 Good morning. Today and on Wednesday we'll look at Kripke's reading of Wittgenstein in these central passages in the investigations. Okay, so today I'll just set out what I take Kripke's basic pictures of what's going on here and start out by looking at what he calls the skeptical paradox. So, suppose you consider your own present use of the word plus. Three plus four is seven and all that. The notion you first learned all those many years ago. So, I guess we have all done a certain number of actual additions in the past, but for any of us there will be some large number such that we haven't done any additions beyond that number, if you see what I mean. You might have added a million and a million and got two million, but there's going to be some number that you never got up to, because some numbers are very, very large. So, Kripke's suppose is that it's 57 that you've never added beyond 57, but I know that we're all hyper sophisticated, and many of you will probably have added beyond 57 at some point, but if you think 57 is not the right number for you, then just pencil in 10 Googles or whatever your favourite large number, whatever you think the number might be that you never got beyond. So, here are two hypotheses as to what you meant by plus when you've used it in the past. One is, you meant addition. The other is, when you were using plus in the past, what you meant was cos. And here's what cos means. x cos y is the same as x plus y if x and y are both less than 57. Follow me very closely. And if either x or y is identical to or greater than 57, then x cos y is 5. Okay, so let's just do a couple of exercises. What's 5 plus 3? It was 5 cos 3. Very good. What's 56 plus 1? What's 56 cos 1? Very good. What's 57 cos 1? Very good. Right. So, here are two hypotheses as to what you meant in the past when you were using plus. That you meant addition or that you meant cos. And the trouble is to see how we're going to decide between them. I suppose, for example, that in the past you always meant plus. Suppose we just take for granted that you always meant plus. But I assigned the meaning cos to the plus sign. Is there going to be any difference between us? Let's break it down. Would there be any difference in the answers that you and I had given to any particular calculations? There would. Above 57. Right, remember that we're taking 57 to be our very large number. If you don't like 57 for that purpose, take 10 to the 57. Okay. So, would there be any difference in the answers we'd given to any particular calculations? No, because for everything below 57, then we will only add that up below 57. And for everything up to 57, x plus y and x cos y give the same answer. So, there'd be no difference there. So, there's no difference in what answers we've given to any particular sums. You mean plus, I mean cos. But when we use the cross sign, we'd both have given the same answers to any particular calculation. So, where would the difference between us be if you meant plus and I meant cos? Because after all, I might say, well, by the cross sign, I mean addition. You say, oh, that's funny. By the cross sign, I meant addition two. We use the same word, addition. Right? So, what is going through our mind when we do all these calculations is just the same? You look at 56 plus 1 of A57. I look at 56 plus 1 and I say 57. And what's going through our mind is just the same when we do those calculations. So, everything that went through our minds in the past, when we were using the cross sign, was just the same. So, we agreed in the answers we gave to every particular calculation. And what was running through our minds as we did it was the same. And Kripke says, well, was there any past fact about me that mandates what I should, that you meant, about what you meant by plus, that mandates what you should do now? Well, there's no telling the difference between you and me, whether you meant plus or whether you meant cos. What is the difference? There isn't any difference between you and me. You meant one thing and I meant the other. So, if you ask, was there anything in your past life that commits you to saying 59, when you're asked what 57 plus 2 is, rather than 5? The answer is no. Cos I meant plus all the time and it's fine for me to say 5. But you only had it going through your head or in the answers you gave, just the same thing I did. So, there wasn't any past fact about you. That was committing you. That was mandating you to give one answer rather than another. No. So, there's nothing that makes it the case that you meant plus rather than cos. That isn't any difference between us. And similarly, that's what's true about the past. The same is true right now. If you ask about what is running through your head right now, there's nothing about you right now that makes it so that you currently mean addition rather than co-addition by the plus sign. But this is pretty far reaching this because you could construct this kind of example for any sign, whatever. Here's Kruppke. Wittgenstein thinks that any construal that looks for something in my present mental state to differentiate between my meaning addition or co-addition or that will show that in the future I should say 125 when asked about 68 plus 57. Is there anything in my present mental state that will differentiate addition or co-addition, meaning addition or co-addition or implying that I've got to say 125 rather than 5. That's a misconstrual. That's attributing to the ordinary person an ocean of meaning that's refuted by these sceptical arguments. By the sceptical argument he means the kind of thing we've just been working through already today and last week where you say well what could there be in the actual calculations you made in your actual behavior or in what was running through your mind. Could there be anything there that would differentiate meaning addition or co-addition or dictating that the right answer is 125 and saying there's nothing there. But that is a really radical conclusion. The argument is very very simple and very very forceful but it implies that there's nothing in your mind right now that constitutes your meaning one thing rather than another by the sign. I just said it up for