The main problem for me is, why can we read a→(b∧¬b) as "a implies a contradiction" but not ¬(a→(b∧¬b)) as "a does not imply a contradiction? — Lionino

In general, the consistency of an axiomatic system isn't provable in an absolute sense due to Godel's second incompleteness theorem; the upshot being that consistency is a structural property of the entire system that isn't represented as a theorem by the system if it is sufficiently powerful.

Suppose that the logic concerned is weaker than Peano arithmetic, such that it can prove its own consistency. Then in this case, a proof of ¬¬a metalogically implies that ¬a isn't provable, i.e that a does not imply a contradiction.

But if the axiomatic system contains Peano arithmetic such that the second incompleteness theorem holds, then a proof of ¬¬a does not necessarily imply the absence of a proof of ¬a, since Peano arithmetic cannot prove its own consistency.

Sure, but that's not really what the example is there to assert, as is clear from the rest of the paragraph. They mentioned replacing the fact about dogs 2+2 = 4 in the next line. It's "if a statement is true, then that statement is implied by any statement whatever," which is straightforwardly counter intuitive. — Count Timothy von Icarus

That's true of classical logic, and more specifically it's fragment known as intuitionistic logic, due to the fact that the respective rule of implication is essentially the logic of functions (including those that ignore their arguments to produce a constant value), rather than the logic of causality - which is described by relevance logic and linear logic.

As for "Modern Symbolic Logic", it doesn't have a well-defined meaning since it refers to a plurality of logics that are separately described in terms of different mathematical categories.

But consider the fact that the halting behaviour of two identical algorithms stands and falls together. So although there does not exist an infallible universal halting tester, there exists an infallible special-case halting tester for any given algorithm, namely a copy of that very algorithm.

Although an epistemic limitation falls short of a metaphysical proof, I am sympathetic to the idea of free will, because in my opinion the conceptual distinction between free will and determinism rests upon a belief in absolute infinity, which i reject.

In my view, to say that "A => B is necessary true" in the sense of material causation, is to say that there exists a Z such that "A => Z is necessarily true" and "Z => B is necessarily true". If we reject the idea that this definition can appeal to actually infinite recursion, then the use-meaning of " A => B is necessarily true" in any given context must eventually bottom out to a finite chain of implicative reasoning, in which the meaning of "necessarily true" is left undefined.

A simpler way of putting it, is to say that we make up the meaning of " A => B is necessarily true" as we go along. This proposition doesn't have precise a priori meaning, and so isn't contradicted by a future discovery that A => B fails to hold, rather the proposition meant by the sentence "A=> B is necessary true" changes on discovery that A => B fails to hold.

Bertrand Russell's Principia Mathematica was nominalist; he treated sets as merely a means of referring to groups of particulars, partly in response to Set theoretic paradoxes, but the approach made it impossible to describe all of mathematics.

This demonstrates an unconscious tendency of nominalism; why do nominalists have a tendency to appeal to an ontology based on the existence of particulars, as opposed to an ontology that starts from a united whole?

In Bertrand Russell's case, it was in the hope of making analysis tractable in piecemeal fashion, in contrast to the British Idealists who might also be described as nominalist, but who considered reality to consist of a single holistically unified entity. But this makes analytics impossible, since it implies that a local material change to reality causes the meaning and hence definitions of the rest of reality to change.

A related example is Godel's trick in his ontological proof of God as discussed in the other thread, which was to define a property P so as to enforce the condition

¬(g → P(g) ∧ g → ¬P(g))

i.e. ¬¬g, which is a classically acceptable proof of existence.

Just because a set S is declared to be "infinite", doesn't imply that S possesses a literally non-finite number of elements, rather it only implies that a bijection between {0,1,2..n} and S isn't or cannot be specified a priori and that n cannot be bounded a priori.

And even when a bijection between S and the set of natural numbers is specified, this only implies that S must be understood in terms of a monotonically increasing process rather than in terms of a completed basket of goods.

In both cases, the use-meaning of "infinity" should be understood to mean "finitization is decided by circumstances that are external to the specification logic".

So philosophical or practical questions about "infinite facts", as opposed to mathematical questions concerning the definition of mathematical infinity, should always be decided by elaborating assumptions until the facts concerned are "finitized". The presence of infinity in a non-mathematical question is only an indication that the question concerned isn't well-posed.

I don't see where that is implied in the argument.

P(ψ)≡¬N(ψ) — sime


If N is supposed to mean necessary existence, that is a rejection of axiom 5. — Lionino

N was supposed to mean the possibility modality (N standing for Negative Properties, in order to stand for the opposite of Positive Properties). The question here I was interested in, is how to give a syntactical definition of Positive Properties such that the resulting argument follows as a valid tautology in some modal logic. This was partly in order to help clarify the the definitions Godel provided, even his assumptions need to be altered slightly and the resulting argument and its conclusion aren't quite the same.

