The Logic of Wittgenstein's Tractatus

The Tortoise I didn't tell you is a notoriously obscure work still not well understood ninety years after its initial publication so one motivation for this research is simply to understand what the tract re-envision for a logic was in particular for a perfect logical notation and I like to think that we actually made substantial progress in that direction another motivation is to see whether such a vision can actually be carried out in in a way that somehow successful that's perhaps not an alternative to standard ways to doing predicate logic and it turns out that that's also true and I'm hoping to convince you of that today okay this works the logic ho notational innovations of the truck tada sebaceous basically - they both occur in the in the five fives and they pertain to a the notion of identity the Kings time wants to eliminate the Equality sign from logical notation not just by dropping it right because then obvious that you lose expressive power sorry he has to find a way to retain the expressive power of logic with identity without actually using a symbol for it and the second idea is to make do with a single logic constant the the so called an operator and well we'll see the textual data from the Tractatus they are unfortunately somewhat which has allowed two well substantial confusions in the secondary ledger okay so let's see what he actually tells us right so it's the famous programmatic slogan identity of the object I expressed by identity of the sign and not by means of a sign of identity difference of the object by difference of the signs some of this is not surprising right obviously you can express identity of the object by using the same sign again what's forbidden is the use of the Equality sign and what is somewhat remarkable is the idea that as soon as you use different signs they have to stand for different objects and as it turns out that pertains not just to what we call names right or constants it also pertains to variables in fact that's where the power of the proposal really resides so then we get a bunch of examples how VidCon style would like to rewrite for sealian logic the first exam going to Panther with the first example which incidentally isn't particularly well chosen but I'm not going to elaborate on now on today the example in five five three two is a very interesting go right so let's see so in wrestling notation right we can write the exist X Y such that f x y and x equals y and so we can sign it wants to write instead there's this next FXX well that's no big deal right we can do that in standard predicate logic to however the second part is interesting right to say that they're distinct objects x and y that satisfy f Vidkun Steinman simply wants to use distinct variables and that's supposed to suffice to indicate that there are two objects satisfy okay which then necessitates so here it looks like this is actually shorter than the rustling expression right but then turns out in other cases it's clumsier in fact it's usually clumsier so in standard notation right we can leave it open whether x and y assume the same value or different values in the VidCon Stinney n' system we have two distinguished cases we have to say either there are two objects x and y such as FX y or one object x satisfies FM in both places ok and then in a similar vein we have the final example here I'm not going to discuss it in detail this example is there I'm pretty sure mainly in order to demonstrate that Russell's theory of descriptions is available even in the identity free logic that Vic and Stein and visitors here there's one sort of textual exegetical point that's mildly interesting about this very last sentence only one X satisfies F and then the open parenthesis this is going to matter a little later in the proto-truk tardes precursor manuscript of the book written Stein here has a notation f of X hat the haps the circumflex is the Principia notation to indicate the independent variable of a propositional function I'm going to talk more about that later that has disappeared from the final manuscript and in fact the circumflex doesn't occur in the truck taught us at all well one could say a lot more about this but I won't so that's the data on identity what's the data on the N operator well first of all NF y bar is the negation of all the values of the propositional variables I basically that means n is a very general joint negation operator so a propositional variable is a variable whose values of propositions if you take all the values of some additional variable together prefix it with n somehow you get their joint denial the Greenstein gives us two well actually three concrete examples right if the variable the propositional very has only one value P then n FXI bar is just the negation of P if does the two values P and Q then is the standard Scheffer stroke more interestingly if the propositional variable is specified with the help of an open sentence f of X we can negate all the instances of f of X with the help of n all right we let C be a propositional variable that has all these instances as values then NF y bar is the negation the joint negation of all the instances for all X not FX or equivalently not there exist in X FX because sure here is this this is trans foliage translator I'm sorry this is translated from DJ well that's the Ogden version of the book that was published in 1923 he has the I can't leave even the the eighth word of 5.5 what is the identity ah yes well um it's yeah um no that's bad let me desert maybe humans do that there's it's not like right he feels compelled to write his own book in the notation that he thinks would be ideal right I mean there are many examples where talks about formulas that are actually written in resilient notation using the standard quantifiers using identity etc well that's somewhat tricky right because what we have here is functional notation and the identity free notation that of the constant envisages