• Mathematical Conundrum or Not? Number Five
    ↪andrewk

    I think there are two issues.

    First, getting conditionalizing on being awake right. It's clearest perhaps to imagine the coin being tossed after the first interview. Could that coin toss have come up heads? Did it have a 50% chance of doing so? Yes and yes, but Beauty will not be asked about it if it did.

    Second, should Beauty conditionalize on being awakened, since she already knew she would be and cannot know how many times she is awakened. I think it only makes sense to consider the odds, in the absence of knowledge. What's more, if you don't, you're in the position of saying, "I'm right about the odds and can prove it by consistently losing money on tosses of this fair coin."
  • Mathematical Conundrum or Not? Number Five
    Since Beauty has no available information distinguishing the three states from her point-of-view, she is simply indifferent about which state she is currently in, and so assigns a probability of 1/3 for each awake state. — Andrew M

    I didn't follow this part of the thread, so sorry for the late reply.

    Beauty isn't assigning 1/3 from a principle of indifference. (I made the same mischaracterization earlier,)

    Monday splits 50:50 because the coin is fair; Tuesday happens or not on a toss of the same fair coin, so each quadrant is 25%, by stipulation.

    Beauty conditionalizes on being awakened, so the values change to

    .
  • Mathematical Conundrum or Not? Number Five
    The probability of first awakening is 0.5 + (0.5 * 0.5) = 0.75. — Michael

    There is a 50% chance that Beauty will be interviewed once, and a 50% chance that she will be interviewed twice, determined by the toss of a fair coin.

    Because the second interview, if it happens, takes place on Tuesday, the chance of being interviewed on Tuesday is also 50%.

    There are two orthogonal partitions of the space, one by Heads/Tails, and one by Monday/Tuesday. Each divides the space into subspaces of equal probability. Thus

    .

    If you now restrict the total space to occasions when Beauty might be awakened, according to the rules of the experiment, you lose , and now you have

    .

    It is this identity that Beauty will rely on, given that she has been awakened.
  • Mathematical Conundrum or Not? Number Five
    My calculation is that expecting that profit is the same as if there were a single $1 bet at even odds that the result of a coin toss will be tails and the coin had a 3/4 probability of coming up tails. To see this, note that in that case the expected profit is

    $1 x 3/4 + (-$1) x 1/4 = $0.50     — andrewk

    I think I see why this is happening, even though the odds are 2-1 against heads.

    By betting tails, I get to double what I risk only when my profit is guaranteed to double. — Srap Tasmaner

    The $1 payoff matrix for a fair coin , betting at even money, is just
              Bet
          H    T
Toss  H   1   -1
      T  -1    1

    If the coin is biased 3:1 tails:heads, we can multiply to get
               Bet
           H     T
Toss  H   .25  -.25
      T  -.75   .75

    Betting tails you win $0.50 on average; betting heads you lose $0.50 on average.

    Here's the $1 payoff matrix for our Sleeping Beauty, at even money:
               Bet
           H    T
Toss  H    1   -1
      T   -1    2

    It's a fair coin, so multiplying gives you:
               Bet
           H    T
Toss  H   .5   -.5
      T  -.5    1

    So betting tails again gives you a profit of $0.50 on average; betting heads you break even.

    So it is because your tails profit is disproportional to your tails risk.
  • Mathematical Conundrum or Not? Number Five
    ↪Michael

    Here's a slightly different argument.

    If, when awakened, Beauty knew it was Monday, she would answer 50%; if she knew it was Tuesday, she would answer 0%. There's a 100% chance of being interviewed on Monday, but only a 50% chance of being interviewed on Tuesday, based on the fair coin toss. That means it is twice as likely that the interview is being conducted on Monday.

    Thus 50% has a 2/3 chance of being the right answer (because the interview is being conducted on Monday) and 0% has a 1/3 chance of being correct (because the interview is being conducted on Tuesday).

