Griceian Quantification: The Implicatures

Speranza

A running commentary on:
http://southalabama.edu/philosophy/loomis/Loomis/About_Me_files/theoria2.pdf

with courtesy to Mr. Loomis.

-----

Wittgenstein's "Tractatus Logico-Philosophicus" carefully distinguished the concept "all" (cited by Grice in "Logic and Conversation") from the notion of a truth-function, and thereby from the quantifiers.

Loomis argues that Wittgenstein's rationale for this distinction is lost unless propositional functions are understood within the context of his picture theory of the proposition.

Using a model Tractatus language, Loomis shows how there are two distinct forms of generality implicit in quantified Tractatus propositions.

Although the explanation given in the Tractatus for this distinction is ultimately flawed, the distinction itself is a genuine one, and the forms of generality that Wittgenstein indicated can be seen in the quantified sentences of contemporary logic.

In 1919, not long after he had given Russell a copy of the Tractatus, Wittgenstein wrote to Russell,


Dear Bertie,
   ---- I suppose you didn't understand the way how I separate in the old notation of
generality what is in it truth-function and what is purely generality.
---- Let me explain:
--- a general proposition is a truth-FUNCTION of all propositions of a certain form.
         Love,
                Your Austrian engineer.

The separation of truth-function from pure generality was clearly important to Wittgenstein.

He had expressed it in the Tractatus at 5.521:

I separate the concept "all" from the truth-function.

If Russell hadn't seen this separation in the notation, the oversight is understandable, not
least because the "old notation" that Wittgenstein was using was Russell's own.

Indeed, Loomis argues that the fact that the notation was not perspicuous in the way that Wittgenstein
thought it was reflects a tension in the Tractatus' account of generality.

Nonetheless, Wittgenstein's treatment of general propositions ("all S are P", or, in better grammar, "Every S is P") did give expression to a distinction among types of generality within quantified propositions that Loomis claims is justified.

Loomis examines how Wittgenstein derives propositional functions, truth-functions, and the general propositions formed from them, within the context of a simplified model language, based on a proposal of John Canfield.

Loomis's intention is to make perspicuous how Wittgenstein regarded propositional functions and quantified formulae as emerging from actual, used propositions of a language, and how his doing so
enables us to see distinct forms of generality in quantified propositions.

Making this case requires showing how Tractarian propositional functions and quantified formulae are importantly different from the propositional functions and formulae of System G (Grice's System).

Grice's conception of propositional functions one has in mind is one traceable to the work of Hilbert and Ackermann.

According to it, a propositional function originates as an uninterpreted syntactical object, and it is specified by first classing signs according types, such as sentential variables and predicate variables, and then stipulating rules for the construction of formulas from the typed signs.

Applying this conception of propositional functions to the Tractatus distorts Wittgenstein's approach.

 For example, consider how Wittgenstein's assertion:

“A function cannot be its own argument."

This fares when we understood the functional sign as a syntactical object along broadly Hilbert-Ackermann lines.

The function sign itself does not declare what can and cannot complete it to make a sentence.

If the function sign is thought of strictly as a sign and not as a sentence form (i.e., an implicitly stated formation rule), then the 'x' in

f1(x)

is merely a place marker, showing that some other sign must be placed there in order to make
a sentence.

Wittgenstein’s restriction on the arguments taken by functions is used by him to block Russell’s Paradox.

But on some reading of function signs as first and foremost syntactical objects, nothing blocks

f (f(x))

from being a sentence, contrary to Wittgenstein's assertion.

And this is just the conclusion one may draw in claiming that, Wittgenstein is led to believe, mistakenly, that the type rule, which is a formation rule, follows from the type distinction [the division of signs into classes based upon their syntactical shape] itself.

This objection presupposes a way of regarding propositional function expressions in the Tractatus that is not uncommon.

However, that the mistake that some claim to find in Wittgenstein is an artifact generated by reading the contemporary notion of a propositional function into the Tractatus.

Wittgenstein has well-grounded reasons for imposing his constraints on function signs, and these appear when we understand how function signs are propositional variables formed from
significant propositions.

But we must see Wittgenstein's ideas about propositional functions as emerging from framework different from the model-theoretic one.


