Modern science can find no such thing that answers to 'the sensation of being in pain'. That's the problem I'm attempting to address. — Isaac

Yes, as was Wittgenstein throughout his entire career.

I would first suggest reading Bertrand Russell's Analysis of Mind. Without reading this book, it isn't easy to fathom the ideas Wittgenstein was criticising and the problems he was attempting to solve, especially with respect to his remarks regarding 'private language'. Wittgenstein's 1930's transcript known as 'Philosophical Remarks' is also necessary reading to understand his intermediate thinking and to put his later remarks into the right context. Both of these works are freely available online.

Notably in PR, Wittgenstein gives what is to my understanding the first 'zombie argument' against behavioural understanding of 'Other Minds', predating David Chalmers zombie arguments by 65 years. But unlike Chalmer, he phrased the zombie problem in terms of the indistinguishability of sense when interacting with a person versus a zombie, as opposed to Chalmer who posed the problem in a more realist fashion in terms of metaphysical substances.

Ultimately, the late Wittgenstein was an anti-realist who didn't think of 'other minds' in the sense of other substances or in terms of property dualism. Rather he emphasised the role that ones own sensations play when one attributes so-called 'other' minds to other people.

Determinism and non-determinism aren't real properties of the universe, for as demonstrated with the Middle-Earth fictional universe, these terms have no descriptive value with respect to any complete data-set. Therefore it makes no sense to speak of the universe as a whole as being either determined or not.

Probability theory hints at what those terms describe; they describe semantic relationships between data-sets or theories. For example, the sequence {1, 2, 3} is "determined" in relation to the sequence {a,b,c} under the assignment a --> 1, b --> 2, c-->3. But it is undetermined with respect to the set of sequences beginning {1,2,...} which include it as a special case.

Consider historical counterfactuals; Must Hitler have invaded Poland? In spite of appearances, the meaning of this question isn't about the literal existence of a possibility available to the German government in the year 1939, rather it concerns the relation of a model of the actual event to a hypothetical set of "similar" circumstances, such as that defined by a historical simulator, where it is the notion of "similarity" that is actually the focus of the question.

In Automata Theory, non-determinism is the existence of two applicable state-transitions from a given state. This isn't an epistemic notion.

The author JR Tolkien determined what happened in the world of Middle-Earth, but he didn't specify what would have happened in the story if alternative courses of action were taken. So is the world of Middle-Earth deterministic , non-deterministic, or neither?

Is even the world of a choose-your-own adventure story, where alternative courses of action can be chosen by the reader, describable with either of these adjectives?

Are we saying the same thing only you maybe more technically correctly? Or something different?
All I'm saying is that if I throw that fair die, I shall get any of 1 through 6, and that I shall expect each to occur about 1/6th of the time. — tim wood

I was responding to the common belief that chance represents ignorance. We know from experience that the odds for a "fair die" landing on any side, if tossed in a typical fashion, are roughly 1/6 in the sense of relative frequencies. This is essentially the definition of what a "fair die" is.

But a priori, we don't even know that. For in the case of an unknown die that isn't necessarily fair, all we know a priori is that the odds for any outcome is between 0 and 1. Nevertheless, there persists a convention which assigns a uniform distribution in the case of an unknown dice. But this is misleading for it conflates knowledge of a fair die with ignorance of an unknown die, and also leads to inferential biases in the case of an unknown die that aren't warranted.

If all you know is that an event has n possible outcomes, there is nothing more than can be said, and chance cannot be quantified.

What does this mean? In words? — Neoconnerd

It is saying that the physical propensity for obtaining the respective possible outcome, is above zero and less than 1. Without additional information, or assumptions, a more precise set cannot be assigned to the outcome.

What is imprecise about assigning 1 - 6 as possible outcomes of the throw of a die? — tim wood

It is imprecise because probability intervals are assigned to outcomes, rather than numbers.

e.g. P ( dice throw = six) = (0,1)

Which only express the fact that throwing a six is possible, but not certain.

If I throw a dice the chance that I throw any of the six numbers is 1/6. The dice rolls determined towards its destined number.