    From that we can calculate a value for my conditional expectation of heads, given that I am being interviewed:





    The wagering argument confirms that this is the proper degree of belief.
  • Mathematical Conundrum or Not? Number Five
    The probability that it's Monday given the fact that it's heads/tails — Michael

    I know what the notation means. I don't understand what these probabilities represent. Are you sure they're even defined?
  • Mathematical Conundrum or Not? Number Five
    So put your money where your mouth is. What should you bet? I say bet £1 on heads. — Michael

    Sure, but it doesn't mean that tails is twice as likely to occur as heads, which is why these betting examples miss the point. All the betting examples show is that it's better to bet on whichever outcome provides more payouts, which is obvious. — Michael

    The betting matters, and be honest: you were happy enough to use wagering arguments when it suited you. (What's more the examples don't show what you say is obvious; they assume it and it's irrelevant here anyway.)

    Why does the betting matter?

    What I'm saying is that there is no reason for her to have a greater belief that it was tails than heads. When she's asked what her belief is that it was heads the rational answer is 50:50. — Michael

    Because of this point of @andrewk's:

    in interpreting the term 'degree of belief' it seems reasonable to make it the probability of tails when betting on a single flip of an unfair coin — andrewk

    Suppose Beauty is not told all the rules, the amnesia pills, etc. She's told she'll get even money -- she has to be told the odds to bet. Given enough trials -- more than is reasonable, I know, it's a story -- she'd conclude from her accumulated payoffs that the coin is unfair.

    Now that's quite curious. If betting is a stand-in for hypothesis testing, it'll support a theory that we, in our omniscience, are inclined to call "false" but it will serve Beauty perfectly. Beauty will have plenty of evidence to support her theory.

    It'd be interesting to flesh out competition between the two theories. If Beauty doesn't learn what's really going on, she would be incredulous that we would claim the coin is fair. Even if we explained, she might see that our theory is as adequate to the evidence gathered so far as hers, but she'd have no reason to switch to our view without new evidence, something like witnessing new trials showing the coin to be much closer to fair than she thought. That whole process is really cool.

    PS: Our explanation is spectacularly more outlandish than a coin being unfair.
  • Mathematical Conundrum or Not? Number Five
    ↪andrewk

    Yeah, that's right. When you win, you get your stake back and the odds represent your profit. So 2-1 against heads gives you back the $1 and pays you $2 more, total of $3. On tails, you'd get your dollar plus $0.50, total of $1.50. Losing, you lose your dollar. (I only know as much about gambling as I need to to understand philosophers who talk about probability this way!)
  • Mathematical Conundrum or Not? Number Five
    ↪Michael

    I don't understand at all what you mean by P(Monday|Heads) and P(Monday|Tails).
  • Mathematical Conundrum or Not? Number Five
    This is Betting Game 2 from this post. — andrewk

    Nearly.

    Game 2: At each interview, Beauty bets $1 to guess what coin came up, and loses that dollar if wrong or wins $2 if right. — andrewk

    This is why I said I couldn't understand how you were setting the odds. Doing it this way is paying off 2-1 on both heads and tails, which is incoherent. (Unless you meant both tails interviews would taken together pay off $2, but that's still incoherent.)

    My calculation is that expecting that profit is the same as if there were a single $1 bet at even odds that the result of a coin toss will be tails and the coin had a 3/4 probability of coming up tails. To see this, note that in that case the expected profit is

    $1 x 3/4 + (-$1) x 1/4 = $0.50

    Hence, in interpreting the term 'degree of belief' it seems reasonable to make it the probability of tails when betting on a single flip of an unfair coin, at which one would have the same expected profit. So under this interpretation of 'degree of belief', the answer is 3/4 for tails, and hence 1/4 for heads.     — andrewk

    Yeah that's really interesting. On the one hand, my pass at making a fair book says the odds should be 2-1 against heads. But the expected profit on betting tails clearly matches the 3-1 version.

