifference applied to bells, which a non-halfer might object to, even though he agrees to use your sample space.

So don't use that as a model, use the well-established methods of conditional probability. Ring a bell at noon of both days. An awake SB hears it, but a sleeping SB is unaffected in any way.

The prior probabilities of a specific bell-ring being on any member of {H+Mon, T+Mon, H+Tue, T+Tue} is 1/4. If SB hears it, H+Tue is eliminated. Conditional probability says:

Pr(H+Mon|Bell) = Pr(H+Mon)/[Pr(H+Mon)+Pr(T+Mon)+Pr(T+Tue)] = 1/3. — JeffJo

Your bell is just a label for the event {Monday OR Tuesday} which is independent of the coin flip, and so you are merely repeating the same appeal to indifference as before.

What I was pointing out is that this application of the principle of indifference isn't consistently applied to SB.

Lets start by assuming the credence that you insist upon:

P(Monday,Heads) = P(Tuesday, Heads) = P(Tuesday,Tails) = 1/3

To verify that you are happy with this credence assignment, you need to check the hypothetical credences that this credence implies. In the case of P(Monday | Tails) we get

P(Monday | Tails) = P(Monday, Tails) / P(Tails) = (1/3) / (1/2) = 2/3.

Are you happy with this implied conditional credence? If SB is told that the outcome is tails when she wakes up, then should she believe that it is twice as likely to be monday than tuesday, given her knowledge of tails?

I agree if I understand your position correctly as being deflationary. I would simply put it by saying that an interrogated subject isn't in the epistemically exalted position to distinguish his 'beliefs' from what he 'knows' about the world.

To find out what somebody believes, don't ask them for a self report of the form "I believe that X is true with n% confidence", but rather, ask them what they know about the world, because what a person is prepared to assert about the world is a more accurate measure of what their actual "beliefs" are, and can be expected to be at odds with what they say about themselves when introspecting unreliably.

The next question should concern the extent to which beliefs exist internally within a person in the sense of a mental state, versus externally of the person as behavioural hypotheses that society projects onto the person. (Since we have no reason to assume that people understand themselves).

In any way that SB can assess her credence, that does not reference her position in the map, the answer is 1/3.
Using four volunteers, where each sleeps though a different combination in {H&Mon, T&Mon, H&Tue, T&Tue}? On any day, the credence assigned to each of the three awake volunteers cannot be different. and they must add up to 1. The credence is 1/3.
Use the original "awake all N days, or awake on on one random day in the set of N" problem? N+1 are waking combinations, only one corresponds to "Heads." The credence is 1/(N+1).
Change the "sleep" day to a non-interview day? It is trivial that the answer is 1/3.

I'm sure there are others. The point is that the "halfer run-based" argument cannot provide a consistent result. It only works if you somehow pretend SB can utilize information, about which "run" she is in, that she does not and cannot posses. — JeffJo

No, the Halfer position doesn't consider SB to have any information that she could utilize when awakened, due to the fact that SB's knowledge that it is either Monday or Tuesday doesn't contribute new information about the coin, which she only observes after the experiment has concluded.

Also, your reasoning demonstrates why we shouldn't conflate indifference with equal credence probability values. Yes, an awakened SB doesn't know which of the possible worlds she inhabits and is indifferent with regards to which world she is in and rightly so. No, this doesn't imply that she should assign equal probabilty values for each possible world: For example, we have already shown that if an awakened SB assigns equal prior probabilities to every possible world that she might inhabit, then she must assign unequal credences for it being monday versus tuesday when conditioning on a tails outcome.

To recap, if P(Monday) = 2/3 (as assumed by thirders on the basis of indifference with respect to the three possible awakenings), and if P(Tails | Monday) = 1/2 = P(Tails) by either indiffererence or aleatoric probability, then

P(Monday | Tails) = P(Tails |Monday) x P(Monday) / P(Tails) = (1/2 x 2/3) / (1/2) = 2/3.

So let's assume that SB is awakened on monday or tuesday and is told, and only told, that the result was Tails. According to the last result, if SB initially assigns P(Monday) = 2/3 on the basis of the principle of indifference as per thirders, then she must infer having learned of the tails result that monday is twice as likely as tuesday, in spite of mondays and tuesdays equally occcuring on a tails result.

As this demonstrates, uniform distributions have biased implications. So if SB insists on expressing her state of indifference over possible worlds in the language of probabilty, she should only say that any probability distribution over {(Monday,Heads),(Monday,Tails) ,(Tuesday,Tails)} is compatible with her state of indifference, subject to the constraint that the unconditioned aleatoric probability of the coin is fair.

However, if she really must insist on choosing a particular probability distribution to represent her state of indifference, then she can still be a halfer by using the prinicple of indifference to assert P(Monday | Tails) = P(Tuesday | Tails), and then inferring the unconditioned credence that it is monday to be P(Monday = 1/2), which coheres with the halver position.