really is not geared towards a treatment of complex terms built up with functions so right you really want to reformulate function in terms of relations and either well that's a very good point I think that's how this has standardly been read I don't think that's true I don't think that what Vidkun Stein has in mind here is a infinitary truth-functional operator very briefly that's the cause this case of applying n is clearly intended to display what Vidkun Stan calls the symbolism of generality so generality is supposed to play a role here if this could be written simply as a as a large disjunction or conjunction what Vidkun Stein called the characteristics of the symbolism of generality would not be there right our generality ah is sort of distinguished by by displaying prototypes all right and the prototype FX would disappear if it were just if he says something like there's an X there is a Y right and it's clear that your signal Y is different from the value of y so if I was if I were to say something like there is an X P of X and there is a Y P of Y so that the wire is long does go I'll talk about that in a moment all right so far I've just given you the data I haven't told you how this is actually going to work okay I'm going to talk about that too in fact that's one of the things I am most proud of in this talk ok so thanks for making people curious okay so it's a joint denial operator it can be applied in very general ways and here's the claim right every proposition can be constructed by successive application of this one operator to elementary propositions so well you know that has been disputed in the literature all right it's been affirmed in the literature um we'll see about that in a moment okay so what's the plan I'm going to try to tell I'm going to tell you are the different ways in which victim steins identity convention could be interpreted I'm going to tell you which one was actually the intended interpretation I'm going to develop an explicit notation for a logic based on the N operator we're going to defend that notation and then I'm going to briefly show you how the two conventions the two logical innovation of the TARDIS can be jointly implemented in a logical system that actually has a reasonably Pleasant proof systems are in this case tableau system which we take to be congenial to the general idea of the trip lattice okay let's start with identity you've already seen that the right the maxim of having different reference for different names extends to bound variables right that was the second example in five five three two if we just write exists X exists Y F X Y right we know these the values of x and y must be to stick okay now one might think that this is somehow just due to the simple fact that x and y are type of gravity distinct variables but that doesn't really mesh with a third example in five five three two if we want to express their just X Y F X Y as read in the ricean way in between Stein system we have to write it this disjunction and it's really not well it's not even coherent to say that the value of this X has to be somehow constrained by the value we pick for the Y and the first disjunct this exit has to range over the entire domain in order to give the rewriting the right meaning so it can't just be a matter of having typographic distinct variables the concurrent prohibition must somehow be due to structural features of the formula in fact it must be due to the fact that the quantifier scopes overlap now that can actually be implemented in different ways well two ways really right both of those were identified by him to commit well 55 years ago one of them is called hinting I called it the weakly exclusive interpretation so the idea is when you reach a quantifier and you want to know which object are excluded from the range of the variable that it binds you look inward from that qualifier you check which objects are mentioned within the scope of the quantifier right where basically the objects mentioned are the values of variables that occur free in the scope of that quantifier so the somewhat lengthy way of saying it the range of a bound variable Z contains all objects except the values of any variables that occur free within the scope of the quantifier binds Z we'll see very few examples in a moment it obviously gives the right result with respect to Vidkun Stein's example right we look at exists X it Just's Y if X Y the range of X is unconstrained because nothing occurs free in the range of X but then right forget about exists X in the scope of X exists Y we have the X free so whatever value is picked for X can't be a value of y that guarantees that there must be two objects x and y such that XY okay the other the other way of implementing this is not by looking inward from a quantifier but by looking outward so when you see a quantifier all right for all Z you check within which scopes that quantifier occurs and any values that have been picked for variables in who scoped the quantifier occurs are excluded from the range of Z so the range of Gaunt variable Z contains all objects except the values of any variables within whose scope Z occurs the scope of a variable being the scope of the quantifier that binds it and this also gives the right results for the example right again nothing is excluded from the range of exists X that has maximal scope doesn't occur within any other scope but exists Y occurs within the scope of because it's X and so the value would pick for y must be distinct from that big 4x does that make a difference right it doesn't make a difference whether I look inward or outward as it turns out it does so here's a simple example right for all X if for all Y P Y then P X if we take the weekly exclusive interpretation this from you has exactly the same meaning that it has classically because nothing occurs free in the scope of anything else right well nothing occurs in the scope of for all Y nothing occurs free and discover for all X so both have unrestricted