If Wittgenstein's notion of a propositional function was not that proposed by
Hilbert and Ackermann, what was it? To answer this, it is worth first looking at Russell's
notion of a propositional function in the Principia Mathematica, and seeing what
Wittgenstein did -- and didn't -- accept in it. In the Principia, Russell described
propositional functions as expressions containing a variable that become a proposition
upon that variable's being given a fixed determined meaning (ibid, 14). They are
expressed by the carat notation 'ˆx ', which distinguishes the propositional function "ˆx is
4 In Loomis 2005, I look at a variety of other attempts, from Carnap to the present, to read the Tractatus
through the lens of contemporary logic.
5 Several recent Tractatus commentators share my desire to resist reading the Tractatus through the prism
of the contemporary, model-theoretic conception of logic. My own understanding of the logic of the
Tractatus is especially indebted to Baker 1988; Varga von Kibed 1993; Ricketts 1996; Hylton 1997; and
Floyd 2002.
5
hurt" from the "ambiguous" open expression "x is hurt".6 Russell did not regard
propositional functions as a species of mathematical functions, but to the contrary took
propositional functions as "the fundamental kind of function from which the more usual
kinds of function, such as 'sin x' … are derived."7 Mathematical functions, which he
called "descriptive functions", were introduced separately. Descriptive functions
"describe a certain term by means of its relation to their argument. Thus 'sin π/2'
describes the number 1."8 Russellian propositional functions, on the other hand, are not
descriptions of terms but rather are compound, structured entities that share their
structure with the propositions that are their values. One can see in the values “Caesar is
hurt” and “Brutus is hurt” a common shared structure that shows them both as a value of
'ˆx is hurt’; a kinship clearly absent between the descriptive function 'sin π/2' and its value
1.9 For Russell, the propositional function expressed by 'ˆx is hurt' is thus not a bare, open
syntactical formula, but a structured compound formed from the previously given
propositions that serve as its values.10
The Tractatus follows this account of propositional functions in an important
respect. Wittgenstein says directly at 3.318 that he conceives of the proposition, as Frege
and Russell do, as a function of the expressions contained in it. And like Russell,
Wittgenstein regards his propositions not as names for objects, but as complexes
6 Russell and Whitehead 1960, 15. Russell thought that 'x is hurt' is an "ambiguous value" of the function
'ˆx is hurt'.
7 Russell and Whitehead 1960, 15. Hylton has shown that the priority propositional functions over other
types of functions is also apparent also from the PM definition of non-propositional functions at *20.01. Cf.
Hylton 1993, 342, and 1990, 264ff.
8 Russell and Whitehead 1960, 232.
9 Russell goes so far as to say that a propositional function is "more complex that its constituents";
meaning by "constituents" the propositions that constitute its values (Russell and Whitehead 1960, 6).
10 Russell was aware of this feature: "[T]he values of a [propositional] function are presupposed by the
function, not vice versa" (Russell and Whitehead 1960, 39). It is because "ˆx is hurt" becomes a proposition
when x is given any fixed meaning that it is a propositional function.
6
consisting of elements combined in a definite way (3.14). This is essential to his picture
theory. In the case of elementary propositions, a proposition's being a complex of
elements is required for its being a picture of a possible state of affairs, which is itself a
complex of elements (2.0272). As with Russell, Wittgenstein's propositions are thus
structured compounds, and this informs his conception of propositional functions.
Wittgenstein's propositional functions are introduced as Satzvariablen –
“propositional variables”.11 At 3.313, Wittgenstein indicates how a Satzvariable is
formed, by taking any part of an elementary proposition that contributes to the
proposition's sense and changing that part into a variable. The result of this change is "a
class of propositions which are all the values of the resulting variable proposition"
(3.315). As with Russell, Wittgenstein regards the value of the Satzvariable to be
determined by the propositions that are its possible values, or as he puts it, by "indicating
the propositions whose common mark the variable is" (3.317). And, as with Russell's
propositional functions, the Satzvariable shares a form with these propositions by
presupposing all of the propositions in which it can occur (3.311).
Behind this similarity between Wittgenstein’s and Russell’s conceptions of
propositional functions there nonetheless lie two important differences. First, Russell
freely introduces negation, conjunction, and other truth functions as propositional
functions that take propositions as arguments.12 Wittgenstein rejects this, and indeed at
3.332 claims that propositional functions cannot take propositions as arguments at all.


Russell is committed to claiming that truth functions such as negation and conjunction characterize the sense of a proposition, for like all propositional functions,
these share a structure with the "aggregations of subordinate propositions" from which
they are formed.13 Thus for Russell, any two instances of that aggregation (values of the
function) have some structural commonality. Wittgenstein to the contrary flatly rejects
the supposition that truth-functions, which he calls “operations”, might characterize the
sense of a proposition (cf. 5.25).

Wittgenstein's rationale for these claims about propositional functions is carefully
grounded in his conception of the proposition, and unintelligible apart from it. This is
best seen through the analysis of elementary propositions and the propositional functions
formed from them, and the non-elementary and general propositions formed from the
elementary ones, in the context of a simple model akin to one proposed by Canfield
(1972). The model consists of a world with two different color objects, primary green,
named by the symbol “g”, primary blue, named by “b”, and four points of a miniature
field, named “p1”, “p2”, “p3” and “p4”. The points are arranged as follows:
p1 … p2
p3 … p4

Concatenating a color name with a point forms an elementary proposition. Thus, “gp1”
says that primary green is at p1. I shall call the model language used to describe the field
“L0”.
A qualification is necessary here. Elementary propositions such as “gp1” in L0
describe the model world, but they should not be understood as reports of what is visible.