addition and co-addition but obviously you could do this again and again for endlessly many different possible meanings for the sign or consider the notion of zaddition which you get when you substitute 58 for 57 or vaddition when you substitute 60 for 57. You see what I mean? There is endlessly many alternative possibilities here. So there's nothing in your mental state that constitutes you meaning one thing rather than another by the sign. But if that's right then there's no such thing as anyone presently assigning any particular meaning to the cross. But the argument's obviously quite general. You could do this for any sign at all. If you take an ocean like chair where you mean by chair just what we regulate by the word chair you just mean a regular chair. When I say chair I mean a word that applies to anything if it's a chair that's been observed before the year 2013 but come 2013 it will apply only to zebras. Is there anything in our minds that will differentiate between what you mean by chair and what I mean by chair? Nothing in which objects we've applied the word chair to in the past and nothing in what was running through our minds when we were using the term in the past. So you could do this for any term whatever. So there's no such thing as anyone presently assigning any particular meaning to the plus sign or indeed to any other sign given the great generality of the argument. The argument is so simple. So here's Kripke. Wittgenstein's main problem is that it appears that he has shown all language and all concept formation to be impossible, indeed unintelligible. I mean it's not that there's something very difficult here that we wish we could do is that when you think through what it is that would be success would be you see you can't make any sense of it. There's no such thing as being successful. There's no such thing as meaning. I mean certainly there's nothing in anything being before 57 that would be different. But when faced with the plus sign and we can test the different, the kinds of additions that would be important on that are past 57 and we can see that they do yield different answers. And so it seems just like that anything prior for anything below 57 is just ambiguous but that doesn't mean that we mean that there's no practical matter about it. Yeah, well once you and I actually do use plus or the plus sign in connection with numbers that are above 57 then you'll go one way and I'll go another. But the question is was there anything in what was in our minds before we got above the 57 that dictated are going one way rather than another? The previous examples on which we agreed don't dictate are going one way rather than another. And for any general formula like well by the plus sign in addition, you and I can both agree on that. That's like that thing about the X plus one that just is dragged around after the particular examples. If you were asked what you meant by addition you'd say well 0 plus 1 is 1, 0 plus 2 is 2 and so on you'd give lots of examples. If you asked me what I meant by addition I'd say 0 plus 1 is 1 and I'd give the same examples as you below 57. So then the question is before we get up to these high numbers above 57 was there anything in what was happening then that dictated that you should go one way and I should go the other. And that's the point if the answer to that is no then it seems like we have a completely free choice when we get to 57 you and I. There was nothing that dictated that you meant one thing rather than the other. But that's already to give you the conclusion. You can still have in mind plus addition rather than quite addition. That's a lot, yes. Right. In the case that you're worried about you're going to use plus and in the case that below 57 it just happens to be a plus that is the same as plus. Your task is to show that there's something different going on in your mind and in my mind even before we start adding the high numbers. Right. So what was the difference? The numbers below 57 as a part of a larger set. Right. And so even though in the case below 57 it doesn't make a material difference I still have I guess in the background somewhere that I'm using plus instead of plus. In my mind. Well the thing is for me to be using plus I need not be using an explicit definition like that. When you say addition I mean it is possible to define addition. But usually when a regular person is using addition they're not using some fancy set theoretical definition of what addition is. Addition is just a primitive. So for me, cost is just a primitive. It's not defined in terms of anything. When we find out what's going on I might say oh when you're using the plus and you're using some really weird notion you mean what you mean is x plus y is x plus y x plus y if they're less for x, y less than 57. And you've got some really weird notion here if x, y are greater than or equal to 57 then you're going to say this is x zadition y for zadition zadition is x cos y zadition is 0 if x and y are less than 57 and x plus y if x and y are greater than or equal to 57. So I see you're using this really weird defined notion I was just thinking of the whole set of numbers and adition for all of them you had this funny defined notion we're both regarding them as primitive but you regard the addition as primitive I regard the co-addition as primitive neither of us regards it as a defined notion but each of us can regard the other as defining some weird counter-intuitive defined notion because you can articulate that you can give a rule in most normal cases you might think that it is simple just because most people don't get a definition for an addition or something but it's still possible it is possible to articulate these things but you've got to rely on a primitive notion of addition but for you is addition if you see what I mean and for me is co-addition you take addition as primitive I take co-addition as primitive so it's possible in both cases as definition but definition always depends on what stock of primitives you have do you mind responding to that? yes sure in the lower numbers what I'm saying the cluster still has some consequence lingering in the back of the head right but I think that's