    Did I get the fair book wrong, or is there some other explanation for why they're coming out the same? (The payoff system here is so wonky, I think this could be another side effect of that, but I can't see how.)

    Either way -- 2-1 or 3-1 -- offering even odds in this game loses money. We agree on that, right?

    How to interpret this as credence or degree of belief, I'm not sure yet.


    Can you please expand on it — andrewk

    I guess I could do this again -- I wouldn't mind, and will if you're really interested, but to me it's no longer relevant (if it ever was). I was still trying to figure out how the thing works when I wrote that.
  • Mathematical Conundrum or Not? Number Five
    I understand how it works, now that I finally got a moment to do the wagering argument.

    By betting tails, I get to double what I risk only when my profit is guaranteed to double. I should not be allowed to do that.

    I also understand now why @andrewk remembered that it all turns on what you mean by "credence". The weirdness here is that even though my own description has the two scenarios arising half the time each -- fair coin, after all -- I should bet "as if" the odds are really 2-1 against heads.

    This is annoying, because we (Bayesians, subjective probabilists of all stripes, and the Bayes-curious like me) want to use wagering to measure degree of belief. If you can show that I must wager in a way that systematically deviates from my degree of belief, that's trouble.

    So there's a good meaty philosophical issue waiting at the end of this one, which I will now ponder (and probably read Lewis). Again, nice thread @Jeremiah.
  • Mathematical Conundrum or Not? Number Five
    1 point for each would also be a Dutch book wouldn't it? — Michael

    Missed this.

    Yes. That was exactly my argument: if you offer even money, I am guaranteed a profit.
  • Mathematical Conundrum or Not? Number Five
    ↪Michael

    You should be paying out $2 on heads, for a wager of $1. That's what 2-1 (against) means. On the favorite, you only pay $0.50, because the odds are 1-2 against.
  • Mathematical Conundrum or Not? Number Five
    1 point for successfully guessing heads and 0.5 points for successfully guessing tails (because you get two opportunities — Michael

    I don't understand this. What you're doing there is giving even money on heads and 2-1 for on tails. You can't do that.

    My policy is to wager $1 on tails whenever I'm asked. You're paying even money.
    If the toss is heads, I lose $1; this happens half the time, so my expected loss is $0.50.
    If the toss is tails, I make $1 each time I'm asked; this scenario happens half the time, so I have an expected profit of $1.
    So I make at least $0.50 on average each time I play, no matter how the toss goes.
    I have made a Dutch book against you.

    Suppose instead you're offering 2-1 against heads, and I still wager $1 on tails each time I'm asked.
    On heads, it's the same: expected loss of $0.50.
    On tails, I only make $0.50 each time, for an expected profit of $0.50.
    My profit and loss cancel out, so 2-1 is a fair book, representing the true odds.

    At least that's how I think it works.
  • Frege's Puzzle solved
    ↪Belter

    Frege's view is that "Hesperus" and "Phosphorus" have different senses but refer to the same object. Since he splits meaning into two "components", it's not clear if you and he agree on what "synonymy" means here.

    Still not sure where you stand, sorry.
  • Mathematical Conundrum or Not? Number Five
    When awakened Beauty does not know if it is Monday or Tuesday. — Jeremiah

    On this issue of whether there's new information: it's true that all she gets is that she's been awakened. She knows no other new facts, but she knows two things:

    • She's more likely to be awakened on Monday than Tuesday;
    • she's more likely to be awakened if the toss came up tails.

    In the absence of new (certain) knowledge of what has happened, shouldn't she go with the probabilities of what has happened?

    One difference between the days and the coin toss is that the days are just clues to the coin toss; it's the toss that determines whether she's awakened a second time.