For example, taking Positive properties to refer to what is necessarily true of all individuals in every possible world, turns Axiom A2 into the definition of a functor, which is rather tempting. It also makes the possibility of god follow as a matter of tautology.

Also, Godel's definition of essences seems close to the definition of the Categorical Product. So why not take the essence of an individual to be the conjunction of his properties?

One thing I overlooked was that God was defined as referring to the exact set of positive properties, which would mean that according to my definition of P, all individuals would be identical. But then supposing we weaken the definition of "Godliness" to refer to a set that contains all the positive properties and possibly some of the negative (i.e contigent) ones?

I think there is quite a few pedagogically useful questions here.

It seems to me you are thoroughly confused — Lionino

about what?

I think the most remarkable and amusing part of Godel's argument, is in the beginning before the use of modal logic, in which he argues for the existence of a 'god term' by turning the principle of explosion on its head.

Constructively speaking, an existential proposition is proved by constructing a term that exemplifies the proposition, as per the Curry Howard Isomorphism. Classically speaking, an existential proposition can also be derived by proving that it's negation entails contradiction, as per the law of double negation.

In Godel's proof however, he defines a so-called Godliness predicate G, where as usual ~G(x) corresponds to the principle of explosion

G(x) --> B(x)
G(x) --> ~B(x)

where B is any predicate.

But in Godel's case, he defines G as only implying properties that satisfy a second-order predicate he calls "Positivity", which is a predicate decreeing that G(x) --> B(x) and G(x) --> ~B(x) cannot both be true.

So in effect, Godel crafted a non-constructive proof-by-absurdity that implies the existence of a god term on the basis that non-existence otherwise causes an explosion! this is in stark contrast to the normal constructive situation of proofs-by-absurdity in which a term exemplifying a negated existential proposition is constructed in terms of a function that sends counterexamples to explosions.


The rest of Godel's proof is unremarkable, since he defined G as implying it's own necessity, meaning that if G is said to be true in some world, then by definition it is said to be true of adjacent worlds, which under S5 automatically implies every world.

The irony of Modal Logic is that there are so many alternatives to choose from, corresponding to the fact that Logic and a forteriori modal logic, has no predictive value per se. But modal theologicians aren't using Modal Logic to derive or express predictions, rather they are using Modal Logic to construct a Kripke frame with theologically desired properties. So ontological arguments aren't necessarily invalid for achieving their psychological and theological purposes, provided they aren't construed as claims to knowledge.

In fact, i'm tempted to consider Anselm's argument to be both valid and sound a priori, and yet unsound a posteriori. This is due to the fact that although our minds readily distinguish reality from fiction, I don't think that this distinction is derivable from a priori thought experiments.

S5 is the logic of epidemics in which every possible world is infected by a virus whose transmission is symmetric and transitive.

As for Godel's argument, if we take the special case of his argument in which the positive properties P are taken to be the properties that are true for every possible individual, i.e by taking

$P(\psi) := \Box \forall x, \psi (x)$

and if we replace axiom A1 above with

$P(\psi) \equiv \neg N(\psi)$

where

$N(\psi) := \Diamond \exists x, \neg \psi (x)$

Then i expect that the resulting argument reduces to a trivial tautology of S5 in which all individuals are infected by the godliness virus.

I don't think it has anything to do with mathematics. This is perhaps clearer if we don't consider the button to turn the lamp on and off but instead consider it to alternate between two or more colours.

What number would you assign to the colour red, and why that? What number would you assign to the colour blue, and why that? Shall we use e and i, because why not?

The logic of the lamp just has nothing to do with numbers at all. — Michael

Yes, and that sounds identical to the philosophy of intuitionism :) It doesn't matter what type of object is associated with the lamp's output.

By contrast, Benecerraf et al argue along more classical lines, by defining an abstract completion of the sequence that doesn't contradict Thompson's premises, but which requires changing the original problem by adding an point at infinity to accommodate such a completion.

Thompson's views sound intuitionistic, in that he didn't apparently consider the "completion" of button-pressing to have a meaningful answer in relation to his thought-experiment. Indeed, if one formally treats Thompson's Lamp as being an unfinishable choice-sequence, which is an object equivalent to what computer-science calls a "stream", then Brouwer's weak axiom of continuity explicitly forbids the construction of any theorem that postulates a property of Thompson's stream that isn't decidable in a finite number of steps. Indeed, Brouwer can be understood as inventing intuitionism to explicitly forbid the informal interpretation of total functions as representing super-tasks, which are concepts that are incompatible with temporal intuition.

By contrast, Benacerraf changed the goal posts by giving Thompson's Lamp a formal treatment from the perspective of classical analysis, in which he interpreted Thompson's Lamp as being an incomplete description of an extended abstract function with a point at infinity, whose value can be chosen as being continuous with some property of the original sequence. However, Bencerraf's formal interpretation doesn't have a temporal interpretation in the sense demanded by the intuitionists. And his methodology runs into problems in situations where the function to be completed has conflicting notions of continuity, as in The Littlewood-Ross Paradox.