Our lack of knowledge gives rise to chances. If we somehow could know the initial state of the dice and the exact interactions with its environment then the final numbeŕ could be known.

Well, that's the naive argument. In practice, it can't be known in priciple. Which doesn't mean that the process is not determined. It is.

You can say that chance is a subjective feature that we project to, for example, the world of dice. A dice has a 1/6 chance of showing one of the six (if the dice is ideal). — Zweistein

The subjective interpretation of probabilities as representing ignorance which you appear to be assuming, is a logical fallacy in my understanding. Lack of knowledge should give rise to possibilities only. Moreover, it is impossible for anyone to distinguish ignorance from objective uncertainty before the fact. Such distinctions can only be drawn after the fact.

To represent uncertainty in terms of probabilities isn't to assign a particular distribution, but to assign a set of distributions, which is sometimes referred to as assigning "imprecise probabilities". In the case of complete ignorance of a thrown die, one should assign the entire set of distributions with six possible outcomes, which amounts to saying that one only knows that there are six possible outcomes.

Furthermore, having described one's state of ignorance by using a set of distributions, one shouldn't then average over the set, as Bayesians often do, to obtain a precise "ignorance prior", for this fallacious practice amounts to an attempt to extract information from ignorance.

It makes sense to say " I am not my body ", even if it doesn't make sense to say "He is not his body". Only by conflating the subjects of these sentences does a contradiction arise.

Wittgenstein as a phenomenologist. Presumably not of the Heideggerian school? — Banno

A precise answer to that question can be found here.

Wittgenstein was not promulgating coherentism. But I have an interest in reconciling Davidson - Quine's intellectual son - and Wittgenstein, so I'm interested, if confused.

Wittgenstein's so-called "Anthropological Holism"
— sime
Start there. What is it? Who called it that? — Banno

Fodor et al. as described here

For Wittgenstein, even the concept of a machine implementing an algorithm is relative to human customs, rather than what machines themselves are doing per-se. He is a logic anti-realist. Unfortunately, his understanding of rules as being ontologically dependent upon the background context of human customs for their recognition is perpetually misinterpreted as referring to meaning and rules being epistemically dependent on social feedback, something which isn't helped by the terrible wikipedia article on the subject.

I don't know any other word with which to describe Wittgenstein other than a coherentist, given his abandonment of foundationalism and recognizably Quineian stance towards semantics, nothing the 1951 publication date of the Two Dogmas of Empiricism that was two years prior to the posthumous publication of PI in 1953. There have been some recent attempts to divide their points of view, but I personally find them unconvincing.

It might be considered cheating to a phenomenologist, but a useful short-cut to clarifying Wittgenstein's ideas is to read Quine's Word and Object, The Two Dogmas of Empiricism, and Truth by Convention. A scientific analysis of semantics helps to dispel the same myths that Wittgenstein dispelled through phenomenological investigation, including the myth that meaning is founded upon either convention or upon 'stimulus-synonymy' as envisaged by the logical postivists and the earlier Wittgenstein. It is generally believed among the academic community, that the later Wittgenstein's so-called "Anthropological Holism" isn't reducible to either social or personal convention or to 'Tractatarian names'. Yet people insist on jumping to conclusions through quote mining , and then attributing nonsensical views to the author.

When under a local anaesthetic for dental treatment, it is certainly possible to feel ambiguity or uncertainty as to whether or not one is in pain. And self-reinforcing beliefs and observer-effects come into play, where one asks "am I really in pain, or just imagining it?

Recall Wittgenstein's analysis of Moore's proposition later on in the book, namely "It is raining, but I believe it is not raining", which Moore had previously considered to be nonsensical when considered in the present tense. Wittgenstein, if i recall correctly. envisages the possibility of this sentence making sense in a situation in which one finds oneself consciously disbelieving that it is raining whilst observing oneself to be behaving otherwise. Our thoughts can belie our actions.

Both intelligence and stupidity refer to behaviour that is interpreted to be goal-driven. In the case of intelligence, the goal state is considered to be desirable, whereas for stupidity the goal state is considered to be undesirable, thereby making the distinction between the two concepts relative and subjective.

