# The Logic of Wittgenstein&#39;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

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