ranges if we look at the strongly exclusive interpretation right we see that Y occurs within the scope of X and so we would read this as the resilient formula for all X if for all Y other than X P Y then P X and unlike the first reading right this is not a logical truth so it does make a difference actually uh there is an additional complication that well or at least a bifurcation that hint ago didn't at least didn't address I don't know if they noticed it probably not because he was loath to speak about free variables at all the weekly exclusion temperature interpretation well we write once we're given a domain and a variable assignment we can interpret any formula according to the weak excuse interpretation that's not true for the strongly exclusive interpretation because it's not clear what we should take the scope of a free variable to be all right if we take for instance exists exists X px and py we could think of the free verbal Y as name like right is not having any scope in which case the range of X is unrestricted because it doesn't occur within the scope of any other variable we could also think of Y as being implicitly universally say bound right over the entire for in which case X would be within the scope of Y and so Y would be excluded from the range of X okay so I'm calling the view on which names of free variables have no scope the narrow strong exclusive reading sorry if we think of free variables of having maximal scope we call it the we're going to call it the broad strong exclusive reading on this reading our example here has this translation into resilient logic right there's an X other than Y px and py um and so we have a total of three possible interpretations of Vidkun stein slogan at five five three so weekly exclusive narrow strongly exclusive broad strong exclusive and the question arises can we tell which one Wittgenstein had in mind now there are some technical reasons that actually favored the we clicks to interpretation the the bookkeeping effort that's needed for the strong exclusive interpretation is somewhat tedious because you have to keep track of material that occurs outside the sub formula that you're looking at all right I don't know if that's a strong consideration when we were convicting Stein but we can do better than just that so first of all the broad strongly exclusive reading can actually be excluded on the basis of the text Hindi call already noticed that if we look again at the example in five five three two one all right at most a satisfies X for all X of F X then x equals a is translated by victim sign in this way if right if anything is F then a is and no two things right now if the broad strong exclusive reading row effect then a would have scope over taya formula this would translate into receiving logic into something like this never mind the first contract here that's not a problem but the second contract would read there are two objects both distinct from a that satisfy elf alright and that's compatible with a and another object one other object being F so I wouldn't have the right truth conditions if it's supposed to translate this okay I'd a has scope over the entire formula so all quantifier all quantifier ranges are constrained by a x and y both both have to be distinct from a all right now unfortunately both the weakly exclusive reading and the narrow strong explicit reading are compatible with all the examples in the book so we have to decide between these as it turns out we have circumstantial evidence to settle the matter there's a posthumously published manuscript by Frank Ramsey that addresses Vidkun Stein's identity convention Ramsey writes the following an apparent variable so recall right the print capilla jargon an apparent verbal is a bound variable free variables are free or various other real variables so a bound variable cannot have the value of any letter which basically any variable occurring in its scope unless that variable is a variable bound in that scope which says exactly that the values of free variables are excluded from what the values of variables that are free in the scope of a quantifier are excluded from the range of that quantifier Ramzi in fact does more he also explicitly rules out the strongly exclusive reading he tells us that never mind whether this is true right but he says it's clear that we must be able to treat frolics FX as a unit having a fixed meaning independent of what else occurs in the proposition and that's not the case under the strong exclusive readings right because for instance if you encounter for all X FX by itself the range of X is unrestricted but if it occurs as a sub formula right within the scopes of other variables then are the value that we pick for Y can't be a value of x and so it's not that its meaning the meaning of frolics FX is not independent of what else occurs in the proposition now why is Ramzi are such good witness in this case well um he wrote a he wrote a review of the tracked artists in early 1923 that was published in mind that's a really remarkable paper to this day probably one of the very best introductions to the tract autism really impressed by how this extremely young man at the time penetrated the book without ever having spoken to Vivian Stein at that point that's not what makes him a good witness he went to see between Stein in Austria in the summer of 1923 for two weeks during which they discussed the truck TARDIS everyday for five hours that must have been fun after that he had extensive correspondence with Wittgenstein among other things about the identity convention there's a letter from December 1923 where they discussed these questions so it's very likely that this manuscript here was written shortly after that correspondence probably in the first half of 1924 Ramsey was obviously in the best position to know right besides Vidkun Stein well maybe he knew better than the ensign how this was supposed to work okay so we believe that the historical evidence here settles the