The question is really about the realism of counterfactuals as well as the question of backward causation, and the reality of the temporal order.

For instance, suppose that If Bob lights a fuse, then a bomb will explode. At first glance, this appears to imply a temporal order in which Bob's action as cause precedes the bomb exploding as effect. But this clause can also be restated in reverse by saying that if the bomb is observed not to explode, then Bob couldn't have lit the fuse.

Presumably, if a bomb is defused in a state of ignorance as to Bob's earlier actions, one can at least conclude that defusing the bomb didn't alter the earlier fact as to whether or not Bob previously lit the fuse. Or does it? For there isn't a way of testing the counterfactual as to what would have happened earlier in the past had the bomb been allowed to detonate. Furthermore, if the bomb is very big then any potential evidence as to Bob's earlier actions might get destroyed by it's detonation. According to a presentist interpretation of history, there aren't any facts about the past that transcend the state of the present. This logically implies that in a finite universe where history cannot be preserved indefinitely, if a sufficiently big bomb explodes, then it must explode for no reason.

The dead, however, present as being utterly devoid of consciousness. This goes beyond reflex action or stimulatory response...one can tell when the brain has died, and cellular metabolism has ceased. From this state, this "death", there appears in my experience to be no regaining of any level of consciousness whatsoever. — Michael Zwingli

Yes, that might be true from your perspective when it comes to you appraising the mental state of other people. But can you speak meaningfully and authoritatively about past cases of your own unconsciousness? what empirical criteria are you using in this case? Is it really possible for you to infer necessary conditions for the existence of your own mind via analogical reasoning from your understanding of the necessary conditions for other minds?

Rationalists have a tendency to overlook and equivocate the vastly different empirical criteria used for defining unconsciousness of the first-person, as opposed to when defining the unconsciousness of another human being.

In the latter case, a human being is publicly defined as 'being unconscious' according to behavioural criteria, such as bodily reflexes failing to respond to stimulation. But in the former case, when we speak of personal unconsciousness of ourselves, we aren't using a public behavioural definition and we only speak of personal unconsciousness in the past tense. Ironically a person makes this past-self judgement according to observations they presently make.

For example, a person wakes up in the morning from a deep-sleep and concludes that they didn't exist during the previous night , due to not remembering anything over that period. But how does the person determine that they don't remember anything about the previous night? They ask themselves what they remember and they observe that "nothing" comes to mind, but "nothing" here means that whatever is presently sensed or comes to mind is considered to be irrelevant to the question. Yet this purported 'evidence' for past-personal unconsciousness in effect constitutes present empirical criteria for defining what is meant by past-personal unconsciousness, which might suggests to an empiricist that the concepts of past-personal-unconsciousness and present personal consciousness aren't ontological opposites.

This article provides relevant context regarding the history and evolution of Wittgenstein's later thought. The SEP's article on private language is also recommended reading.

In my opinion, it only makes sense to discuss Wittgenstein's remarks when they situated in the context of the traditions of both analytic philosophy and phenomenology.

In my opinion, yes, on my understanding that it is linguistic convention which ultimately decides whether a sentence is true or false (for on any physical understanding of verbal behaviour, every assertion can be understood to be a true representation of it's physical causes, and therefore true).

Every person has their own linguistic convention and associated agenda, and conventions come into either conflict or cooperation for political reasons. Discussions and debates which on the surface look like passive disputes over the objective nature of shared truth, are ultimately analysable in terms of the resolution of socio-political objectives. But i don't see this as a nihilistic conclusion and more like an alternative view of philosophy.

Recall that the later Wittgenstein was negatively reappraising the phenomenalist doctrine of logical atomism that he, Russell and Mach previously assumed, where names are interpreted as stand-ins for sense data as suggested by the picture theory of meaning. But this solipsistic doctrine implies a fixed and idiosyncratic relationship between word meanings and the perceptions or mental states of a solitary speaker of the language, which in turn implies that the public meaning of such names in a community of speakers is synonymous to the use of indexicals like this, him, here, it, with names only communicating raw presence among speakers without a shareable accompanying description of what is perceived. Such a conception of names essentially ignores context, and in particular the (overwhelmingly complicated) inferential semantics that constitutes the descriptive function of names, both in the case of public communication and in the case of private introspection. As Wittgenstein illustrates, the actual usage of words , both in public and in private, ultimately defies any purported private or public definitions, except in the most trivial and useless cases. As with scientific and mathematics terminology, the definition of a natural language is forever playing catch-up after the facts.