    If I'm making a mistake, it's somewhere around here.
  • Mathematical Conundrum or Not? Number Five
    Do you mean that if both people who are asked if it's tails correctly guess tails then that should only be counted as 1 success rather than 2? — Michael

    Not exactly. If you're calculating wagers and payoffs, you'd need to add up all the gains and losses for the odds to make sense.
  • Mathematical Conundrum or Not? Number Five
    ↪Jeremiah

    Gotcha. I'm at work so I'll read later. Maybe much later since I'm enjoying fighting through this "on my own".

    Another good thread.
  • Mathematical Conundrum or Not? Number Five
    These cases are different because being asked (again) provides you with additional information, whereas it doesn't in the original case. You're going to be asked regardless, and you have no idea if you've been asked before. — Michael

    One more quick note.

    The point was that being asked doesn't have to be a simple fact/non-fact, but can have a probability, and we're told that the probability of being asked is driven by the results of the coin toss.

    You might still be right that this doesn't matter, I'm not sure.
  • Mathematical Conundrum or Not? Number Five
    Would you consider the 2/3 debacle an argument against the 1/3 argument and for the 1/2 argument? — Jeremiah

    Don't know what you mean.

    ↪Michael

    The multiple person argument is really interesting and might persuade me. (One thing that occurred to me is that if there's betting, splitting the payoffs among multiple people is wrong.)

    ((Sadly it'll be a little while before I can come back to this.))
  • Mathematical Conundrum or Not? Number Five
    ↪Michael

    Variation 2

    Suppose I toss a fair coin; if it comes up heads, I ask you if it was heads or tails; if it comes up tails, I don't. When asked, you should always guess "heads".

    Now move the likelihood that I'll ask in each case.

    Variation 3

    If the coin comes up heads, I ask you once to guess; if it comes up tails, I ask you twice.

    This is different from our case because for each round you know the first question is the first question. That one's 50-50, but once I ask again, you know to answer "tails".

    Now suppose there were a way to fix it so you didn't know how many times you were being asked or whether a question was a first or a second. All you know is that on tails you'll be asked twice. What do you guess?
  • Mathematical Conundrum or Not? Number Five
    ↪Michael
    ↪andrewk

    Here's a variant that is structurally similar:

    No sleeping theatrics.

    Examiner tosses a fair coin, and then tosses another. If the first toss was tails, she asks Beauty her credence that the first toss came up heads; if the first toss was heads she only asks for Beauty's credence if the second toss was heads as well, otherwise the round is over.

    Done this way, Beauty will know that when she was not asked the first toss was heads, but she can do nothing with that knowledge. She's not asked and the round is over. What matters is that she's always asked when it was tails and asked half the time when it was heads. So her credence that it was heads should be 1/3.

    and in the case that it's tails it's only her bet on the last day that's accepted — Michael

    I don't understand this provision. Beauty knows that she will be woken more often on tails than heads and should be allowed to show that in her betting behavior, even if she can't at the time know how many times she's been woken.

    ((Will try to put together a betting argument when I have time, because I haven't understood the ones presented so far. If the odds of heads really are 2-1 against, a fair book with no vig should give an expected payout of 0, and Beauty should be able to make a dutch book against anyone who thinks the odds are even. I don't understand how you and andrew are setting the odds.)
  • Mathematical Conundrum or Not? Number Five
    The trouble is that we cannot use conditional probabilities. — andrewk

    Finally got some time to come back to this and I think you can use conditional probabilities.

    I have Beauty treating Monday-Tuesday as another 50-50 coin toss. Not conditional on her being awakened, mind you -- will come back to that -- just something she has no way of knowing by some other means so she forms no opinion either way.

    Here's the simple table @Jeremiah posted:
           Mon   Tue
Heads   A     S
Tails   A     A





    That makes sense to me. Am I missing something?

    We can also ask, what is the chance that it's Monday, given that she's been awakened?





    That looks right, and also makes sense. Of course it's twice as likely to be Monday given that she's been awakened, just as it's twice as likely that the coin toss came up tails, given that she's been awakened.