a mistake because there's a difference between defining the way the function works on all inputs and then applying that function to specific inputs to give an output right so if I say x plus 3 and then I give you 2 you say 5 right so in addition you have different rules for adding fractions right you're thinking about finding the lowest denominator all this stuff but when I'm adding 2 and plus 3 is 5 I'm not thinking about what I do with fractions no that's true so similarly a cluster isn't thinking about what they do with numbers greater than 6 to 7 right if I'm following you then the point is that someone using QoS might be just how should I say if you're asked what's 23 plus 2 most of us would just as you might say unthinkingly say 25 it's not that you run through some backup operation here and you're saying similarly for the QoS when they say what's 23 QoS 2 they don't do some calculation they just say 25 they don't need to explicitly look and say are they less than 57 they don't consult a definition they don't do anything like that they just unthinkingly say 25 I think that's exactly the picture that's right so I understand the difference between a function and a line a function in particular cases but all the same you can have a function so the point you made here is that you can have different notions about what's primitive you can still articulate what's primitive you can still have that in your mind and you could test that against what somebody else articulates as primitive so you say I'm weird because I'm using plus and really the primitive thing is QoS but I could still say to you that really the primitive thing is plus and not QoS and so we can have an argument about that right well we could if we could get to that point but the thing is when you say you can make that explicit or you can articulate it what the primitive thing is well the thing is what do you say is the primitive thing if you're how can I put this all regular guy how would you put what your primitive is you see addition right well the thing is that's what I say I say it's addition and there isn't any definition to appeal to here because we're both taking you say when I say addition I mean something primitive and I say sure when I say addition I mean something primitive so it's not clear what articulate means here with a primitive or it doesn't mean give a definition so the people behind you wanted to comment on just this I think yeah one two you can say that there is QoS you can still sorry can you do that louder I feel like you may have answered me I'm not okay there was one other person behind you wanted to QoS has inherent contradictions no wait a minute oh 50 plus 7 is identical to 57 yes that's right no no wait a minute yeah you have to make up your mind there are two different calculations you might mean by 50 plus 7 plus 1 okay it's 58 right is that right what's the contradiction contradiction is P and not P right okay do it just go back where you go 50 plus 1 50 plus 1 is 51 right this thing is hopeless the best you can hope for as I get confused but look if you look at the just a second if you go back to the definition of QoS right this is perfectly well defined there really isn't a contradiction in this if you just think of that as you really might want to construct a function like that in doing some mathematical problem some simple mathematical problem no reason why not it just is well defined and I think that I don't want to force you to go in this too long but I really don't think you can see how you could be getting a contradiction there it would be an absolute revolution if you did if from a simple definition like that that looks perfectly sensible you could generate a contradiction there is more to say about this but I think it's not going to take there's more to say about how you might get into difficulty here trying to use QoS but I don't think it's going to take the form of just generating something in the form P and not P I don't see how that could happen I want to I don't see why not I mean so long as you have an ocean like Z addition well X and Y X Z Y X Z Y is a zero if X and Y are less than 57 right and if otherwise X Z Y is X plus Y so then ordinary addition that's right that's right addition that's right on the understanding of primitive on which addition is primitive then addition is more primitive sure but the whole trouble is who gets to say what's primitive yeah and it's kind of more basic than that it's that once you acknowledge that whatever you do some signs are primitive what gives those primitive signs their meanings and the thing is what happens in a finite set of initial cases isn't enough to give those signs their meanings and since they're not defined signs there's no general formula you can appeal to as what gives them their meanings so it really is a very simple problem but it's very general one two yeah actually by considering the finite so in particular when you think back to how you learned to add larger numbers in grade school you always do with this you know you have 571 and then you write the like 20 below it and you write the line and the plus you know and then you do that addition by doing addition all on just two digits seems like the only rules you really need are or at least possibly the only rules you really need are the addition of the numbers 0 to 9 this rule of carrying over and maybe the tricky part from defining this is getting someone to understand that no matter how many places there are you have to keep applying the same rules but it seems like you can then possibly explain that yeah in one sense of course you're perfectly right that we do seem to manage to explain meaning by just a finite number through just a finite number of examples but the trouble here is coming when you say but what's going on when you say now just carry on in the same way if you remember the add one example yeah I mean it doesn't have this kind of breakdown you see what I mean well maybe I should maybe there's this kind of breakdown but the thing is that kind of breakdown there clearly doesn't help because you give us finite number of examples and then you say just keep doing the same thing and then the deviant insists I mean