SB's answer: "Because the protocol ties one lamp to Heads-runs and two lamps to Tails-runs, among the awakenings that actually occur across repeats, the lamp I'm under now will have turned out to be a T-lamp about two times out of three. So my credence that the current coin toss result is Tails is 2/3." (A biased coin would change these proportions; no indifference is assumed.)

The coin's fairness fixes the branches and the long-run frequencies they generate. The protocol fixes how many stopping points each branch carries. Beauty's "what are the odds?" becomes precise only when she specifies what it is that she is counting.

Note on indifference: The Thirder isn't cutting the pie into thirds because the three interview situations feel the same. It's the other way around: SB is indifferent because she already knows their long-run frequencies are equal. The protocol plus the fair coin guarantee that, among the awakenings that actually occur, the two T-awakenings together occur twice as often as the single H-awakening, and within each coin outcome the Monday vs Tuesday T-awakenings occur equally often. So her equal treatment of the three interview cases is licensed by known frequencies, not assumed by a principle of indifference. Change the coin bias or the schedule and her "indifference" (and her credence) would change accordingly. — Pierre-Normand

Thirders who argue their position on the basis of frequential probabilities are conflating the subject waking up twice in a single trial (in the case of Tails) for two independent and identically distributed repeated trials, but the subject waking up twice in a single trial constitutes a single outcome, not two outcomes. Frequentist Thirders are therefore overcounting.

There is only one alleatorically acceptable probability for P(Head | Monday OR Tuesday) (which is the question of the SB problem) :

P(Head | Monday OR Tuesday) =
P(Monday OR Tuesday | Head) x P(Head) / P(Monday Or Tuesday)

where

P(Head) = 0.5 by assumption.
P(Monday Or Tuesday) = 1 by assumption.

P(Monday OR Tuesday | Head) = P(Monday | Head) + P(Tuesday | Head) = 1 + 0 = 1.

P(Head | Monday OR Tuesday) = 1 x 0.5 / 1 = 0.5.

Then what would you say it is? If you say Q, then your credence in Tails must be 1-Q, and you have a paradox. — JeffJo

If you insist that credence must be expressed as a number Q, then in general I would refuse to assign a credence for that reason - cases like SB in which credences are artificially constrained to be single probability values, doesn't merely result in harmless paradoxes but in logical contradictions (dutch books) with respect to causal premises. Likewise, I am generally more likely to bet on a binary outcome when I know for sure that the aleatoric probability is 50/50, compared to a binary outcome for which I don't know the aleatoric probability.

In order to avoid unintented inferences, the purpose for assigning credences needs to be known. For example, decisions are often made by taking posterior probability ratios of the form P(Hypothesis A | Observation O )/ P(Hypothesis B | Observation O). For this purpose, assigning the prior probability credence P(Hypothesis A = 0.5) is actually a way of saying that credences don't matter for the purpose of decision making using the ratio, since in that case the credences cancel out in the posterior probability ratio to produce the likelihood ratio P(observation O | Hypothesis A)/P(Observation O | Hypothesis B) that only appeals to causal (frequential) information. This is also the position of Likelihoodism; a view aligned with classical frequential statistics, that prior probabilities shouldn't play a part in decision making unless they are statistically derived from earlier experiments.

An acceptable alternative to assigning likelihoods, which often cannot be estimated as in single experiment situations, is to simply to list the possible outcomes without quantifying. Sometimes there is enough causal information to at least order possibilities in terms of their relative likelihood, even if quanitification of their likelihoods isn't possible or meaningful.

The SB problem is a classic illustration of confusing what probability is about. It is not a property of the system (the coin in the SB problem), it is a property of what is known about the system. — JeffJo

Then you are referring to subjective probability which is controversial, for reasons illustrated by the SB problem. Aleatory probability by contrast is physical probability and directly or indirectly refers to frequencies of occurrence.

That is, your credence in an outcome is not identically the prior probability that it will occur. Example:

I have a coin that I have determined, through extensive experimentation, is biased 60%:40% toward one result. But I am not going to tell you what result is favored.
I just flipped this coin. What is your credence that the result was Heads?
— JeffJo

It is correct to point out that credence does not traditionally refer to physical probability but to subjective probability. It is my strong opinion however, that credence ought to refer to physical probability. For example, my answer to your question is say that my credence is exactly what you've just told me and nothing more, that is my credience is 60/40 in favour of heads or 60/40 in favour of tails.

Even though you know that the probability-of-occurrence is either 60% or 40%, your credence in Heads should be 50%. You have no justification to say that Heads is the favored result, or that Tails is. So your credence is 50%. To justify, say, Tails being more likely than Heads, you would need to justify Tails being more likely to be the favored result. And you can't. — JeffJo

I definitely would not say that my credence is 50/50, because any statistic computed with that credence would not be reflective of the physical information that you have provided.