    Am I being completely stupid about this?
  • Mathematical Conundrum or Not? Number Five
    ↪andrewk

    Presented in a way that suggests we should look for a conditional, but that won't work. I could see that, but I'm still mulling it over.

    Everything else seems to point to 1/3. Twice as many possible awakenings are on tails, leaving just 1/3 for heads. 2/3 of the awakenings are on Monday and half of those are heads, so 1/3 again.

    But now I don't know if that's misdirection too.
  • Mathematical Conundrum or Not? Number Five
    ↪Jeremiah

    Yeah that's pretty clean.

    I realized right after I posted that the question is almost literally what is the probability we got heads given that you have been awakened and are being interviewed? So we want the conditional P(H|A).

    That's what I'm working at. Not sure what I've got so far. Nice puzzle for me because my probability skills be weak, so thanks.
  • Mathematical Conundrum or Not? Number Five
    ↪Jeremiah

    So there's a conflict between guessing which interview this is -- and you weight by the number of possible interviews -- and guessing which day this is, right? It's not Wednesday, because you're being interviewed; if it's Monday, you're interviewed either way; so what are the odds that it's Tuesday?
  • Frege's Puzzle solved
    ↪Belter

    I wouldn't have even thought of "... refers to ..." as being an identity relation. It's clearly not. A symbol is not the thing it symbolizes. I don't think Frege anywhere suggests that reference is an identity relation.

    How about this one?

    (1a'') "Hesperus" refers to Venus.

    Frege's approach here is to say that this has the same truth value as (1a) and (1a'), but that there are real semantic differences between these sentences. The referring expressions in each refer to the same object, the planet Venus, and that's what determines truth value; but each expression specifies or determines the reference in a different way -- i.e., they have different senses but the same reference. Anyway, that's where Frege lands, as I understand it.
  • Frege's Puzzle solved
    But we do not use natural language identities to say non-informative tautologies but to make identifications between words (synonymy relation) and a word and an object (reference relation). — Belter

    Okay, I think I understand what you're saying now.

    Frege analyzes "Hesperus is Phosphorus" as asserting the identity of the objects picked out by the names "Hesperus" and "Phosphorus" -- they refer to the same thing.

    On your analysis, "Hesperus is Phosphorus" is one of these:

    (1a) "Hesperus" refers to Phosphorus.
    (1b) "Phosphorus" refers to Hesperus.
    (2) "Hesperus" and "Phosphorus" are synonyms.

    At least one of the names is implicitly being mentioned rather than used, and that blocks substitution. But it blocks substitution only within the quotation marks, so you still need a way to block

    (1a') "Hesperus" refers to Hesperus.
    (1b') "Phosphorus" refers to Phosphorus.

    and this you get by arguing that we don't (cannot?) know which of (1a) or (1b) is intended. (Is "intended" right? Or is it that there is no "fact of the matter" as to which analysis is right?)

    If we're unable to use (1a) or (1b), we must take "Hesperus is Phosphorus" as (2).

    (2) asserts the synonymy of the names used; Frege asserts the identity of the things named.

    But what is the meaning of a name?

    If the meaning of a name is its reference, then asserting two names have the same meaning is asserting they have the same reference, refer to the same object. Isn't that exactly where Frege starts?

    Sorry, still a little confused.
  • Mathematical Conundrum or Not? Number Four
    The Russell predicate most definitely picks out a class: the class of all things that are not members of themselves. This class just doesn't happen to be a set. It simply turns out to be the case that some collections defined by predicates are sets; and others are not. — fishfry

    Yeah, that's a good point. I had forgotten about the proper class stuff and was reaching for "class" as the generic term. (I guess here we have to use "collection" for a generic term?) Not an overly satisfying distinction, this, but it is what it is.