This qualification is made to avoid placing the model's elementary sentences at odds with
Wittgenstein's 6.3751 remark that, "the assertion that a point in the visual field
["Gesichtsfeld"] has two different colors at the same time, is a contradiction" (my
emphasis). Now, it is essential to elementary propositions in the Tractatus that they be
independent (cf. 4.211), as are the elementary facts they represent. Hence “gp1 & bp1”
must be a consistent proposition stating that both primary green and primary blue are at
p1. There is no tension here with claim made at 6.3751, however, provided that we
observe a distinction between two colors being combined at a point, on the one hand, and
two colors being co-exemplified by a point, on the other. The latter would occur if a point
were visibly green and visibly blue at a single time. The former would occur if we
combined green and blue paint at a point. At 6.3751, Wittgenstein excludes only coexemplification
as logically impossible. He does not exclude the possibility that the
combination of colors is both possible and expressible as a logical product.14 As such,
the sentence 'gp1 & bp1' in the model language L0 can be understood as a significant
proposition expressing the combination, but not the co-exemplification, of two colors at a
point. It should thus be understood as part of a possible analysis of visible colors, and not
as itself a description of what is visible.


14 Indeed, that Wittgenstein had countenanced the combination of two colors at a point is clear from his
"Remarks on Logical Form", in which he says that he had assumed in the Tractatus that a complete
analysis would demonstrate the impossibility of two colors appearing together at a point in the visual field
by showing how statements of differences of color would analyze into conjunctions of elementary
propositions. Each such proposition would express different degrees of brightness or shade, and the
analysis would expose "some sort of contradiction" in the joint assertion of two colors at a point in the
visual field; cf. Wittgenstein 1929, 168. As this proposed analysis reveals, Wittgenstein clearly did not
intend the Tractatus to exclude the combination of colors at a point, and indeed presupposed the possibility
of such a combination, such as that elementary propositions expressing different units of brightness b' and
b'' could be combined, and the combination expressed as a logical product (ibid). Infamously, Wittgenstein
acknowledged in the same essay that the combination of two colors cannot be finally analyzed as a logical
product, contrary to what Tractatus had assumed. This problem, however, is a defect intrinsic to the
Tractatus itself, and is not a feature imposed by the model language L0.


In L0, the expression 'gp1' is a Tractatus symbol.

It is a sign, consisting of
perceptible marks, coupled with a significant use, namely, the use specified by the
elucidations that I have given above for “g”, “p1”, and their concatenation.15 The signs
“g”, “p1”, and their concatenation are also Tractatus expressions. Expressions are
everything essential for the sense of a proposition that propositions can have in common
with one another (3.31). “gp1” can have something in common with, for instance, “gp2”,
“bp1”, and “bp2”. We understand this commonality from the elucidations, and grasping it
is a necessary condition of understanding these propositions. Put otherwise, not seeing
that “gp1” has something in common with expressions like “gp2” and “bp1” entails not
understanding “gp1”. Similarly, not to see that that the expression “g” is a color name and
so something that can be at “p2”, and that “b” is a color name and so something that can
be at “p1”, is to fail to understand “g” and “b” as expressions. As an expression, “g”
"presupposes the forms of all propositions in which it can occur. It is the common
characteristic mark of a class of propositions" (3.311).

The class of elementary propositions for which “g” is a common characteristic
mark in L0 are “gp1”, “gp2”, “gp3”, and “gp4”. Following 3.312, we can represent the
form of this class by a variable, “gy”. Here gy's values are the propositions that contain
“g” (cf. 3.313). Likewise, we can replace “g” in “gp1” with a variable to determine
another class of propositions thus: “xp1”. Following Wittgenstein's instruction at 3.315,
we can further form the variable expression “xy”, which has as its values all of the colorpoint
propositions. We can still further represent gp1 by means of the variable “r” (cf.,
4.24). In “xy”, we grasp what the substitution instances are for the variables, and in doing
so, that they must be distinct. The four variable expressions: “gy”, “xp1”, “xy”, and “r”,
are examples of Wittgenstein's Satzvariablen.

Satzvariablen expose that an elementary
proposition is a function of its names by showing us what elements are expressions, that
is, are essential for the sense of the proposition, and what propositions have in common
with one another (cf. 3.31, 4.24).

These Satzvariablen are constructed from meaningful propositions, as we see
when we construct them according to Wittgenstein's instructions in the 3.3s. It would be
wrong to say that Wittgenstein requires that every Satzvariable be so constructed, for at
5.501 he indicates that the description of the variables for a proposition can be given by:
(1) “direct enumeration”, (2) “giving a function fx, whose values for all values of x are
the propositions to be described”, and (3) ”giving a formal law, according to which these
propositions are constructed.” The Satzvariablen formed from sentences of L0 are of the
second form. Such Satzvariablen are unlike the propositional functions common in
contemporary logic, for they are not formed by first giving independently-specified
syntactical schemata, such as “xy” or “Fx”, and then subsequently assigning an
"interpretation" that specifies the possible values such schemata might take. The
difference is highlighted by Wittgenstein’s assertion that, "The rules of logical syntax
must follow of themselves, if we only know how every single sign signifies" (3.334), and
by his dismissal of the possibility of enumerating logical forms a priori as “arbitrary”
(5.554-1).