But recall that Wittgenstein regarded the ordinary meaning of "to know" to not imply infallibility, in the sense that even if a fact P necessarily implies another fact Q, "knowing that P" does not necessarily imply Q.

This stems from his epistemic consideration that in a literal sense nothing is knowable in the sense demanded by a philosopher. And yet he appreciated that everyone including himself ordinarily use the verb "to know" all the time. Therefore he concluded that the ordinary meaning of "to know" isn't an insinuation of ideal knowledge.

If Moore's knowledge of his hands is interpreted in that light, then had Moore later discovered that he didn't actually have hands, his discovery wouldn't contradict his earlier ordinary claim to "know that he had hands"

I imagine that a supertaskist might agree with the formalist or intuitionist that supertasks aren't mathematically or logically defensible, whilst nevertherless insisting that reality contains supertasks, by arguing that mathematical logic is the map rather than the territory, and by conceding that they are only using mathematics as a descriptive tool for expressing their beliefs, rather than as a prescription for justifying their a priori beliefs in super-tasks.

For instance, a supertaskist might appeal to the fact that one cannot say how many moments of time has passed during a minute, or how many physical operations took place in one's computer to sum 1 + 1, and they might appeal to this inability to measure, divide and count experience or events as grounds for being open minded to the idea that space-time is a literal continuum.

But in that case, how does the supertaskist propose identifying what isn't a super-task? If super-tasks are to have empirical meaning and inferential value, the supertaskist must delineate task from super-task, but how can they delineate them on a non-ad hoc basis?

The inability for dimensionless points to be reconciled with the continuum is what motivated Whitehead's point-free geometry, a precursor to the field of Pointless Topology, as for instance formalised using Locales whose distributive law characterizes the meaning of a "spot". (It might be useful to test this law in relation to the SB tree, for both the truncated and infinite version).

See that phrase, "perfect information"? That's why I say formalism attempts to do the impossible. In other words, it assumes an ideal which cannot be obtained, therefore it's assumption is necessarily false. — Metaphysician Undercover

Perfect information isn't an assumption of formal reasoning, rather it is regarded to be a necessary condition of the meaning of "formal" reasoning in that it is by definition finitely deducible and does not require appealing to unformalized intuitions about infinite and ideal objects. Most importantly, the condition of perfect information ensures that formal reasoning cannot interpret an expression such as {1,2,3,...} as representing an abbreviation of some ideal object; the former expression must either be formally treated as a finite object of some type, else the expression must be considered illegal.

It is actually by sticking to formal reasoning that the illusion of the ideal is never obtained. The opposite impression is due to Platonists disguising themselves as formalists, which might be said to even include Hilbert himself.

Formalism makes the reasonable demand that whatever informal intuitions originally motivated the construction of an axiomatic system, and whatever informal interpretations one might subsequently give to the signs of that system, the methodology of theorem-proving should be purely algorithmic and make no appeal to such intuitions, whether such intuitions be rooted in platonism or in Kantian intuition.

I view formalism as a form of Platonism. It's a Platonist game in which the participants deny their true character, that of being Platonist. Notice "perfect information" is the foundational feature of Platonist idealism. That perfection is the only thing which supports the eternality of Platonic ideals. So formalism and Platonism are really just the same thing, even though the formalists will claim otherwise. — Metaphysician Undercover

The irony of Hilbert, is that his formalism ultimately led to the rebuttal of his own informal intuitions about infinity, namely his presumption that a closed axiomatic system must possess a finite representation of it's own consistency. Had Hilbert better understood the implications his formalism, and especially the finite formal meaning of The Law of Excluded Middle which he apparently accepted for instrumental purposes, then Godels incompleteness theorem might not have come as a shock to him. It is evident that Hilbert was a methodological formalist who didn't mean to insinuate that mathematics was a meaningless game void of semantics, but only that the terms used to denote sets, formula and constants shouldn't require interpretation for the purposes of theorem proving. Unfortunately, his intuitions misled him regards to the outcome of his formal program.

If we inspect the finite activity of theorem proving in a formal system, we see that every term that is informally interpreted as denoting an "infinite object" only possesses finite conditions under which the term is introduced into a theorem and under which the term is eliminated from a theorem.

Different formal systems can be regarded as differing only in regards to their ability to distinguish types of finite object. E.g Intuitionism that formalizes choice-sequences can distinguish uncompleted finite sets from ordinary finite sets, whereas ZFC as a theory of first-order logic can only distinguish finitely defined functions from finite sets - so whilst ZFC might be informally said to be a theory about "infinite sets", this isn't the proof-theoretic formal meaning of ZFC, and so a formalist is free to reject the platonic myths that surround ZFC.