    Wikipedia has the quote from Russell's letter:

    there is no class (as a totality) of those classes which, each taken as a totality, do not belong to themselves. From this I conclude that under certain circumstances a definable collection [Menge] does not form a totality

    It's interesting how unlike something like Russell's set is. If it were "the set of sets that contain themselves and don't contain themselves", it would not be so peculiar. Instead it's simply "the set of sets (that are sets) and don't contain themselves". All you have to do to get into trouble is take the collection as itself an object, something that could be the value of .

    You can almost hear the other sense of "totality" in our English-language descriptions: there's no problem defining a set that contains only sets that don't contain themselves; you just can't define a set that contains all and only sets that don't contain themselves. You can tell a barber to shave only men who don't shave themselves, just not all of them.***

    Pages back I mentioned the word "heterological" but got no takers, Right before the quote above, Russell says this:

    Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself? From each answer its opposite follows. Therefore we must conclude that w is not a predicate.

    "... is heterological" sure looks like a predicate, and this time there's no use saying it picks out a "proper class" instead of a "set". In Russell's opinion it's not a predicate at all. What's up with that?

    *** ADDED: Nah, it works the other way around too, "all" but not "only".
  • Mathematical Conundrum or Not? Number Four
    ↪Akanthinos

    I disagree.

    The Russell set is manifestly pathological. That it is, I could know even without knowing its role in the history of philosophy, but the converse does not hold: I have to understand what it is to understand why it played the role it did.

    The more interesting question might be why it is pathological, and that could be a matter of historical context. If, in the fullness of time, dialetheism wins out, the Russell set might no longer seem pathological.

    As things stand, we have a predicate composed of simple, well-behaved elements to all appearances assembled in an acceptable way, and yet this predicate cannot possibly be predicated of anything. If we could say why this abomination is no predicate at all, we could regain the Paradise in which predicates always pick out classes.
  • Mathematical Conundrum or Not? Number Four
    ↪Akanthinos

    I can't find anything in your post I'd agree with. Certainly not the "amused pity" or the treatment of Russell's argument as an artifact.
  • Frege's Puzzle solved
    ↪Dawnstorm

    That's all dead on. Sometimes when he and I sit on the front porch, I teach him a little philosophy.

    With "the go ahead run" I was imagining the case of someone learning about baseball and misusing that phrase as meaning "run".

    "Did we score another go-ahead run, Daddy?"
    "No, sweetie, this one was tacked on -- only the first one this inning was the go ahead run."

    So there's your hiccup.

    (The example I really wanted to use was, "Did we score another equalizer, Daddy?" but I don't even know if football announcers still say, "It's the equalizer!" like Toby Charles did.)
  • Frege's Puzzle solved
    More weirdness with definite descriptions. A run in baseball is only "the go-ahead run" if the game is tied before the run scores. If the game's not tied, that description just doesn't seem to have a reference.

    With Venus, we cross into territory where the object is persistent and the description can be capitalized into a Name. Maybe others feel the description still bleeds through, but I wish I had a clearer example.

    ADDED: Sometimes it refers to a possible run because you can do the math. Announcers talk this way: if your team's down by one with a runner on base, they might describe the player coming up to bat as "the go-ahead run". You still can't score the go-ahead run if you're already ahead.
  • Frege's Puzzle solved
    ↪Dawnstorm

    Agree completely that the use of "another" implies some stuff I was ignoring, much of which you described nicely.

    There are so many ways to do this. Some run into the issue you and @BlueBanana raise that these terms are only (supposed to be) used in certain situations.

    (1) You can call me "Pat".
    (2) You can call the Morning Star "the Evening Star".

    (1) might turn out only to apply in a range of situations, not all, or not, but it's tempting to say (2) is just, well, if not wrong, then really weird.