In L0, we can observe that the symbol “gp1” is also a fact, namely the fact that “g”
left-flanks “p1”. Seeing “gp1” as a fact, and not simply as an object, a compound name, or
a cluster of names, is essential to seeing it as saying that primary green is at p1 (cf.
3.1432). Here the left-flanking relation has to be noticed in order for the elementary
proposition to describe a state of affairs. There is, of course, no necessity that the sign “g”
left flanks “p1” in order to describe this state of affairs. L0 involves arbitrary agreement,
as we see when we consider that in a different language L1, with a different logical
syntax, we might have expressed that there is primary green at p1 by saying “p1g”. In L1,
the Satzvariable xy must assume different values for its constituent variables, which, per
3.316, are thereby different variables. As Wittgenstein tells us at 3.315, “r” can function
as a Satzvariable for a proposition of either L0 or L1, since in r all arbitrary determination,
including the conventions governing the concatenation of expressions, is removed. Yet r
still determines a class of propositions.

Both “gp1” in L0 and “p1g” in L1 thus share something, which Wittgenstein calls
the "form of representation" (2.17). This form is what any picturing fact must have in
common with what it represents, namely the constraints on possible configurations of
colors and points (cf. 2.15). Unlike the languages of everyday life, whose ambiguity
allows for errors (3.323), the simple languages L0 and L1 do not allow for the expression
of what is not possible. For instance, in neither L0 nor L1 is it possible to say anything
illogical like "green is green at p1". This impossibility is embedded in the logical syntax
of each language, and manifests in the elucidations of the primitive signs and the
concatenation relation. And while this constraint on possible formations is common to
both L0 and L1, it is not stated by either one of them but shown. One does not understand
the possible facts of the model world if one does not see that a color of a color of a point
is not among them. Likewise, one does not grasp either “gp1” or “p1g” as depicting the
fact that primary green is at p1 unless one also excludes “ggp1” or “p1gg” as nonsense.

The claim that an elementary proposition like “gp1” is a fact is connected with the
claim that it shares something in common with what it represents. As with a picture, the
concatenation of names in an elementary proposition must mirror the concatenation of
objects in a possible fact (Sachverhalt), such that facts are depicted by facts (cf. 2.13-5).
Moreover, the elementary proposition must be articulate in exactly those places where the
represented fact is articulated; it must possess "the same logical (mathematical)
multiplicity" (4.04f.). The names in an elementary proposition, which "go proxy for"
(vertreten) their objects in sentences, must have logico-syntactically admissible
concatenations in a sentence which mirror the combinatorial possibilities of the objects
named. Only thus does a proposition represent (darstellen).

There is thus an internal reciprocity between a proposition's sense and the fact it
represents.

Which fact is indicated by a proposition is, of course, determined by what the
proposition says. Yet equally, that the proposition says what it does is determined by the
possibilities intrinsic to the fact it represents. We cannot, for instance, have a proposition
which asserts what is impossible, for Wittgenstein makes it a necessary condition of
something's being a proposition that it have truth-possibilities corresponding to the
possibilities of the existence or non-existence of possible facts (cf. 4.25, 4.3). This
condition is justified by his notion of picturing. A picture of reality must have some
conditions of agreement with the world that may or may not obtain (cf. 4.462). A picture
that agrees with the world in every case (or no case) is not a proposition but a tautology
or a contradiction. By 4.2, the agreement or disagreement of a proposition with these
possibilities is the proposition's sense.

So a sentence to which there corresponds no possible fact is eo ipso senseless and not a proposition.

And a sentence in which any sign exhibits combinatorial possibilities that do not correspond to combinatorial possibilities had by the object for which the sign goes proxy corresponds to no possible fact.

Consider this in the context of the attempt to say something illogical by substituting propositions as values of variables in Satzvariablen. Russellian propositional functions require type-theoretic restrictions on such substitutions such that one cannot predicate a first-order function of a propositional function of type 1, for instance.

Wittgenstein regards such restrictions as unnecessary.

We see at once in L0 that "xy" cannot be a value for "y" in "xy".

The pseudo-symbol “x(xy)” does not predicate a colour of a colour of a point.

A coloured point, such as gp1, is a constellation of objects – a fact.

There is no issue of predicating a colour of a fact.

Likewise, the significant symbol “gp1” is a fact.

There is no issue of a fact being chained together with an
object, like the name “g”, to form another fact. Indeed, no fact can right-stand any
substitution instance of “x” in ‘xy”, because no fact can right-stand anything. There is no
symbol “x(xy)” possible in L0 or its extensions that is a configuration of objects such as is
required for picturing (cf. 2.031-15).19 Here it is important to distinguish sign from
symbol (cf. 3.32- .326).

There is no significant use for the perceptible marks (the sign)
“x(xy)” in L0 by which it is a significant fact (a symbol). This point generalizes beyond
L0; no propositional function can take another propositional function as its argument on
pain of its ceasing to be a chaining of objects (names) into a fact.