question of what was supposed to be the correct interpretation now of course as I mention at the beginning of the idea was the apse we would eliminate the Equality sign in a way that doesn't cut down on the expressive power of receding logics well does the weekly exclusive reading or in fact any of the other readings accomplish that which turns out that in a sense yes they do I'm not going to dwell on this for too long since I've written about that elsewhere but here's a quick theorem right for every resilient formula well think for every formula or the logic with identity there is a formula an identity free formula that has exactly the same models under the weekly exclusive reading as the original formulas under at our skin reading provided that all the variable assignments we consider are one one on the free variables that occur in the formula as long corollary right if we look at sentences closed formulas formulas contain no free variables well we have a full or translate ability result we can translate every resilient sentence into an identity free sentence that has exactly the same models as the original resilient sentence and so since we all know right that individual constants are not essential you know in some logical sense this really is a pretty strong result it's not it's not explosive so so it's something polynomial right as a second corollary right if we by the way I'm sort of fudging the distinction between names and free variables here a little bit that's basically for purposes of exposition right I'm thinking of names as free variables under a fixed variable assignment so if we look only at languages that don't contain Co referring names then the original theorem gives a full interpretability result right for such languages resilient logic can be fully translated into the weakly exclusive reading in fact any of the other exclusive readings as well the actual translations Sai here will of course be different depending on which exclusive reading we're looking at for website is w/e true in marching you and site has some complicated structure then of course you will have to start referring to to its substrate lateral refers to assignments those assignments have to be objective as well well yeah that's true but they will automatically become injective because of the quantifier clauses right the because for all X Phi means for every object other than those that the assignment assigns to free variables right and so the yes exactly right I'm I'm sort of putting this off until the very end where I'm incorporating both conventions into one system right but that's the basic idea so if you start with a sentence any variable assignment that comes into play is automatically going to be injective okay now these results in a sense right can indeed be taken to be a justification for a bit consigns claim at five five three three the identity sign is therefore not an essential constituent of logical notation hmm well for you so if we play with the semantics of the quantifiers a little bit we can make it go away okay ah so much for identity I hope I'm not running too much overtime but you guys are used to that okay let's talk about the N operator briefly again right NF sidebar is this joint negation of all the values of oxide we've seen that the interesting thing here is that when victims time talks about the N operator the arguments to the operator always have this generic form side bar it doesn't exactly say something like well if we have this function FX then n of FX bar equals not exists X FX right and that's significant he's actually been misread as meaning that which really cannot be the case but we'll talk about that for a moment what does sidebar and closing parentheses mean well he actually does say something about that so there's a long remark at five five zero one I'm not going to read through all of this but the gist is that we can write a sidebar in parentheses when SCI is a variable that has propositions as values to denote basically the set of all the values of that propositional variable alright if clients three values PQ R then we can write the set like this basically in standard set notation but somehow we must of course say which values we must indicate which values this propositional variable has and that can be done in at least three ways and this is where it gets interesting the first way is easy we've already seen the example if we can just enumerate the values of the propositional variables if they're finally many we just write them down as a list right he has already done that up here but we can also give a function FX basically so that's a propositional function right so that's basically an open sentence containing the free variable X we can specify a propositional function FX whose values for all values of X are the propositions to be described and then there's a third way that's actually very intriguing what I'm not going to talk about it so I'm not even going to read it although pestering Soviets although we might have occasion to talk about it afterwards but I don't know okay so how can we put this together into a reasonable logical notation well as I said right here because a myself doesn't tell us how this notation is supposed to work concretely but really the the correct way of doing this is not is hidden not very far below the surface of five five zero one except that one has to look a little bit into the intellectual environment of the practice in particular of the tradition of pre capilla so there is a notational policy stipulated very early in Principia that pertains to writing expressions for functions that's where the famous circumflex comes in all right this is just one of the places where they talk about it when we wish to speak of the propositional function corresponding to the open sentence X is hurt we shall write X hat is hurt thus X hat is heard is the propositional function and X is hurt is an ambiguous value of that function so what is an ambiguous value well it's something