this is what eccentric solvers say I'm just doing the same thing I don't understand why other people act so weird I say look I am doing the same thing I'm doing just what you showed me in these initial examples that's right they can still claim to be right that's right so it seems like we can you know finite amount of time show someone how to add the number is zero to nine that's right it's just zero to nine that's right you've given a finite number of examples yeah but it's when you say and there's a strategy for generalizing those examples to the new cases that's where there's nothing that makes you go one way rather than the other because when you explain what the strategy is for generalizing to future cases you're just using more signs and those signs themselves have to be interpreted and the only way of interpreting them is to look to see what someone does in particular cases this is really basic and we'll keep coming back to this one, two, yeah if you had meaning yeah I don't really understand what you were saying do this again, you said you can make addition like plus if you then they're addition of those two numbers that's right so if I add that meaning to addition yes, if you said I'm going to define addition in these terms yeah well, okay, when I set this up I'm taking it for granted that we all know what we mean by addition and it's not quas yeah but that's really just for the sake of the example it's like a ladder, I'm now going to kick away because let's take for hypothesis that you mean addition and I mean quas and then look is there anything intrinsically different between us and what I'm saying is take those systematic rules that the last questioner was talking about, about what you do if you add one right, you get that going through your head I get that going through my head too yeah, and what we say about all particular cases we've had so far that's just the same so what's destabilizing here is what it comes to that you meant addition because there's nothing different between you meaning addition and you meaning quaddition it actually turns out to be just the same thing when you look at the hard facts about what your mental states were that's the argument, so you start out with these two different hypotheses and then you realize there's actually no difference between them they're assuming just the same things plus one so the student actually sees what the machine goes through and it adds plus one and it can check if they're doing it right it's actually out there in the world as an objective fact okay, I agree it's a really good idea to think about machines in this context maybe even a machine that can learn you can think about how you'd train it the thing is it depends exactly what machine you have in mind can it handle 27 digits can it handle a million digits can it handle a Google of digits any machine has its limitations yeah I don't know what an actual you mean a physical thing that is realizing a Turing machine you mean an actual concrete object yeah, right yeah we'll actually come on to this kind of thing in just a second but the thing is I mean not to evade it the thing is suppose you took a sum that it would take from now until the heat death of the universe to compute what asset does a machine give to that is it, well two questions the answer well sure there's a right answer but it can't be that what makes it the right answer is that's what the machine would do because the machine wouldn't do it the machine would have evaporated along with everything else you'll have an opportunity to come back to this it is more mileage okay but you see that this is a difficult conclusion to live with I mean saying maybe the stuff we've been talking about is just a mistake I mean if this is right then the whole class has been basing an even more profound mistake namely there's no such thing as meaning in the first place for us to study yes but that is yes that is a bombshell okay well there's an answer Kripke explores some length that is actually connected to this thing about machines he says couldn't you appeal suppose we accept that was different between you and me when you mean addition and I mean quaddishion there's no difference in which answers you actually did give in any past case and there's no difference in what you had running through your head but none the less your dispositions were different even back then before I ever dreamt of adding numbers as big as 57 if you had asked me what 57 plus 68 is I would have said 125 that was always true of me so even though I never did actually get up to those big numbers it was always true of me that had you asked me I would have given the addition rather than the quaddishion reply I mean that's true of all of us we have endlessly many tendencies to act in lots of different ways so if you just take is it clear what I mean by a disposition just some of the threats and promises some of the possibilities that we all have if you do this to me I'll give you that as my reaction so for lots I mean just to take I mean you don't really see my full range of possibilities when you just think about addition sums but they are among the potentialities that you or I have to answer lots and lots of addition sums so far and indeed through your life you only ever get to answer to deal with finitely many addition sums but your potentialities your dispositions to act have always been there and that gives you a way of expanding the talk about what might be fixing meaning beyond just what answers you actually did give and what general formula you had running through your mind oops so you can say look there's all these dispositions was 1 plus 1 was 68 plus 57 right you right now you all along all these dispositions and you might say my dispositions are different because of a quadr that's what makes me someone using plus sign to mean add rather than to mean quad but the trouble with this is that when you look at what your potentialities are what you will do to answer what answers you really would give I mean after all you might have blind spots with some particular numbers it might be that there are some numbers such that if you were to try to add them your brain actually has a weak spot there and