These considerations form the grounds for Wittgenstein's rejection of type theory
at 3.332:

No proposition can say anything about itself, because the propositional sign
cannot be contained in itself (that is the whole “theory of types”).

Wittgenstein calls a property "internal" if it is unthinkable that its object not possess it
(4.123). The representational form of a significant Tractatus expression is an internal
property in this sense, since it is intrinsic to that expression's making a contribution to a
proposition as a picture of a state of affairs. In L0's world, the internal property of colors,
which requires that they be predicated of points and not of facts, shows itself by means of
the internal property of the propositions of L0, which requires that no fact stands to the
right of an object. This is the mirroring of internal properties by internal properties that
Wittgenstein demands (cf. 4.124). Here the syntactical constraints on expressions are not
something specified in terms of criteria of sentence composition that might ignore the
application of those expressions.
I think that this is behind Wittgenstein's insistence at 4.126 that "formal concepts
cannot, like proper concepts, be presented by a function." Rather, formal concepts, or
concepts of internal properties, are signified by Satzvariablen (4.127, 4.1272).
“Function”, for example, signifies a formal concept. A sentence of the form "f is a
function" is not an expression about a mathematical or logical object in the way in which
one of the form "x is an even number" is. The sentence "f is a function", even in the
common, mathematical sense of “function”, presupposes the application of Satzvariablen
such as “f(x) = y”. The sentence "f is a function" does no more than elucidate the role of
“f” in such an application. Contravening Wittgenstein's requirement at 4.1272 that we
represent functions as variables and not functions would require attempting to understand
"f is a function" as itself a function “g(f)”. Once this move is made, special restrictions are
required to prevent the application of g to itself to yield “g(g)”, and thereby Russell's
paradox. But from the perspective of Wittgenstein's picture theory such restrictions are
unnecessary because functions are presented as variables, not as functions. The variable
“f(x)” expresses a form with its restrictions built in, for "the functional sign already
contains the prototype of its own argument and it cannot contain itself" (3.333).
Tractatus Operations and Compound Propositions
As I noted above, Wittgenstein and Russell regarded propositional functions as
constructed from propositions, and as distinct from “descriptive functions” such as
mathematical functions. Wittgenstein assigned a fundamental importance to this
difference. He expressed it by distinguishing between operations, which exhibit the
features of mathematical functions, and propositional functions (5.25). Wittgenstein's use
of the word "truth-function" in the Tractatus, while carefully explained by him, invites
misunderstanding if it is understood in the contemporary sense. Wittgenstein's truth
functions are neither Fregean functions, nor functions of names as elementary
propositions are. Rather, truth-functions are the results of operations, as Wittgenstein
makes clear at 5.234: "The truth-functions of elementary propositions are results of
operations which have the elementary propositions as bases" (cf. also 5.3).
Wittgenstein defines an operation at 5.23 as "that which must happen to a
proposition in order to make another out of it." Operations generate propositions from
other propositions by being "the expression of a relation between the structures of [the
proposition's] result and its bases" (5.22). However this generation occurs, it must obey
Wittgenstein's dicta that an operation does not characterize either the sense of a
proposition (5.25), or its form (5.241), but rather only indicates differences between
forms of propositions (5.24). Furthermore, Wittgenstein's general form of the proposition
requires that every proposition be the result of some one truth operation on elementary
propositions (5.3). The one truth-operation is joint denial, which Wittgenstein indicates
by the operation sign “N”. The application of the N-operator to a single elementary
proposition p returns its negation ~p. The application of N to two propositions p, q
returns their joint denial, ~p & ~q (5.51). Wittgenstein thus requires that conjunction and
negation be operations (cf. 5.2341). These constraints must be met by elementary
propositions, since they are the bases of operations. What features of elementary
propositions allow operations to satisfy these constraints? The answer is disarmingly
simple.
Recall that for Wittgenstein elementary propositions are independent of one
another.20 This independence insures that the conjunction of any two elementary
propositions can be treated truth-functionally. Independence requires that there be no
logical import internal to the structure of an elementary proposition, for if there were then
whether one elementary proposition is true might follow from (or contradict) another's
being true – a possibility that Wittgenstein denies (cf. 4.211, 5.134). Since they are
independent, the joint assertion of any two elementary propositions p, q is equivalent to
their logical product. Conjunction is thus intrinsic to the structure of elementary
propositions as logically independent pictures.


Wittgenstein also regards negation as intrinsic to the structure of elementary propositions.

The sense of a proposition is its agreement or disagreement with reality, that is, its bipolarity.

The possibility of denial is, as Wittgenstein says, pre-judged in the affirmation of a proposition (5.44).