You can say that I have a problem with formalism, because I do. Like claiming that accepting certain axioms qualifies as having counted infinite numbers, formalism claims to do the impossible. — Metaphysician Undercover

Formalism as a philosophy considers mathematics to be reducible to a finite single-player sign game of perfect information in which proofs refer to deterministic winning strategies, and hence Formalism does not support the Platonic interpretation of abstract mathematics as denoting actually infinite objects, whatever the formal system concerned.

So I think your problem is actually with Platonic myths that have become psychologically wedded to innocent formal definitions, and in particular the formal definitions of limits and total functions that are ubiquitously misinterpreted in both popular and scientific culture as denoting a non-finite amount of information, E.g as when the physicist Lawrence Krauss misleads the public with nonsense about the physical implications of Hilbert Hotels.

There is a fundamental problem with identifying supertasks with series limits — sime


This is the kind of mistake that Benacerraf makes in his response to Thomson, as explained here.

The lamp is not defined as being on or off at particular times; it is turned on or off at particular times by pushing a button.

This is an important difference and is why so many "solutions" to Thomson's lamp (and other supertasks) miss the point entirely.

If the lamp is turned on after 30 seconds then, unless turned off again, it will remain on for all time. This is why if you claim that supertasks are possible then you must be able to give a consistent answer as to whether or not the lamp is on or off after 60 seconds. If you cannot, because no consistent answer is possible, then this is proof that the supertask is metaphysically impossible.

It is necessary that the lamp is either on or off after 60 seconds, and for it to be either on or off after 60 seconds it is necessary that the button can only been pressed a finite number of times before then. — Michael

My impression of Benacerraf is that he is defining Thomson's Lamp as a boolean valued function

$l : \overline {\mathbb {N}} \rightarrow \mathbb {B}$

on the domain of the extended natural numbers $\overline {\mathbb {N}}$ which introduces an additional point $\omega$ at "infinity", and then arguing that the value at $\omega$ can be chosen arbitrarily and independently of the function's limiting value, if any. But if this the case, then he isn't engaging with Thomson's argument and has merely shifted the goal posts to declare victory in an incomparable axiomatisation.

But the point about Frege's Law Vb also applies to the extended natural numbers; Thompson's lamp when defined as the function $l$ has a domain consisting of two definite and maximally separated points 0 and $\omega$ and a number of points between 0 and $\omega$ that is intensionally described as being countably infinite. However, if Frege's Law Vb is rejected for reasons mentioned previously, then although $l$ still has the aforementioned intensional properties, it does not possess an extensionally well-defined number of points, in which case it cannot be considered to represent the metaphysical notion of a supertask.

Essentially, mathematical analysis will fail to persuade unless one is already a true believer of supertasks.

There is a fundamental problem with identifying supertasks with series limits, namely the fact that literally infinite summations are not expressible in calculus, given that they cannot be written down.

A formalist is free to use the name "1/2 + 1/4 + ..." to denote 1, but the formalist cannot interpret "1/2 + 1/4 + ..." as an expression implicitly representing part of an infinite summation, because the formalist considers expressions to have no meanings other than being finite states of a syntactical parser when proving a theory in a finite number of steps.


Frege fell into a similar trap as the supertaskers in the Grundgesetze when he proposed his law V. He wanted there to be a one-to-one correspondence between every function and it's representation as a table of values, even in the case of functions with infinite domans. So he proposed Basic Law Vb with disastrous consequences:

{x∣Φx} = {x∣Ψx} → ∀x(Φx ↔ Ψx).

To a finitist or potentialist, Law Vb can be interpreted as introducing fallacies of induction into Set Theory, since they will likely interpret the sets-as-extensions on the left hand side as denoting a finite amount of observable information, and they will likely interpret the function on the right-hand side as denoting an unbounded amount of implicit information, meaning that they cannot regard Law Vb to be a reliable rule of induction. Furthermore, according to their reasoning Law Vb cannot be regarded as constituting a definition of the right hand side, unless one gives up the idea of functions having infinite domains).

Wittgenstein himself warns in the preface that PI isn't a very good book and not the book he intended to write. The unfortunate consequence of it not being a good book, and yet being a book of tremendous importance for analytic philosophy, is the necessity of gatekeeping and elitist assholes, partly in order to rectify commonplace misunderstandings of Wittgenstein that were promulgated in the secondary literature by a significant proportion of the previous generation of gatekeeping and elitist assholes.

You first need to distinguish evidence of Christianity from interpretations of "Christianity", in order to clarify the extent to which your argument is grammatical and theological rather than inferential.

Do you really wish to argue that mystical visions are externally related to Christian concepts and present inferential evidence that those Christian concepts denote 'facts'? For how could such an argument ever get off the ground?

didn't think he proposed a solution. Rather, it was an example to show that it is impossible to complete a supertask. — Michael

Yes, in other words rejecting iii), namely the idea that one can finish counting an infinite sequence.