that trying to add those particular numbers you'd have an epileptic fit right I mean that could happen it could be that we'd have to add some numbers might be so big that we'd have to add more stuff to your brain before you could cope with them yeah so that's perfectly consistent intuitively with you meaning plus rather than plus so if you run through if we take you right now and we say what answer would you give if we asked you four plus two well you know your friends gather around and they say well that's a safe bet you could marriage that six nope not a problem let's do something bigger 68 plus 57 that wouldn't be a problem but let's suppose we asked what's 37,829,458 plus four seven three eight two nine one well with all due respect you just might not make it your brain might blow up in you in that point the thing is that seems perfectly consistent with you knowing perfectly well what addition is and what matters is not what answers you would give but given your current understanding of the sign what answers you ought to give that's the basic problem that given the meaning you currently assign to plus there are right answers to these endlessly many literally infinitely many questions of our addition sums it doesn't matter if you're not perfect if you would in fact give the wrong answers because you're not very good at big sums because your brain gets tired because I've got a mathematician friend and whenever he introduces himself to people they say oh I was never any good with numbers you can be not very good with numbers so you just give very bad answers once the questions get a little bit complex but you still know perfectly well what add means what plus means so a little bit kind of obvious that I think Kripke's interpretation here the negative point that he's making is really consistent with the kind of picture we were talking about the last couple of times where if you ask what makes it the case that you and I we could put it in Kripke's terms by saying what makes it the case that you and I both mean the same thing by successor by add one there are the examples you give there are the examples I give but the thing in the middle doesn't do any work because if we deviate at any point we could both have that same thing running through our heads and our dispositions to give particular answers I might be disposed to go off the tracks compared to you around a hundred but that might just show that I have a tendency to give the wrong answers once we get to around a hundred there seems to be something about what we both mean by the sign that isn't being caught by what answers we'd be disposed to give hope we've got running through our head all the answers we give in particular cases but certainly the thing in the middle that X plus one bit drops out as irrelevant all we're left with is our tendency to give particular answers particular cases okay so let's just go back one more time over this thing about what the implications are of this picture for what you say about truth and reference the intuitive picture we've been working with is that the meaning of a word is its contribution to the meanings of sentences and the meaning of a word fixes what it is for a sentence containing it or wrong so if you take an old example Sally is tall you get two parts to that Sally refers to Sally and is tall applies to something that thing is tall but in these terms Wittgenstein's challenge is what does it come to that you know that the predicate is tall applies to something just if it's tall and what is it you'd be meaning one thing rather than another by the sign is tall what do you have to appeal to well you got a string of words running through your head you got an image of somebody very tall but you could interpret either of these in endlessly many ways any string of words you got running through your head the question is still but what do they mean or if you have an image of tallness an image of the elderly person going uphill with a stick or sliding gently downhill any kind of image you have could be interpreted in many ways so you don't have anything running through your head that constitutes you knowing this that is tall applies to something just if it's tall this is really just like a kind of shimmering fantasy the idea that there's something like that in your mind a Kirke's version of this challenge is how do you know that in the past you use the predicate is tall in such a way that it applies to someone that person is tall after all maybe you meant stall where stall applies to someone just if it's before Monday November the fifth and they are tall or is Monday November the fifth and they are three feet high right there's nothing in your past use that means you mean one rather than the other of those so if you're now confronted with someone and you're asked are they stall sorry are they tall I mean you wouldn't know what you're talking about but if I said are they tall and you looked back to your previous understanding of the term there wouldn't be anything in your previous understanding of the term that demanded that you had to give one answer rather than another there isn't anything that makes it the case that is tall applies to something if it's tall and similarly for the term salary first to salary all that we have the only hard facts we have are our verdicts to the rightness or wrongness of particular statements we do in fact agree in what we say about particular cases as lucky we do because only because we do agree in particular cases that communication civilization is possible but we don't agree in those verdicts because we understand the term our agreeing in those verdicts constitutes there being such a thing as a meaningful term here so nothing to guide nothing to keep you and me together in saying this one is tall, that one is tall that one over there is tall you say that, I say this one is tall, that one is tall that one over there is tall the image can't be what keeps us together so the image drops out and the string of words is tall applies to something just if it's tall that drops out on that bombshell Kripke tries to salvage the situation and we'll look at Kripke's positive picture on Wednesday, thanks great questions thank you