The actual referents of a belief are it's immediate physical causes; so whenever a speaker asserts a so-called "false" belief, any alleged epistemic error exists solely in the minds of the listeners, due to their misidentification of the causes of the speaker's assertion.

How does a person experience dementia? presumably he finds himself learning about things that he has good reason to believe he has previously forgotten. But how does he classify an experience as being of something forgotten? Whatever the experiential criteria, perhaps an avid learner should consider dementia to be the ultimate learning experience.

I recall an interview in which Sam Harris (atheist, author, neuroscientist) claims that to believe nonexistence (I mean death) is unthinkable is, as he put it, "...for a lack of trying..." He explains: there are people in Paris, his choice of city, who don't know you exist; in other words, you don't exist as far as Parisians are concerned. That, according to Sam Harrris, is to give you a glimpse of what nonexistence is! — TheMadFool

There are people in Paris observing the Eiffel tower, who are not observing the computer monitor you are observing; in other words, your computer monitor doesn't exist as far as Parisians are concerned.

And so presumably according to Sam Harris, he has given you a glimpse as to what the non-existence of your monitor is.

What I mean is that one can look up the entry for the axiom of choice on nLab, without encountering a rant against ZFC. So I think you're the one adding that part, and not your fellow constructivists / category theorists / programmers or whatever direction you're coming from.

Indeed, nLab expresses choice as "every surjection splits," which they note means "every surjection has a right inverse," in set theory. This formulation is easily shown to be equivalent to the traditional statement of the axiom of choice. There is no distance between the category-theoretic and set-theoretic views of choice. — fishfry

Category theory is a useful meta-language for understanding precisely where the mathematical foundation proposed in the early 20th century goes wrong, as well as being helpful for relating alternative theories. CT is itself philosophically neutral in the sense that it only assumes the presence of identity arrows in a category, but places no other constraints on either the presence or absence of arrows, provided the laws of arrow composition and association are obeyed. Therefore disputes between intuitionists, formalists and platonists carry over into the language.

Like a mathematics department, nlab as an encyclopedia is obviously going to disseminate mathematics in a politically neutral fashion. Or perhaps i should have said, unlike a mathematics department. But political neutrality doesn't amount to reasonableness regarding which mathematics should be prioritised.

You lost me there. Absent AC there is a set that is infinite (not bijective with any natural number) yet Dedekind-finite (no proper subset is bijective with the entire set). I don't know what you mean here. — fishfry

I'm talking about stream objects in computer science, or equivalently the isomorphism
S <----> 1 x S in the Set category, where S is a stream object, 1 is the terminal object, i.e the singular set, and x is the Cartesian product. Unfolding the definition:

S <----> 1 x S <----> 1 x (1 x S) <----> 1 x (1 x (1 x S)) ....

Since each of the arrows is invertible, S clearly has, by definition, a surjection onto any finite set, which is what i meant by Kuratowski infiniteness.

On the other hand, recall that in category theory every element of a set S is an arrow of the form
1 --> S. However, these arrows haven't been specified in my above definition of S, and therefore the number of elements of S is currently zero, i.e. S is the empty set, which is another way of saying that S is a completely undefined set until the first observation is made.

Every time an observation is made, an arrow of the form 1--> S is introduced into the above category, and we can denote the current state of the stream by shifting from left to right in the above diagram. But at every moment of time, the arrow S --> S that is implicit in the the product S ---> 1 x S is a bijection, meaning that is S is forever dedekind-finite.

S cannot exist as an internal set of ZF because it isn't a well-founded set, although it can exist in the sense of an "external set" that is to say, inside ZF as part of a non-standard interpretation of an internal well-founded set. And it it cannot exist in any capacity inside ZFC.