Let's first remember the fact that the limit of a sequence isn't defined to be a value in the sequence.

Re : The Cauchy Limit of a Sequence

"When the values successively attributed to the same variable approach indefinitely a fixed value, eventually differing from it by as little as one could wish, that fixed value is called the limit of all the others"

A converging sequence might eventually settle on value equal to its limit, but even then the two concepts are not the same. So it doesn't matter whether we are talking about Thompson's Lamp, or merely a constant sequence of 1s. In either case, a limit, if it exists, doesn't refer to any position on the sequence, rather it refers to a winning strategy in a type of two-player game that is played upon the "board" of the converging infinite sequence concerned.

So it make no literal sense to consider the value of an unfinishable sequence at a point of infinity, so the meaning of a "point at infinity" with respect to such a sequence can at best be interpreted to mean an arbitrary position on the sequence that isn't within a computable finite distance from the first position. In the newspeak of Non Standard Analysis, such a position can be denoted by a non-standard hyper-natural number, meaning an ordinary natural number, but which due to finite limitations of time and space cannot be located on the standard natural number line.

As for the OP, its triad of premises are inconsistent. For only two of the three following premises can be true of a sequence

i) The length of the sequence is infinite.
ii) The sequence is countable
iii) The sequence is exhaustible

For example, Thompson's proposed solution to his Lamp paradox is to accept (i) and (ii) but to reject (iii). Whereas solutions to Zeno's Paradox tend to start by accepting (iii) but reject the assumption that motion can be analysed in terms of a countably dense linear order of positions, either by denying (i) (namely the assumption that the sequence of positions is infinite, which amounts to a denial of motion) or by denying (ii) (namely the assumption that motion can be used to count positions, for example because the motion and position of an arrow aren't simultaneously compatible attributes).

Sure. That is indeed a different take. I'm taking what I like to think of as a traditional scientific approach, otherwise known as a reductionist materialist approach. Like anyone in this field, I'm driven by a particular set of beliefs that is driven by little more than intuition - my intuition is that reductive scientific methods can explain consciousness - and so a big motivation -- in fact one of the key drivers for me - is that I want to attempt to push the boundaries of what can be explained through that medium. So I explicitly avoid trying to explain phenomenology based on phenomenology. — Malcolm Lett

Consider the fact that traditional science doesn't permit scientific explanations to be represented or communicated in terms of indexicals, because indexicals do not convey public semantic content.

Wittgenstein made the following remark in the Philosophical Investigations

410. "I" is not the name of a person, nor "here" of a place, and
"this" is not a name. But they are connected with names. Names are
explained by means of them. It is also true that it is characteristic of
physics not to use these words.

So if we forbid ourselves from reducing the meaning of a scientific explanation to our private use of indexicals that have no publically shareable semantic content , and if it is also assumed that phenomenological explanations must essentially rely upon the use of indexicals, then there is no logical possibility for a scientific explanation to make contact with phenomenology.

The interesting thing about science education, is that as students we are initially introduced to the meaning of scientific concepts via ostensive demonstrations, e.g when the chemistry teacher teaches oxidation by means of heating a testtube with a Bunsen Burner, saying "this here is oxidation". And yet a public interpretation of theoretical chemistry cannot employ indexicals for the sake of the theory being objective, with the paradoxical consequence that the ostensive demonstrations by which each of us were taught the subject, cannot be part of the public meaning of theoretical chemistry.

So if scientific explanations are to make contact with phenomenology, it would seem that one must interpret the entire enterprise of science in a solipsistic fashion as being semantically reducible to one's personal experiences... In which case, what is the point of a scientific explanation of consciousness in the first place?

Let S denote the set of stairs, let N denote the standard natural numbers and let N* denote the nonstandard numbers. We can model the cardinality of S, which is equivalent to the height of the top of the staircase, by using a non-standard natural number h* from N*. Lets assume

i) There does not exist an injection N --> S
ii) There exists a surjection I ---> S, where I is a subset of N.

Condition i) represents the hypothesis that we do not know how many stairs there are, or equivalently that we cannot know the height of the top stair due to assuming that we will never reach the bottom of the staircase.

Condition ii) represents the physically plausible situation that although we cannot count the stairs, there cannot be more stairs than some finite but unboundedly large subset of the natural numbers.

In other words, we are assuming that S is subcountable.

Let s(n) denote the n'th stair that is visited when descending. Using this order of descent on S, we have a total function S --> N* describing the height of each stair as a non-standard natural number, namely

s (0) => h*
s(1) => h* - 1
s(2) => h* - 2
..

which when written directly in terms of the indices denoting the order-of-descent is a function f

f : N --> N* :=
f (n) = h* - n*


This function describes an infinite descent in N*, and is paradoxical because

1) Every nonstandard natural number e* that is in N* corresponds to some standard number e in N, and vice-versa.

2) We have defined an infinitely descending chain of non-standard natural numbers in N*.