It is pre-judged
because there is no possibility of picturing unless there are conditions of agreement and
disagreement with the world.21 To be able to say that a proposition “p” is true or false,
we must be able to call “p” true (cf. 4.063). Negation and conjunction are thus operations
internal to the picturing function of elementary propositions.
This account of how Wittgenstein's N-operator is built-up from the structural
features of elementary propositions reveals how "an operation shows itself in a variable"
(5.24). The variable required to show the operation N is the Satzvariable “p” (or “r”,
etc.). No further logical multiplicity is required once it is understood that this
Satzvariable is only formed from a genuine proposition with the internal properties of
bipolarity and independence that are required for the N-operator.
Given that every proposition is the result of the one truth-operation N applied to
elementary propositions, how is the picture theory to be extended to non-elementary
propositions? This has seemed to some to be less than clear. Michael Kremer, for
instance, has objected that on this account it is "not obvious that a conjunction of pictures
is also a picture." 22 Kremer asks us to consider an example in which we:
"conjoin" a picture of Tom standing to the left of Paul and a picture of Tom
standing to the right of Mary by drawing Tom standing between Paul and Mary.
Now suppose that we "negate" a picture by literally "using it in an opposite
sense," by turning it upside down. How can we conjoin the denials of the two
simple pictures in our example? If we "merge" the upside-down pictures of Tom
standing to the left of Paul and Tom standing to the right of Mary, we get an
upside-down picture of Tom standing in between Paul and Mary, which is the
denial of the conjunction of the two pictures, rather than the conjunction of their
denials.

Kremer's objection rests on his example, which is intended to show the
implausibility of regarding the operators as arising simply from the independence and
bipolarity of elementary propositions. Kremer thinks that "merging" the negation of two
simple (propositional) pictures of:

(A) Tom is to the left of Paul

and

(B) Tom is to the right of Mary

This gives a new picture,

(C) Mary is between Tom and Paul.

This apparent consequence is generated by Kremer's suggestion that we negate
the pictures by inverting them.

Kremer is correct that inversion can be used to accomplish the logical operation of negation, in which case inverting the pictures and the operator sign “~” are the same Tractatus symbol.

However, we must be careful to recognize that in this use, the negation of a picture by inversion is not the same as the assertion of the inverted picture.

Negating picture (A) by inverting it says that Tom does not stand to the left of Paul.

It does not say that Paul stands to the left of Tom, despite the fact that the inverted picture might, in a different form of use, be made to depict this.

Likewise, the negation by inversion of the two pictures (A) and (B) is not equivalent to the assertion of
what is depicted when the pictures are held upside-down and read as if they were
themselves pictures of a new state of affairs.

Yet this is precisely what Kremer must do to
get picture (C); it is only on the assumption that the negation of (A) puts Paul to the left
of Tom, and the negation of (B) puts Mary to the right of Tom, that Kremer is able to
claim that the conjunction of the negation of (A) and (B) yields (C).
Wittgenstein clearly indicates that truth-tables are propositional signs (4.44,
4.442). They are not definitions of logical operators over bare syntactical objects but
rather are, like all propositional signs, facts that stand in a projective relation to the world.
The truth-table for the conjunction of p and q is thus itself a propositional picture,
complete with representational form, truth-poles, and so on. Its negation is clearly distinct
from the conjunction of the negation of p and the negation of q. But regarding truth-tables
as propositional signs for non-elementary propositions prohibits any account by which
non-elementary propositions can be formed by the merging or melting together of the
elements of elementary propositions. Wittgenstein can use only the operator N for the
construction of non-elementary propositions. Non-elementary propositions are therefore
not formed by chaining together names, as elementary propositions are, but rather by
using operations to expose the internal relations among propositions. This presupposes
that in non-elementary propositions, the names of complexes or of the relations among
complexes do not "go proxy for" (vertreten) new objects and relations, but instead
disappear on analysis.
Consider this in the case of the complexes Tom and Paul, and the left-of relation
between them, as expressed by the statement "Tom is to the left of Paul".24 Following
Wittgenstein's instruction at 2.0201, the propositions expressing their relations "can be
resolved into a statement about their constituents and into the propositions that describe
the complexes completely". The statement in the analysis will be a truth-function
(product of an operation) of elementary propositions, each of which will in turn be a
concatenation of names of simple objects (4.22f.).
To see how this can work, suppose we extend the model world of L0 to include
complexes and relations among them.25 The extension consists of a field of six points,
arranged as follows:

p1 … p2 … p3
p4 … p5 … p6

There are also two complex objects.

One is denoted by the name “Tom”, and consists of two vertically-arranged contiguous green points in the field.

For simplicity, assume that
diagonally opposite points are not contiguous, and no color complexes overlap a point.
Another complex is denoted by the name “Paul”, and consists of two vertically arranged
contiguous blue points. Let L2 be an extension of L0 that describes this world, and that
includes the additional names “Tom” and “Paul”, as well as the relation-sign “left of”,
and the logical operations of conjunction and negation (“.” and “~”). “Tom left of Paul”
is a propositional sign in L2 which states that two contiguous green points stand to the left
of two contiguous blue points, as for instance when p1 and p4 are green, and p2 and p5 are
blue.

Following 2.0201,

“Tom left of Paul”

analyzes into two conjuncts, each of which
decomposes into elementary propositions. The first conjunct, α, is a statement about the
constituents of the complexes mentioned in “Tom left of Paul” which gives us a complete
characterization of those complexes and how they could be arranged:
α

The following example expands on Canfield's "blau/grün" example in his 1972, 351f. I have expanded
the field from that given for L0 in order to avoid the trivializing result of having only one possible leftstanding
relation for complexes.