All of which is tantamount to saying that ZF has only partial relevance to modern mathematics in terms of being an axiomatization of well-foundedness, whilst ZFC is completely and utterly useless, failing to axiomatize the most rudimentary notions of finite sets as used in the modern world.

I'm perfectly happy to stipulate so for purposes of discussion. After all, there are no infinite sets in physics, at least at the present time. So, what of it? The knight doesn't "really" move that way. Everybody knows that knights rescue damsels in distress, a decidedly sexist notion in our modern viewpoint. Therefore chess is misleading and unrepresentative nonsense. Nevertheless, millions of people enjoy playing the game. And millions more enjoy NOT playing the game. What I don't understand is standing on a soapbox railing against the game. If math is nonsense, do something else. Nobody's forcing you to do math, unless you're in school. And then your complaints are not really about math itself, but rather about math pedagogy. And I agree with you on that. When I'm in charge, a lot of state math curriculum boards are going straight to Gitmo. — fishfry

So are you agreeing that mathematical infinity has neither philosophical nor scientific relevance and that everyone knows this, or am i right to stand on a soap box and point out the idiocies and misunderstandings that ZFC seems to encourage?

I might point out, though, that assuming the negation of the axiom of choice has consequences every bit as counterintuitive as assuming choice. Without choice you have a vector space that has no basis. An infinite set that changes cardinality if you remove a single element. An infinite set that's Dedekind-finite. You lose the Hahn-Banach theorem, of vital interest in functional analysis, which is the mathematical framework for quantum mechanics. The axiom of choice is even involved in political science via the Arrow impossibility theorem. — fishfry

Obviously, a denial of AC doesn't amount to an assertion of ~AC, given that things are generally undecidable, but i see no counter-intuitive examples in what you present. In fact, many examples you raise should be constructively intuitive if we recall that construction can proceed either bottom-up from the assumptions of elements into equivalence classes, or vice versa, so an inability to locate a basis in a vector space using top-down construction seems reasonable.

As for the sciences, AC is meaningless and inapplicable when it comes to the propositional content. At best, AC serves a crude notation for referring to undefined sets of unbounded size, but ZFC is a terribly crude means of doing this, because it only recognises completely defined sets and completely undefined sets without any shade of grey in the middle as is required to represent potential infinity.

QM has also been reinterpreted in toposes and monoidal categories in which all non-constructive physics propositions have been removed, which demonstrates that non-constructive analysis is dying and going to be rapidly replaced by constructive analysis, to the consternation of inappropriately trained mathematicians who resent not knowing constructive analysis.

Besides, if you have a nation made up of states, can't you always choose a legislature? A legislature is a representative from each state. If there were infinitely many states, couldn't each state still choose a representative? The US Senate is formed by two applications of the axiom of choice. The House of Representatives is a choice set on the 435 Congressional districts. The axiom of choice is perfectly true intuitively. If you deny the axiom of choice, you are asserting that there's a political entity subdivided into states such that it's impossible to form a legislature. How would you justify that? It's patently false. If nothing else, each state could choose a representative by lot. — fishfry

Obviously, the axiom of choice isn't used in the finite case. In the infinite case, the sets of states needs to be declared as being Kuratowski infinite in order to say that the elements of the set are never completely defined, and so a forteriori the size of the set cannot be defined in terms of it's finite subsets.

Secondly, the set should be declared as Dedekind finite, in order to say that the set is an observable collection of elements and not a function (because only functions can be dedekind-infinite).

So, yes, you can choose as many representatives as you wish without implying a nonsensical completed collection of legislatures that are a proper subset of themselves, but formalisation of these sets isn't possible in ZFC, because AC and it's weaker cousin, the axiom of countable choice, forces equivalence of Kuratowski finiteness and Dedekind finiteness.

It upsets some people (Frega, Meta) that mathematical axioms don't necessarily "mean" anything or "refer" to anything. — fishfry

And it should do, for classical set theory and real analysis are misleading and unrepresentative nonsense, unless cut down to the computationally meaningful content. Students who are taught those subjects aren't normally given the proviso that every result appealing to the axiom of choice is nonsensical, question-begging and of use only to pure mathematicians and historians.