like an open sentence of which we don't quite know which value has been assigned to the free variables now the note 8 the hat notation is of course terrible it's terrible essentially because it doesn't include a scope indicator and we'll talk about that in a moment - so really you should think of X hat is heard as lambda X X is hurt now the authors Wikipedia themselves frequently violate this convention it's often not important right in particular well right way back up so if this policy is enforced Vidkun Stein's should strictly speaking have written we can describe the values of propositional variable by giving a function FX hat whose values for all values of x are the propositions to be described now it's not really necessary at this point to follow the convention because he tells us that it's the function FX he's talking about he also doesn't need to indicate which values the independent variable and which are parameters because there is only one variable displayed all right so there's no actual need to follow the policy his teacher is cavalier about following the policy the circumflex as I mentioned before interestingly doesn't occur anywhere in the truck tardes although curiously it occurs in one place in the proto-truk tardes namely where I showed you that victim signs victim son rights F and then the parentheses around an empty space the product or tells actually has a circumflex there now the the sort of the publication history of the Travis is so complicated that I wouldn't be surprised if there is actually a typographical reason for the missing circumflex but I'm not sure about that if well okay maybe back up again who was Wittgenstein writing for right he was writing for Russell and for Freya and for the for people like Rammstein very smart people who knew how logical notation was supposed to work well if we look at Ramsey in particular as I mentioned right when he wrote the review of the truck TARDIS in 1923 he hadn't even ever spoken to Vicky Stein he gives the following explanation of the end notation now this - pffft of Khaybar is just an in felicitous alternative notation for NM sidebar that victim son and bat abolishes a little just shortly after introducing it in favor of the end notation so don't worry about that but Ramsey says write da da this notation where side bar is the set of of FX hat here's the Hat right is what is written ordinarily as not exist XFX now vitka stein doesn't even sorry Ranchi doesn't even think it's necessary to explain that he's adding the circumflex or something was just obvious to him that that's what this should really mean right there should be a half okay so why am i making such a big deal about the Hat well because the Hat is one ingredient that is one ingredient in the notation that we think is implicit in the indirect Alice for a notation of course the Hat alone doesn't work right as I said before it doesn't include a scope indicator so we can't just write something like n of n of f of xny hat because we don't know whether the first end binds the X or the Y well that is precisely where the horizontal bar comes in the horizontal bar is the missing scope indicator so why do I like this so much because it actually explains why the horizontal bar is there right I don't know of any other interpretation in the literature that makes sense or in fact even mentions the presence of the horizontal bar alright Fogel in in particular simply ignores the horizontal bar writes right not exist XFX simply as n of FX and then wonders why you can't express multiple a heterogeneous multiple quantification okay so once we have both an indication of the bound variable and of its scope we have a completely adequate quantifier notation so just to give you some examples here I'm just going to play them all at once right and I'm omitting parentheses for readability you see how beautifully this actually works well let's look a let's look at a very simple example all right how do we read a formula like this well we start with FX Y right then we indicate that Y is actually a bound variable we look to the next horizontal bar above its hat that's the scope of that bound variable we look for the first n operator that occurs precisely before the beginning of that horizontal line that's the quantifier that goes with a hat yes exactly exactly yes so we're looking at the set of all propositions FX a for all possible values a and we're negating that and then in the proposition so obtained we're making the the X vary the the X variable all right Justin exactly no not at all I mean there's a there's also no joint occurrence of the Hat and the horizontal bar okay so that's all of that is reckon structured what there is is basically the horizontal bar and we claim the awareness that an astute reader write steeped in the per capita tradition would know that you indicate a function by using the circumflex over its independent variable is the only do you think so this is mine well let me let me try a nuanced answer to this question in principle I think that yes this was exactly what he had in mind one thing that makes me say this is the discussion out somewhere in the 404s where he considers various ways of expressing the symbolism of generality like attaching an index to the bound variable and he says well that's obviously not going to work because we don't know how far the scope of the bound variable extends so he was very conscious of the problems or the challenges involved in establishing a notation for generality so in a salt so I think the correct answer to your question really is yes I'm not sure whether you know this was like a brief flash of brilliance that then again disappeared in a bit of a muddle but he knew that somehow with the horizontal bar would work all right I mean I don't think it's contentious if I say that while vipin Stein was a well sometimes Willian philosopher of logic he was not a logician of anything of the class of fragran I mean the technical details simply didn't have that at his