[(gp1 & ~bp1 & gp4 & ~bp4) v (gp2 & ~bp2 & gp5 & ~bp5) v (gp3 & ~bp3 & gp6 &
~bp6)] & [(bp1 & ~gp1 & bp4 & ~gp4) v (bp2 & ~gp2 & bp5 & ~gp5) v (bp3 & ~gp3
& bp6 & ~gp6)]
Since complexes cannot overlap, the complete characterization of the complexes
mentioned in “Tom left of Paul” requires the explicit exclusion of blue and green at a
point.
“Tom left of Paul” has a second conjunct, β, which "describes the complexes
completely". It is:
β
[(gp1 & ~bp1) & (gp4 & ~bp4) & (bp2 & ~gp2) & (bp5 & ~gp5)] v [(gp2 & ~bp2) &
(gp5 & ~bp5) & (bp3 & ~gp3) & (bp6 & ~gp6)] v [(gp1 & ~bp1) & (gp4 & ~bp4) &
(bp3 & ~gp3) & (bp6 & ~gp6)].
The three disjuncts in β describe the three possibilities in which Tom is left of Paul. For
example, the first disjunct places Tom at points p1 and p4, and Paul at points p2 and p5.
The "indeterminateness" shown by βs having three disjuncts is an inherent feature of any
proposition in which an element signifies a complex (cf. 3.24).
Upon analysis, we see that there is nothing for which “left of” goes proxy in
“Tom left of Paul”. “Left of” in L2 is not a Vertreter. While “left of” makes a contribution
to the sense of any proposition in L2 in which it appears, its doing so does not require that
we go beyond the operations of conjunction and negation and the internal properties of
the simples named by “b”, “g”, “p1”, and so on.26



Satzvariablen in the Tractatus are general.

They range over the class of their substitution instances, which class is not given independently of a particular Satzvariable, but is rather determined by the possible substitutions within the meaningful proposition
from which the variable is formed. Quantificational generality has as a necessary
condition the generality carried by Satzvariablen. Wittgenstein expresses this by stating
that it is peculiar to a symbolism of generality that it refer to a logical prototype (5.522),
and that the generality symbol occurs as an argument (5.523). We see this, for example,
in the Satzvariable “xp1”, in which generality is expressed by the variable (argument
position) “x”. This variable refers to a logical prototype, namely all of the propositions of
this form. Nothing about this kind of generality requires the truth-function, as
Wittgenstein clearly sees (cf. 5.521).


Quantificational generality introduces the truth-function, which is operator N, on
top of the generality of Satzvariablen.

Consider for instance the Satzvariable “xp1” in L0.
Let “ξ” denote all of its values, viz., “bp1” and “gp1”. Then following 5.52, N(ξ) =
~(∃x)xp1. "N(ξ)" is, as noted above, the application of the joint-negation operation to the
values of ξ, viz., “~bp1 & ~gp1”. The range of the values of ξ is set by the Satzvariable,
which itself requires apprehending the application of elementary propositions in L0 to
assert facts about its world. Here we can see how "all logical operations are already
contained in the elementary proposition" (5.47). A grasp both of the range of the variable
“xp1’ required for the construction of a logical product, and of the joint negation operator
N are already implicit in understanding the proposition “gp1” from which “xp1” is formed.

Consider next the non-elementary sentence

“Tom left of Paul” in L2.

From this
proposition we can form the Satzvariablen “w left of Paul”, and “w left of z”. As with the
Satzvariablen constructed from “gp1”, the range of the variables is constrained by the
sense of the propositions from which the Satzvariable “w left of z” is constructed. The
substitution instances for “w” and “z” are restricted to complexes, but there is no need to
mention this restriction because there is no genuine elementary proposition, such as “gp1
left of bp2”, from which the Satzvariable could be formed.

In the Satzvariable “w left of Paul”, the values are given by a list in which every
complex is mentioned that might stand to the left of Paul in the model world (it would
include, among other things, β). Here two possible cases must be distinguished: one in
which a name like “Paul” signifies a type that may be multiply instantiated, and another
in which it signifies an individual particular. As I introduced it, “Paul” is functioning as
the name of a type. Following Wittgenstein's assertion that the Law of the Identity of
Indiscernibles is at most contingently true (5.5302), the set of possibilities indicated by
“w left of Paul” must also include the three possible situations in which one Paul,
understood as a type, stands to the left of another, numerically distinct Paul.