The paradox is resolved due to the fact that the order-of-descent we are using when descending the "infintie staircase" from the top has no recursively definable relationship in terms of the order of ascension when climbing the staircase from the bottom; although Peano's axioms rule out the existence of non-wellfounded subsets for recursively enumerable subsets of the natural numbers, our subset isn't recursively enumerable in terms of those axioms, and is therefore an external subset that cannot be talked about by Peano's axioms.

Sorry Fishfry.

On further reflection the infinite sided die shouldn't need a choice axiom in its construction (e.g a sphere can be painted by working clockwise and outwards from a chosen pole - since there is an algorithm choice isn't needed). But then what of the idea of rolling said die an actually infinite number of times? That surely is equivalent to choice, assuming the rolls are random.

The natural numbers are well ordered in their usual order. — fishfry

Yes, that is true, by Peano's inductive construction of the natural numbers. And a well-order is usually assumed for an infinite sided die, in spite of its construction lacking an inductive specification (for which side should be assigned what number?) - So the assumption of a well-ordered infinite sided die that lacks an inductive definition is the same as a countably infinite set of objects equipped with the axiom of countable choice.

For what it's worth, the fact that we can't put a uniform probability measure on the natural numbers doesn't mean they have to be "all the same number." They're all different numbers. And I can't understand the idea you're getting at. — fishfry

I took the idea to mean that the faces of an infinite die isn't a well-ordered set, unless the Axiom of Countable Choice is assumed. If this axiom isn't assumed, then the sides of the die can only be ordered in terms of their order of appearance in a sequence of die rolls, which implies that unrolled sides are indistinguishable.

I'm quite fond of this potential infinity solution and believe it may be the correct direction to pursue.

However, the die in the paradox possesses an actually infinite number of sides (the set of sides is Dedekind-infinite). What more needs to be said to argue that such a die cannot exist? — keystone

The first problem is one logical inconsistency. In Kolmogorov's treatment, the axioms exclude the proposition; if one introduced such a die as a new axiom, the system wouldn't be consistent. Whereas in my above (very rough) proposal, A Dedekind infinite set is measured directly in terms of its definition rather than in terms of it's cardinality,but which in turn implies that it has lower probability than its subsets, violating additivity, (Here I am assuming that we want to use standard rules for mapping distributions from one set to another. I'm not actually sure if there might be some other workaround than banning Dedekind-infiniteness).

The second problem is one of motive. Is the motive good enough? Consider what it means to say that the Natural Numbers are Dedekind infinite. In type theory, it refers to an object N with an arrow
1 + N --> N that has an inverse ( here 1 denotes zero, and + indicates disjoint union, and the arrow is the successor function). A standard computational reading of this arrow is that it conveys the fact that one can count upwards from zero to an arbitrary finite number of one's choosing and then count downwards to return to zero. In a nonstandard reading one is also allowed to count from an arbitrary position that cannot be reached from zero. But in either case, the arrow doesn't have the extensional significance that set theorists like to assume. That is to say, the arrow doesn't imply that "every member of the natural numbers exists prior to it being counted" , rather the arrow is used to construct as many members as one desires. In summary, we can say that Dedekind-infiniteness is a type of rule that can be used to generate Dedekind-finite extensions of any size that can be freely extended as and when one desires, by applying the rule once more.

In the case of an infinitely sided die, if the die can only be rolled a finite number of times, then its trajectory of outcomes is equivalent to the trajectory of some Dedekind-finite die that by definition is guaranteed to possess an arbitrary but finite number of unrolled sides after the final roll of the die. Is rolling the die a Dedekind-infinite number of times extensionally meaningful? Not according to the functional interpretation of Dedekind-infiniteness, which deems the previous analysis sufficient for the philosophical analysis of the fall of man paradox.

There are two types of infinitely sided dice; those whose set of sides is actually infinite - meaning Dedekind-infinite as in the set of dice sides possessing a countably infinite proper subset, versus those dice whose set of sides is potentially infinite - meaning Dedekind-finite but without having an a priori finite upper-bound on their number of sides.

You need a non-standard probability theory to express the idea of an infinitely sided fair die, but the idea only makes sense for Dedekind-finite dice.

Rather than assigning a real number to a set of outcomes to denote the probability of the set, which doesn't work in the case of infinitely sided dice, we can in the case of a fair and infinite die eliminate the distinction between sets of outcomes and their probabilities, because in this case the probability of a set of outcomes is equivalent to the set itself.

So let P(N) = N

- We simply drop the normalisation factor since it is a constant, and let N directly denote both the set of natural numbers and the probability of choosing a natural number (i.e the value of probability one) .

Let N/a denote the set of natural numbers that is the complement of the subset of natural numbers a. Then

P(N/a) = N/a

P(a) = a

P(a OR N/a) = N/a + a = N

- If N/a is cofinite, meaning that it's complement is finite, then its interpretation as a probability value is larger than the probability value for any finite set of naturals, but is nevertheless smaller than N.