command as much but he did see things and I think this is one of the things he actually did see and then uh-huh so that's the propositional use of n right that's the first way of describing propositional variables so let's let me start with the first example so we have a list of one proposition Phi yes exactly exactly okay are we good all right so let's put it together so what's the language of the of trick camera and logic going to look like well we're going to build atomic formulas as usual from let's say variables and predicate symbols of course now the equality sign is not among the predicate symbols right otherwise it works as usual then we have the the propositional use of the N operator so basically it occurs here as a variable a disappea noir and then we have the quantificational news I'm here reverting to a notation that was at some time proposed by Geach in response to Fogell ins challenge right that the notation wasn't expressively adequate I'm just doing that for well for ease of typesetting also for ease of reading so NX Phi basically means for all X naught Phi ok so we regenerate the formulas inductively in this way we can define free and bound variables in the obvious way and then we can define what it means for a formula to be true in a structure under variable assignment oh the obvious thing for atomic formulas the obvious thing for the propositional applications of n and then as I said an x5 is basically for all X naught Phi except that we now also have to incorporate the identity convention according to which all the objects mentioned within the scope of bound variable are ineligible for the range of that variable so all right that means for all X naught fire really means for every a in the domain except for those assigned by the variable assignment to variables that occur free in the formula all right and so for all those values a famously falls under this variable assignment a close look at the state variables at the same value disable yes yes um well I mean that's uh that really just gives you more flexibility right and I mean you could restrict this to a variable assignments that are one one on the free variables I'm really mostly interested in sentences and for sentences or injectivity of the variable assignments is going to be forced by the state ways to shoot their values be destined yeah um condition three is going to ensure injectivity for sentences what if you want to include it in one that's fine too right if you want to say we're only going ever to consider a variable assignment so that one one on the free variables that actually occur that's fine that would correspond to Vidkun Stein's claim that in a proper logical language you never have two names for the same object right so yes in a in a properly tracked Aryan spirit that should be required yeah here so any either not fine not some yeah oh my god have a good I get that wrong yeah that's quite possible propositional logic is always difficult so so this is yeah yeah so there's a negation this thing right yeah yeah sorry about that Internet yes question few years so with the weak interpretation we give this is a very nice compositional semantics could you do that very strong good very good you can do it but you can't you can't evaluate just a formula you have to evaluate pairs of formulas and sets of free variables I'd say you have to keep track of the variables that occurred earlier quantified all right so that's basically what happens Tom yes absolutely maybe you do just by saying the attractive area valuation Sigma is what which is objective and then just the clauses that go through would just use those restrictions you could do that but of course right you would always you would you would have to you would have to mention the actual formula for which the assignment is bacterial because it's not required that the assignment is 1:1 on all free variables it just has to be 1:1 on the verbs that actually occur free in the formula all right I mean for there's a trivial reason for not wanting the apps if you have a finite domain but you have infinitely many free variables you don't want to require injectivity because nothing can be satisfied okay so that's the syntax and semantics of bacteria and logic we're being first-order here right as I announced in the time of my talk I'm I haven't really talked much about the question of first-order maybe we'll talk about that in the discussion period a little bit okay and right this is not difficult to see then we have the same kind of translate ability result from first-order logic with identity to this criterion language as long as variable assignments are 1 1 on the free variables right we get we can embed first or logic with identity into this logic and we have the corresponding corollaries that I showed you earlier for just for the case of identity okay so can we actually build some kind of proof system for this logic well it's obvious by now that we can in a sense because clearly we can I haven't showed you that but we can we can clearly translate the Const in e'en formulas into first or logic that identity as well and since we have calculi for that right that's going to translate back but we can actually we can do this directly in a rather prettier way there's a complication because Vidkun Stein's idea of logical truth isn't the one we think of we use today right so following tarski for us validity is truth in all non-empty structures for Vidkun stein apparently right there was there was one fixed universe of objects so what could vary is the configuration of these objects so right different relations might obtain between the objects and different models but the domain is fixed once and for all so if that's the case then validity should be truth in all structures over this one domain now unfortunately the cosine doesn't tell us what the size of the domain is right or at least it's not at all clear from the truck tallest there is any kind of decision on that that means it's a little well it's