What seems clear to me is that if we do live out other lives, in order to do this, there has to be a source that maintains the continuity of the self, otherwise it's difficult to make sense of this idea. — Sam26

Why not just deny the possibility of eternal oblivion by denying the existence of a continuous self, even within a single lifetime? That way you circumvent the need for evidence of reincarnation, and avoid all of the scepticism that the begging of evidence entails.

In my understanding, he was pointing out that our inferences can neither be understood nor be given justification on a definitional basis on pain of infinite regress, a point first raised by Lewis Carroll , whom Wittgenstein was evidently inspired by and indebted to.

The impossibility of infinite analysis, coupled with the problem of theory under-determination, means that the inter-translation of rules and phenomena and even rules and rules is under-determined in both directions. Hence Tractatarian approaches to philosophy are profoundly misguided.

Precisely. the proposition ~R --> A isn't in contradiction with the proposition ~R -->C because both denote possibilities, as opposed to probabilities or propensities. To get the latter, a non-logical probability measure must be added.

Or alternatively, since precise probabilities are usually difficult and controversial to assign, one simply ranks ~R --> C above ~R --> A to indicate which they believe is the most likely.

Modus Ponens is a logical rule for the composition of possibilities but not probabilities, since all logical statements are relative to the truth of premises that are non-logical axioms. So it is perfectly acceptable to disbelieve the actuality of a conclusion of Modus Ponens, for non-logical reasons.

Logic specifies what can happen, but not what will happen. After all, if that weren't the case, then an axiomatic system such as Peano arithmetic wouldn't be a forest of proofs, but merely a single proof of one result consisting of a single chain of reasoning.

Needless to say, there is an (unfortunate) temptation among philosophers and mathematicians to mix the concepts of logic/possibility with statistics/probability by considering conditional-probabilities to be a generalised form of logical implication. This is generally disastrous, because possibilities are easier to state and justify than probabilities which are usually ill-defined and whose use is generally controversial.

Consider the fact that

A. Oceans aren't defined in terms of unions of droplets.

This means that atomically constructive definitions of oceans in terms of merging droplets together is irrelevant in terms of the logical characterisation of an ocean that assumes no physics. To mathematically define an ocean is to write it down instantaneously without constraining it's size.

B. Oceans are potentially infinite in terms of their number of droplets, but are not actually infinite.

This means that

1) An ocean is Dedekind-finite; there does not exist a constructable bijection between any number of droplets extracted from the ocean and a proper subset of those droplets.

2) An ocean is not specifiable a priori as a finite object in the sense that there is no a priori specifiable upper-bound on the number of droplets that can be extracted from it. In other words, an ocean, apriori, isn't equivalent to any a finite subset of droplets extracted from it. In mathematical parlance, oceans are therefore Kuratowski-infinite, like an infinite-loop in a computer program that isn't a priori equivalent to any finite number of loop iterations.

Together, 1 and 2 necessitate the rejection of the Axiom of Countable Choice, since that axiom forces all non-finite sets to be dedekind infinite.

Oceans are streams in a type-theoretical sense, which are lazily-evaluated lists

Ocean (0) = Ocean (no droplets so far extracted)
Ocean ( n) := [ droplet (n+1), Ocean (n+1) ] (n+1 droplets so far extracted)


Therefore we can say Ocean(0) > Ocean(1) > Ocean (2) .... without assigning a definite quantity to Ocean (0) and its predecessors, and without assuming that Ocean(i) is evaluated for all i, in the sense that only when we draw a droplet from ocean (i) does ocean (i) expand into [droplet(i+1), ocean (i +1) ].

And when the ocean eventually runs dry, our non-standard mathematical specification that is consciously aware of an a priori/ a posteriori distinction in mathematical meaning, isn't contradicted by reality, unlike in the case of classical set theory that in appealing to AC equivocates the a priori with the a posteriori.

I interpret you as asking: to what extent do acts of violence and destruction satisfy the motives of the terrorist?