    Whether the use of a particular name (or nickname or description) is appropriate may not change the truth conditions of sentences it's used in, appropriately or not. I think if my son pointed at Venus of an evening and said, "Look, the Morning Star has risen," that would be true if a bit arch.
  • Forced to dumb it down all the time
    West Cupcake Weekly Advertiser. — Bitter Crank

    Don't be a hater. This was a great paper until Murdoch bought it.
  • Frege's Puzzle solved
    Kind of my issue with Frege - the assumption is that syntax and semantics is the whole story for natural language, whereas it is not (although it might be for formal languages). — MetaphysicsNow

    I was going to pass this by because you're kinda right, but there's more to say here, coming off my last post.

    Say you want to approach the natural language issue by looking at the pragmatics -- when do we have a use for statements like "Hesperus and Phosphorus are the same object"? That looks fruitful. You'd start by noting that there's "no point" in telling someone that "Hesperus is Hesperus" (which probably isn't quite true), and go from there.

    But something about that isn't quite right. The reason we feel there are different uses for "Hesperus is Hesperus" and "Hesperus is Phosphorus" is precisely because we feel they don't contain the same information. So it is with "4 = 4" and "2 + 2 = 4". It's that sense that these two equations carry different information -- they "say different things" -- that drives their different uses. So the semantics drives the pragmatics here.

    We could then circle back to Frege and look at the truth conditions, either in terms of possible worlds or in terms of verification, and confirm that there's a semantic difference.
  • Propositional Logic
    ↪Rayan

    Yes.
  • Frege's Puzzle solved
    ↪Belter

    I'm still not clear what you're saying.

    Everyone agrees these assertions achieve the same purpose:

    (1) Hesperus and Phosphorus are the same object.
    (2) "Hesperus" and "Phosphorus" are different names for the same object.
    (3) "Hesperus" is another name for Phosphorus.
    (4) "Phosphorus" is another name for Hesperus.

    The puzzle to be solved is that names for the same object should be substitutable salva veritate. No problem saying

    (5) Phosphorus and Hesperus are the same object.

    But this is weird:

    (6) Phosphorus and Phosphorus are the same object.
    (7) "Phosphorus" is another name for Phosphorus.

    So you remember that that names are substitutable salva veritate only in extensional contexts -- it's practically the defining feature of extensional contexts -- and conclude that this is an intensional context somehow, thus akin to

    (8) Pat doesn't know that Hesperus is Phosphorus.

    You can't say this is equivalent to

    (9) Pat doesn't know that Hesperus is Hesperus.

    Everyone's clear on that much. But we don't have an obvious tip-off in sentences like (1)-(7) -- no propositional attitude verb like "know", no "that" clause.

    Frege's solution, you could say, is to say there's always something a bit like an intensional component: the sense (Sinn) rather than the reference (Bedeutung).

    It's not a solution everyone loves, because pure extensionality is much cleaner. [Insert a hundred years of discussion here.]

    For natural language, maybe there's a solution that just treats these types of statements involving names and identity as special. The thing is, Frege was really looking at mathematics, where it seems awkward to say that equations are just a special case. ("2 + 2 = 4" is exactly the same issue.) But hey maybe they are. Mathematics as a domain is a deliberate narrowing of what we might say.
  • Propositional Logic
    ↪Rayan

    Go for it. It looks really nice.

    If you have questions, this is reasonable place to ask first. If the answers you get here aren't helpful, you can head for the logic StackExchange.

    Quora is often a good place to find recommendations for other internet resources like classes posted on YouTube.
  • Frege's Puzzle solved
    ↪Belter

    I don't understand what you're saying.

    (1) "Hesperus" = "Phosphorus"
    (2) Hesperus = Phosphorus

    (1) is just false, and (2) is the claim that "Hesperus" and "Phosphorus" designate the same object.

    Other ways of switching around use and mention, such as

    (3) "Hesperus" = Phosphorus

    might not come out at all like you expect. We say things like this when the quotes indicate something fake about the name, like at the end of a Scooby-Doo mystery; or we end up with a start on Haddock's Eyes.

    So what is it Frege is getting wrong?

    (Btw, I don't think of names, unlike descriptions, as ever being translated, though they do get localized. It's an odd area.)
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