I don't see any questionable appeal to the principle of indifference being made in the standard Thirder arguments (though JeffJo may be making a redundant appeal to it, which isn't needed for his argument to go through, in my view.) Sleeping Beauty isn't ignorant about frequency information since the relevant information can be straightforwardly deduced from the experiment's protocol. SB doesn't infer that her current awakening state is a T-awakening with probability 1/3 because she doesn't know which one of three indistinguishable states it is that she currently is experiencing (two of which are T-awakenings). That would indeed be invalid. She rather infers it because she knows the relative long run frequency of such awakenings to the 2/3 by design. — Pierre-Normand

But the SB experiment is only assumed to be performed once; SB isn't assumed to have undergone repeated trials of the sleeping beauty experiment, let alone have memories of the previous trials, but only to have been awoken once or twice in a single experiment, for which no frequency information is available, except for common knowledge of coin flips. So the SB is in fact appealing to a principle of indifference as per the standard explanation of the thirders position, e.g. wikipedia.

In any case, a frequentist interpretation of P(Coin is Tails) = 0.5 isn't compatible with a frequentist interpretation of P(awoken on Tuesday) = 1/3.

For sake of argument, suppose P(Coin is Tails) = 0.5 and that this is a frequential probability, and that inductive reasoning based on this is valid.

Now if P(awoken on Tuesday) = 1/3, then it must also be the case that

P(awoken on Tuesday | Coin is Tails) x P(Coin is tails) = 1/3, as typically assumed by thirders at the outset. But this in turn implies that

P(awoken on Tuesday | Coin is tails) = (1/3)/0.5 = 2/3.

Certainly this isn't a frequential probability unless SB having undergone repeated trials notices that she is in fact woken more times on a tuesday than a monday in cases of Tails, in contradiction to the declared experimental protocol. Furthermore, this value doesn't even look reasonable as a credence , because merely knowing apriori that the outcome of the coin is tails shouldn't imply a higher credence of being awoken on tuesday rather than Monday.

Credences are a means of expressing the possession of knowledge without expressing what that knowledge is. To assign consistent credences requires testing every implied credence for possible inconsistencies. Thirders fail this test. Furthemore, credences should not be assigned on the basis of ignorance; a rational SB would not believe that every possible (day, coin-outcome) pair has equal prior probability, rather she would only assume was is logically necessary - namely that one of the pairs will obtain with either unknown or undefined probability.

What the SB problem amounts to is a Reductio ad absurdum against the principle of indifference being epistemically normative, a principle that in any case is epistemically inadmissible, psychologically implausible, and technically unnecessary when applying probability theory; a rational person refrains from assigning probabilities when ignorant about frequency information; accepting equal odds is not a representation of ignorance (e.g Bertrand's Paradox).

- It is commonly falsely argued by thirders, that halvers are suspect to a Dutch-book argument, by virtue of losing twice as much money if the coin lands tails, than they gain if the coin lands heads (since the dutch-book is defined as an awoken SB placing and losing two bets, each costing her $1 in the case of tails, one on monday and one on tuesday, versus her placing and winning only one bet rewarding her with $1 on Monday if the coin lands heads). But this dutch book argument is invalidated by the fact that it it equivalent to SB beingapriori willing to win $1 in the case of heads and losing $2 in the case of tails, i..e. SB knowingly accepting a Dutch Book with an expected loss of 0.5x1 - 0.5x2 = -$0.5 before the experiment begins, given her prior knowledge that P(H) = 0.5. So the Dutchbook argument is invalid and is actually an argument against the thirder position.

The (frankly unnecessary) lesson of SB is that meaningful probabilities express causal assumptions, and not feelings of indifference about outcomes.

"The 'ought' you mentioned, as in 'it ought to rain,' is a prediction. In contrast, the 'must' in a normative conclusion is a requirement for action—a behavioral standard that everyone ought to abide by." — panwei

Your definition of 'must' is circular here. Circular definitions are characteristic of speech acts ("Tie your shoelaces! because I said so!") and also of analytic propositions ("Bachelor" means "unmarried man").

In such contexts, it is right to point out that their use is not necessarily inferential, because they might represent instructions, wishes, promises, postulates, conventions, orders etc, rather than assumptions or facts. But the English meaning of "ought" is used both as a speech act and as an inference, depending on the context, which reflects the fact that we often cannot know whether a sentence is meant as a speech act or as a hypothesis, especially when considering the fact that speech acts are often issued on the basis of assumptions.

This also reflects a fundamental asymmetry of information between speaker and listener; When a speaker uses "ought", they might intend it as a speech act or as a prediction but the listener cannot be certain as to what the speaker meant, even after asking the speaker to clarify himself, because we are back to circular definitions.