This raises the question as to how the motives of terrorism, and violence in general, are determined, and the extent to which it is possible to determine motives through the analysis of language and behaviour.

For example, what were the motives of rampaging England fans after they lost to Italy? Is a Marxist analysis of English hooliganism warranted? or were they merely indulging in spontaneous and instinctual acts of self-gratification in the absence of a sufficient deterrent under the influence of alcohol? I'm inclined to believe both.

There might be something lurking in the notion of 'good reason' that has to do with degrees of good reason, which also relates to degrees of confidence in beliefs. And Pfhorrest broaches the matter of lack of certainty. I'm not inclined to it, but maybe a solution does lie in that direction. — TonesInDeepFreeze

In logic, either an arrow A -> B exists, or it does not. And so for logic there exists only possibility or non-possibility. On the other hand, probability measures over a set of propositions in a model of logic are chosen freely in accordance with external beliefs or experiments.

On the left side below are the axioms of OP's problem that specify every possible election outcome. On the right side is an example of a consistent set of degrees of confidence assigned to each possibility that coheres with every premise of the OP.

Andy or Carter --> Andy 0.25
Andy or Carter --> Carter 0.75

Reagan 0.80
Carter 0.15
Andy 0.05

As usual, Modus Ponens holds while saying nothing about the relative likelihood of possible winners.

Probability theory, which is currently the most fashionable calculus for representing and reasoning about beliefs and uncertainty, is only defined up to a measure over a sigma-algebra of sets denoting a collection of propositions. Unfortunately, practitioners of the theory don't normally consider this collection to be a model of any specific set of logical axioms, but rather as representing classes of observables, which means that modus ponens is formally absent from probability theory. Whenever an underlying logical system isn't specified in an application of probability theory (which is nearly all of the time), it is undetermined as to whether conditional probabilities or joint probabilities are the more fundamental epistemic principle.

Nevertheless, it is natural for Bayesian practitioners to assume some implicit underlying logic in an ad hoc fashion and to interpret modus ponens in terms of set intersections, in Venn diagram fashion. But as the example demonstrates, probabilities can behave non-intuitively with respect to modus ponens. Formally, Modus ponens speaks only of logical possibilities and not probabilities which are property of a model of a logic.

MP can be defined generally and abstractly as the composition of arrows in a category. In problems such as the above, the arrows denote conditional probabilities of the form P(B | A) between two propositions A and B , and premises denote arrows of the form 1 -> A, where 1 is a terminal object representing an "empty" premise.

The example also highlights a general problem: given a state of knowledge, is it consistent? and if so, how do you determine what the underlying arrows are?

In the previous example of the OP, the beliefs given are consistent. The arrows are the conditional probabilities of candidates winning given knowledge of the failure of one or more of the remaining candidates, and there is only one premise, namely that a republican wins.

If the principle of non-contradiction is regarded as being be logically necessary, then it cannot say anything apart from asserting a grammatical promise not to contradict oneself, in which case it is merely a normative linguistic principle rather than a empirically descriptive epistemic principle. On the other hand, if the principle is regarded as being empirically descriptive, then it must fallible, in which case it also cannot play a role in any epistemic foundation.

This also seems to be the case for any other suggested foundational principle: either it is regarded as being infallible, in which case it cannot rule out any conceivable possibility and hence is epistemically redundant, else it must be regarded as fallible and therefore not a foundational principle.

In the case of statistics or beliefs which involve probabilities,the standard non-probabilistic version of Modus Ponens is generally inapplicable,since there it isn't generally used as a constructive principle, and so it is neither fair nor surprising to point out the failure of MP in this situation . And yet statistical relations do obey a generalised version of Modus-Ponens with respect to conditional probabilities:

Take for instance, the following beliefs:

P (Reagan wins) = 0.80
P (Carter wins ) = 0.15
P (Andy wins ) = 0.05 (i.e. distant third republican)


P (Reagan or Andy) = 0.80 + 0.05 = 0.85 (i.e. the probability that a Republican wins)

P(Reagan | Reagan or Andy ) + P(Andy | Reagan or Andy) = 1 (i.e, as a logical tautology, Andy must win if Reagan doesn't, relative to the assumption that a republican wins)

But if Reagan doesn't win, then

P(Andy | Andy or Carter) = 0.05/ (0.05 + 0.15) = 0.25, (i.e. Carter remains favourite over Andy)

But notice that although this example contradicts (the misuse of) logical Modus Ponens, it doesn't contradict "probabilistic modus ponens" of the form P (B,A) = P( B | A) * P(A), which when summed over the values permitted for A recovers P(B).