a little difficult to actually say what the validities of the logic are going to be because right if if the domain is infinite then every sentence of the form there exists X 1 up to there exists X N and then some tautology is going to be a logical truth but if the domain is finite then most of those are going to be contradictions so we have to actually look at these options separately right and when I speak of three options that really means truth and all non empty structures and in tar scheme stands truth over a fixed infinite domain and then truth over a fixed domain of finite cardinality em that really makes it infinitely many options but ok I'm going to pass over the propositional or rules in the to blow calculus right that's pretty much obvious let's talk about the negated quantificational and instantiation rule which that is that's going to be common to all these calculi now negated M that's really existential quantification right n nx5 means for all X not 5 if you negate that it's not for all X not 5 so that's an essential instantiation rule and it looks rather different from what we're used to all right and standard to blow calculi we see an existential formula on a branch and then we just append an instance of the quantified form your with a fresh parameter right the idea being of course that if it turns out that one of the objects we were already talking about is the existential witness well we'll just assign the fresh parameter the same value in the castilian system we can't do that because distinct parameters are going to have distinct values so we actually have to distinguish cases here right we have to say well if there's this an X such that Phi it's either one of the objects we've already talked about on this branch all right so we're going to open up a branch a separate branch for all of those possibilities or it's an object we haven't yet talked about and in a final branch we're going to introduce a fresh parameter now of course again because we have the identity convention we can't necessarily use all parameters on the branch we have to exclude those that occur within the scope of the quantifier all right but that's this is actually nice you can do this obviously for for classical logic as well and what happens if you do to blow calculus like this is the calculus will automatically find finite models if the formula you're starting the tableau with has it right that's not guaranteed in the with the standard existential instantiation rule if you think of the typical example for all X there goes to Y F X Y that generates one infinite branch right with the standard rule this guy is going to show you the 1 element model in the classical case in fact it's always going to produce a model of minimal finite cardinality if there is such a model so ok nice and then of course you close a branch if it contains both a formulators negation let's see I'm I'm not going to talk about all these options let's just take this one right suppose the domain is infinite then how does the basically Universal instantiation wall go well it goes in just the way you would expect we append an arbitrary instance to the bottom of the branch provided that the parameter used to form the instance isn't it doesn't occur within the scope of the quantifier because in that case it's not it's not within the range its value is not within the range of the quantifier right that's again the weakly exclusive constraint here I'm going to contrast that really quickly with the case of variable domains where we have to be a lot more careful we can't just append an arbitrary instance of Phi really not Phi right because the model might be finite and if we choose a new parameter satisfiability of the formulas of the branch in the original model might be lost because the model doesn't have enough object to interpret all the parameters right if the model has n objects we've already used n parameters then we can't introduce a new one without destroying satisfied ability in that model because we can't assign it an object and so so we can append only instances for parameters that we've already introduced on the branch except in the one case where we haven't yet introduced any in that case we would just pick one because we know that models are never empty so we can do that once but then that's it okay so let me jump over the fixed final domain case and just come to the conclusion so it's been suggested I'm talking very impersonally here although you're going to see that all the quotes I'm giving you are from one person it's been suggested in the literature that the sparseness of notational detail in the Tractatus is evidence for the view that smooth running system of pure logic holds no interest for Vidkun style that victim signs remarks in fact graphically limit the usual ways in which quantification theory is presented and that he has no interest in presenting a systematic way of deriving patterns of quantificational structure from other patterns of connotation quantificational structure we take ourselves to have shown that none of this is true right in fact the Tractatus with a little bit of digging in its intellectual context contains a smooth-running ideologically minimal expressively adequate system of logic that as a matter of fact seems like an interesting alternative for doing first row logic without a ID in the first place thank you

Channel: Philosophy Overdose

Views: 42,263

Rating: 4.8916254 out of 5

Keywords: Philosophy, Wittgenstein, Tractatus, Logic, First-Order Logic, Predicate Logic, Predicate Calculus, Analytic Philosophy, Formal Logic, Philosophy of Logic, Ludwig Wittgenstein, Early Wittgenstein, Higher-Order Logic, Philosophical Logic, Tractatus Logico-Philosophicus, Mathematical Philosophy, First Order Logic, Formalism, Ideal Language, Philosophy of Mathematics, Logical Atomism, Picture Theory of Meaning, Classical Logic, Logical Positivism, Symbolic Logic

Id: e75U1fgBC6g

Length: 59min 14sec (3554 seconds)

Published: Sun Jan 27 2013