- if a is finite but non-empty, then it's interpretation as a probability value is smaller than the size of any cofinite set and any infinite set, but is nevertheless larger than zero.

- If both N and a are infinite, then their corresponding probability values are of intermediate magnitudes between the two previous cases.

I think this is pretty much all that is needed for a basic non-standard probability treatment of a fair infnitely sided dice, and the resulting measure is both countably additive and normalizes to 'one'.

Lastly, the assumption of Dedekind finiteness is important, because we don't want to derive
P(N) < P(N).

So the intuition stated in the OP, that switching is always the best decision, is represented in terms of the magnitudes of cofinite sets of natural numbers as always being larger than the magnitudes for finite sets, but without succumbing to the false conclusion that the probability of getting a better result is 1, which is a 'bug' of classical probability theory caused by it's insistence upon using standard models of arithmetic.

Huh? Why? Inverted qualia arguments are specifically about different S experiencing different things. The degree of difference is what seems to defeat certain theories. — AmadeusD

Fregean ideas are necessarily perspectival, whereas the public meaning of Fregean sense is a-perspectival. So if by "qualia" you mean to refer to your first-person perspective constituted by your Fregean ideas, then what criteria of comparison to you propose to use to relate your lived and actual qualia with what you abstractly conceive and hypothesize to be my experiences? How could scientific analysis which is deliberately restricted to propositions stated only up to the third person, be of any help here?

On the other hand if by "qualia" you are in some sense referring to both of our experiences, then I presume you are no longer referring either to your actual first-personal experiences or to mine, but to some abstract concept. Which is the starting point of any behavioural, functional or computational third-personal analysis of "shared sensations" "sensation similarity" and so on, whether in type or in token.

Also I don't think language is at all relevant and is in fact a red herring. Presumably deaf, illiterate mutes who aren't blind can see colours. — Michael

Semantics is relevant, due to the aforementioned ambiguity as to what is being referred to when speaking of qualia. In these sorts of discussions, it is often implicitly assumed by participants that "qualia" is meant in some Fregean sense. Which does indeed permit the sort of abstract functional and behavioural analysis that you propose in terms of type-token distinctions and similarity criteria, but which also forgets the reason why "qualia" were included in philosophical parlance in first-place - as a means of bridging the subjective private understanding and use of language in the first person, with the use of physical concepts that speak only in terms of abstract definitions stated in the third-person.

With respect to physicalism, the question is whether or not this difference in colour perception requires differences in biology, and with respect to naive realism, the question is whether or not one of them is seeing the "correct" colour (in the sense that that colour is a mind-independent property of the object). — Michael

Yes, I am in agreement there, although I would say that the question you mention is with regards to the physical analysis of perception, rather than a philosophical analysis of perception which less constrained than physical analysis, since the latter analysis is free to define concepts in relation to Fregean Ideas, which isn't possible in an aperspectival physical analysis.

IMO, when eliminative materialists speak of "consciousness not existing", I interpret them to mean (whether they agree with my interpretation or not), that physical analysis is by definition restricted to the analysis of cognition and perception in terms of Frege's notions of sense and reference which constitute the meaning of "objectivity", but which does not include the meaning of "subjectivity" that refers to the unshareable Fregean ideas that modern philosophers often refer to as "qualia".

First of all, does it make sense to speak of shared sensations?

If the answer to that question is deemed to be negative, then inverted-qualia arguments cannot get off the ground, but in which case aren't necessary for refuting physicalism, for a negative answer to the former question would imply that the set of "Alice's sensations" is both disjoint from, and unrelated to, the set of "Bob's sensations", however they might be labelled. But then again physicalism cannot also get off the ground, since physical concepts are "shareable" by definition.

Recall Frege's semantic distinctions of sense (referring to a term's public usage), reference (referring to what if anything a term signifies) and ideas (referring to a term's private aesthetic meaning that varies from person to person). Then ask if a sensation is shareable in any of those above semantic aspects.

Obviously, Fregean ideas aren't shared by definition, so if by "sensations" we mean the Fregean ideas that each of us subjectively intuits about word meaning, then we can refer to our previous analysis and conclude that that the concept of inverted-qualia is nonsensical.

But if by "shared sensations" we are referring only to Fregean sense (which the word itself might suggest), then we are only referring to the shared public usage of the term "sensation". in which case "inverted qualia" could mean something like when two subjects react equally and oppositely to the same stimulus - a meaning which actually amounts to a physical definition of "inverted qualia", even if the concept says nothing about any underlying Fregean senses that will typically be assumed to exist regardless of the physical concept's silence on the matter. In fact, the very meaning of a "physical concept" might be interpreted to mean a concept that is by definition invariant to the Fregean ideas that individuals privately associate with the concept, in contrast to aesthetic concepts whose definitions are allowed to vary among language users in line with their unique Fregean ideas that they each associate with their shared terminology.