In other words, if we take the conditional probabilities as being fundamental and follow this example in the bottom-up direction using this probabilistic modus-ponens, we recover the initial unconditional beliefs.

The axiom of inifinity is non-controversial, as it merely amounts to the inductive convention of calling a finite tree a "tree", a finite list a "list, a finite set a "set" etc. Nobody who talks about "lists", "trees" or "sets" in ordinary language implies a completed totality of such objects, and neither does the use of the axiom of infinity in a proof, because as we recall proofs by definition have finite derivations and use every axiom finitely.

The real numbers however, are nonsensical with respect to experimental physics and engineering, where their literal definition is at odds with respect to how the formalism is actually used. There, real numbers aren't used literally in the sense of referring to infinitely precise quanitities, but are used non-rigorously or "non-standardly" to refer to indefinite and imprecise quantities and taken together with noise and error terms. For this reason, in conjunction with the rapid ascent of automated theorem proving and functional programming that are based on type theory, the awkward, misleading and practically false language of real analysis can only die fast.

There are different formulations that may have equivalences, and there are complications throughout, but I know of no proof nor mention in the article you cited that shows the equivalence of AC with LEM in intuitionistic set theory. The SEP article does say "each of a number of intuitionistically invalid logical principles, including the law of excluded middle, is equivalent (in intuitionistic set theory) to a suitably weakened [italics in Bell's earlier article] version of the axiom of choice. Accordingly these logical principles may be viewed as choice principles." But the question was not that of various choice principles but of AC itself, and we have not been shown a proof that AC and LEM are equivalent in intuitionistic set theory. — TonesInDeepFreeze

yes, originally I was speaking roughly in relation to that article while making what i considered to be a tangential point in relation to the thread topic. As an axiom, LEM when interpreted in the Set category by the usual Tarskian approach, is an axiom of "finite choice" in the sense of asserting 'by divine fiat' the existence of a choice function for every relation into a finite set, i.e. that every finite set is 'choice '. Stronger choice principles additionally assert the existence of choice-sets that are the non-constructive infinite unions of the finite choice sets.

Observe that the meaning of choice principles are different in constructive logic than in classical logic, and recall that the controversies over LEM and AC concern only their implied non-constructive content.

Bear in mind

1) All of the non-constructive content of classical logic is discarded by jettisoning LEM.

2) The axiom of choice holds trivially as a tautology in sets constructed in higher-order constructive logic, because in this logic existence is synonymous with construction.

So one could even say that absence of LEM implies AC (or perhaps rather, that AC is an admissible tautology in absence of LEM).

But this statement isn't enlightening, because it conflates the difference in meaning that AC has in the two different systems, for AC holds trivially and non-controversially in constructive logic as a tautology, where it doesn't imply anything above and beyond construction.

In the constructive sense, i think it is fair to say that LEM implies AC, when speaking of AC not in the sense of an isolated axiom, but in the commonly used informal vernacular when speaking of choice principles in their structural and implicational senses

"And of course, we know that LEM does not imply AC, since we know that ZF is consistent with ¬AC while LEM holds." (MathStackExchange) :chin: — jgill

Sorry for the confusion. Yes that is true for ZF, since it is built upon classical logic. In set theory, controversial instances of the excluded middle are the result of both the underlying logic if it is classical as well as the set theoretic axioms of choice and regularity.

What i had in mind wasn't ZF, but intuitionistic set theory, in which choice principles and LEM are approximately equivalent as documented in the SEP article on the axiom of choice.

https://plato.stanford.edu/entries/axiom-choice/