Are 'oughts' inferences, and are 'ises' reducible to 'oughts'?

In ordinary language, "ought" is also used to signify predictive confidence, as in "it ought to rain"; so "oughts" aren't necessarily used in relation to utility maximisation. Furthermore, we understand what an agent is trying to achieve in terms of our theory of the agent's mind, which is partly based on our observations of their past behaviour. So an inference of what an agent 'ought' to do on the basis of what 'is' can perhaps be understood as an application of Humean induction. And our description of what 'is' tends to invoke teleological concepts, e.g. if we describe a ball as being a snooker ball it is because we believe that it ought to behave in the normal way that we expect of snooker balls from past experience.

So if descriptions of what is the case are necessarily inferential, and if our understanding of moral obligations are in terms of our theory of minds which in turn are inferred from behavioral observations, then perhaps there is an argument for saying that only oughts exist, even if we are never sure which ones.

In Decision Theory, States and Actions are generally treated as logically orthogonal concepts; an 'is' refers to the current state of an agent, and an 'ought' refers to the possible action that has the highest predicted utility in relation to the agent's 'is'. This treatment allows causal knowledge of the world to be separated from the agent's subjective preferences.

Paradoxically, this can imply that the psychological distinction between states versus action utilities is less clear, considering the fact that agents don't generally have the luxury of having perfect epsistemic knowledge of their worlds prior to taking an action (e.g. as required to solve the Bellman Equation).

Also, an action is only as good as the state that it leads to - rewards are related to (state,action) pairs, so utility values can be thought of as equivalence classes of states quotiented with respect to action utilities. This is practically important, since agents don't generally have the memory capacity to store perfect world knowledge even if it were available. Agents tend to visit and focus their learning on the state->action->(reward,state) chains that correspond to highest reward, and then learn compressed representations of these visited states in terms of a small number of features that efficiently predict utility. E.g Chess Engines estimate the utility of a board position by representing the board in terms of a managebly small number of spatial relations between pieces, especially in relation to the Kings. So the representational distinction between states and action reward values in the mind of an agent is muddied.

In order to fully dislodge the Cartesian picture, that Searle's internalist/introspective account of intentionally contentful mental states (i.e. states that have intrinsic intentionality) indeed seem not to have fully relinquished, an account of first person authority must be provided that is consistent with Wittgenstein's (and Ryle and Davidson's) primary reliance on public criteria. — Pierre-Normand

Quine provided the most useful conceptual framework for both scientists, technologists and philosophers, since LLMs can be naturally interpreted as physically instantiating Quine's web of belief, namely an associative memory of most public knowledge. The nature and knowledge of LLMs can then appraised in line with Quine's classification of sentence types.

(A short paraphrase of (Quineian sentence types returned by Google Gemini))

Theoretical sentences: Describe things not directly observable, such as "Atoms are the basic building blocks of matter". They require complex background knowledge and cannot be verified by a simple, direct observation.

Observation categoricals: Sentences that involve a relationship between two events, often derived from theory and hypothesis together, such as "When the sun comes up, the birds sing".

Occasion sentences: Sentences that are sometimes true and sometimes false, like "It is raining". An observation sentence can also be an occasion sentence, as "It is cold" is true on some occasions and false on others.

"Myth of the museum" sentences: Traditional view of language where sentences are like labels for pre-existing meanings, which Quine rejects because it assumes meanings exist independently of observable behavior.

They are the "Chinese room" types of sentences that bear no specific relationship to the sensory inputs of a particular language user, that are encoded in LLMs, by constrast to Quine's last category of sentences, namely the Observation Sentences, whose meaning is "private", in other words whose meaning reduces to ostensive demonstration and the use of indexicals on a per language-user basis.

I find the the appeals to Wittgenstein as a gold standard of philospohical writing ironic, considering how indispensible AI is for the layreader who wishes to engage with Wittgenstein's thinking in a historically accurate fashoin. This is all thank to Wittgenstein's apparent inability to articulate himself, and because of a greater irony that the anti-AI brigade of this forum overlook: Wittgenstein never quoted the philosophers he was targetting or stealing from, leading to great difficulties when it comes to understanding, criticising and appraising the originality of his ideas. (I'm not aware of any idea of Wittgenstein's that wasn't more precisely articulated by an earlier American pragmatist such as Dewey or Peirce, or by a contemporary logician such as Russell or Frege or Ramsey, or by a post-positivist such as Quine) And yet these more articulate philosophers are rarely discussed on this forum - I would argue because precise writing is more technical and therefore more cognitively demanding than giving hot-takes of aphorisms .

Wittgenstein's standard of philsophical writing wasn't publishable in his own time, at least not for the standards required by anayltc philospohy, let alone our time. So if AI should not be quoted because of source uncertainty, then what is the justification on this forum for allowing people to quote Wittgenstein?