Lastly, the common-sense of naive realism and the classical psychology of perception might lead us to consider "sensations" as lacking any Fregean referents. Indeed, we tend to speak of an object as "looking red to an individual" but not as being red per-se. However, the later Wittgenstein remarked that different types of sensation vary as to the degree that the subject of the sensation attributes the sensation to himself versus the object of his perception. In Wittgenstein's example of a green stinging nettle, he points out that we will tend to refer to the nettle as possessing "green leaf patches" but not as possessing "painful leaf patches", and seemed to imply that the degree to which a sensation-type is attributed to the object of perception was determined by the Fregean sense of the sensation type.

According to Externalism, knowledge is merely true belief, in which the truth-maker (reality) is external to whatever justifications one might offer in the defense of their beliefs. So externalism avoids the Gettier problem of false justifications that produce true beliefs, because it doesn't consider beliefs per se to be truth-apt. Or alternatively, if it is assumed that truth is internally related to beliefs, then externalism denies the existence of beliefs. Either way, externalism eliminates the normative dimension of epistemology, an elimination which many philosophers find problematic, and which is a common characteristic of naturalised epistemology.

"Now this is eternal life" : Was Saint John a presentist?

According to this (possibly unreliable) answer as to the meaning of "eternal life" :-

"
....
It is a mistake, however, to view eternal life as simply an unending progression of years. A common New Testament word for “eternal” is aiónios, which carries the idea of quality as well as quantity. In fact, eternal life is not really associated with “years” at all, as it is independent of time. Eternal life can function outside of and beyond time, as well as within time.

For this reason, eternal life can be thought of as something that Christians experience now. Believers don’t have to “wait” for eternal life, because it’s not something that starts when they die. Rather, eternal life begins the moment a person exercises faith in Christ. It is our current possession. John 3:36 says, “Whoever believes in the Son has eternal life.” Note that the believer “has” (present tense) this life (the verb is present tense in the Greek, too). We find similar present-tense constructions in John 5:24 and John 6:47. The focus of eternal life is not on our future, but on our current standing in Christ. "

In which case the so-called "after-life" of his Christianity is a misnomer, in that it's conditions of verification aren't considered to transcend the present.

Experimental synthetic languages such as Ithkuil and Lojban were designed to improve upon the semantic deficiencies and limitations of natural languages, in the knowledge that many of those deficiencies being responsible for the creation of philosophical pseudo-problems. However, a a learner will inevitably rely upon their mother tongue as a meta-language when learning those synthetic languages, so it is hard to see what the payoffs are in learning such languages in the short to medium term, especially considering the fact that one can reason and communicate poorly in any language.

Also, the more powerful a language is in it's ability to express and disambiguate information, the harder it is to master the language due to the increased complexity of it's semantics. There were no speakers of Ithkuil for that reason, given that it might take hours for a human to compute a sentence. Hence the invention of New Ithkuil

I think 'Kripke's Wittgenstein' was tainted by Kripke's semantic foundationalism. Kripke was correct IMO to conclude that Wittgenstein's concept of rule-following involves appealing to semantic criteria that are independent of the psychological facts of the rule-grasper (roughly speaking), due to the trivial implication that "to grasp" something entails a distinction of the grasper and the grasped thing. However, Kripke was wrong to conclude that the external criteria referred to assertibility conditions laid out by social convention. I think Kripke's incorrect conclusion about the later Wittgenstein's views was due to the fact that Kripke, unlike the later Wittgenstein, could not accept the non-existence of a universal and shared semantic foundation.
For Wittgenstein, any assertibility criteria can be used for defining the meaning of 'grasping' a rule, and not necessarily the same criteria on each and every occasion that the rule is said to be 'used'. And a speaker is in his rights to provide his own assertibility criteria for decoding what he says, even if his listeners insist on using different assertibility criteria when trying to understanding the speaker's words.

I'm not sure I follow. Can you reword this? — Tom Storm

I'm basically pointing to the ancient debates regarding the question as to what grounds personal identity. Does the ground consist of essential criteria, or not? And is the ground context-independent or not? The ghosts of folklore suggest to me, that humans ordinarily do not appeal to essential criteria when identifying a person.

In interviewing people who have experienced ghosts, what I find interesting is how often hauntings come with sound effects and beings present as fully dressed, often in period clothing. I get the theory behind a spirit appearing in some form, as an entity, but in clothing seems a stretch to me. Why would clothes also survive death? And sometimes there are ghost trains, cars and horses and dogs with their drivers or masters. What makes animals or machines come along for the undead journey? — Tom Storm

Isn't our very concept of a person made entirely out of the clothes of contextual accident?

Robots that make "perceptual errors" are only epistemically wrong in the sense of behaving in a fashion that their owners find undesirable. So if humans are robots, then humans don't really make epistemic errors when they fall victim to optical illusions.