The denial of the Identity of Indiscernibles alters how we understand a given
propositional function because it widens the possible substitution instances for variables
in propositional functions formed from non-elementary propositions. However, I do not
think that its denial is essential to the Tractatus' account of quantification. If, contrary to
5.5302, we wish to name numerically distinct individuals that cannot be multiply
instantiated, then the analysis reveals that no numerically distinct complex can possibly
stand to the left of itself, for any such complex reduces to a collection of simples whose
internal properties insure different substitution instances for the variable positions in “w
left of z”. For instance, if we denote by “Paul1” any vertically arranged Paul configuration
occupying point p1, so that Paul1 = bp1& bp4, then Paul1 is never left of Paul1. To see this,
suppose for a reductio that “Paul1 left of Paul1” is a proposition. Then on its analysis we
are met with a contradiction, since in attempting to give complete characterization of this
sentence akin to the α component of “Tom left of Paul”, we would at once be met with
the sentence, "(bp1 & ~bp1 & bp4 & ~bp4)" (similarly with all of the other disjuncts in α).

“Paul1 left of Paul1” thus fails the bipolarity condition on propositionhood.

As a
consequence of the fact that “Paul1 left of Paul1” is a pseudo-proposition, the Satzvariable
formed from “Paul1 left of z” is different from that formed from “Paul left of z”. The
former prohibits the substitution of Paul1 for “z”, while the latter does not.
We can once again represent the values of the Satzvariable “w left of Paul” by
“ξ”, and apply the operator N to it. Doing so gives us the joint denial of these values.
Unlike functions, operations can take their own results as arguments (5.251), so applying
N again to this result gives us N(N(ξ)), which is the logical sum of the Satzvariable’s
values. For simplicity, we can represent the logical sum of a set of arguments with “Σ”,
and the logical product of a set of arguments with “Π”.28 Then “Σ(w w left of Paul)”
states that there is something to the left of Paul (the subscripted “w” indicates the scope
of the “Σ” operator in front of it). From this proposition we can form another Satzvariable
thus: “Σ(w w left of z)”. Here “z” ranges over any object that can left-stand any
substitution instance of “w”. We can take the logical product of these z-values, and

represent them as: “Π(z Σ(w w left of z))”, which in modified Russellian notation
corresponds to “(z)(∃w)(w left of z)”. Similar constructions allow for all the other
quantification possibilities, including all mixed quantifications.29
Quantifier notations such as “Π(z Σ(w w left of z))”, or equivalents formed with
the N-operator, appear to require conventions governing the variable-names within the
scope of the operators. This is required by Wittgenstein's elimination of the identity sign,
and by his insistence that identity of object be expressed by identity of sign (5.53). Yet
Wittgenstein did not tell us how these restrictions might appear.30 Why not? Because, I
suggest, he believed that the construction of Satzvariablen from genuine propositions
would make the relevant restrictions clear, and that that attempting to state these
29 My analysis of this sentence here follows Ricketts (ibid), and Varga von Kibéd,'s 1992. As von Kibéd
illustrates in detail, R. Fogelin's claim (in Fogelin 1982), that there is a "fundamental error" in the Tractatus
which prohibits the construction of certain mix-quantifier formulae such as “(z)(∃w)(w left of z)”, is
incorrect and ignores the variability possible within Wittgenstein's Satzvariablen. Against Fogelin, von
Kibéd shows that we can negate such formulas as faa, fba, fac using operator N, and further construct the
joint negation of sets of these sentences (von Kibéd 1992, 89f; cf. also Jacquette 2001). Thus, we can apply
N to sets of formulas like: {faa, fab, fac… }, {fba, fbb, fbc, …}, {fca, fcb, fcc, …}, …, to yield: {~faa & ~
fab & ~fac & …., ~fba & ~ fbb & ~fbc, …. , ~fca & ~ fcb & ~fcc, …, …} . Applying N to this set in turn
yields: (faa v fab v fac, v …) & (fba v fbb v fbc v ….) & (fca v fcb v ….) & …. This is equivalent in the
Tractatus to (z)(∃x)fzx, which is precisely one of the formulas that Fogelin denies can be constructed.
In the case of L2, a modification would be necessary to insure that sentences like “(z)(∃w)(w left of
z)” aren’t necessarily false in virtue of there being no points to the left of p1 and p4. One such modification
would involve understanding the points as forming a loop, so that p3 and p6 would appear to the left of p1
and p4. With such a modification, “(z)(∃w)(w left of z)” would be analyzable along the same lines that von
Kibéd has proposed, with the w and z variables ranging over possible cases of complexes like Tom and
Paul left-standing one another.
30 For some possible interpretations of how the required restrictions might appear, see Hintikka 1956, 228-
9, and Floyd 2002, 324-7. Ricketts proposes the following restrictions for his 'Σ' and 'Π' notation that I have
used above:
(1) Variable-names with embedded scopes must be distinct.
(2) If one variable lies within the scope of a second, then the first variable cannot simultaneously
take as a value the same name that the second does.
(3) No variable may take as a value any name within its scope.
Cf. Ricketts, "Generality and Logical Segmentation in Frege and Wittgenstein", 26.
26
conditions by means of a genuine Tractatus propositions would be neither possible nor
necessary. We see, without being told, what the Satzvariable ranges over.31
The Limits of Showing
In his post-Tractatus philosophical career, Wittgenstein readily employed the
notion of a linguistic rule as something wholly or partially constitutive of the meaning of
an expression. Yet the notion of a rule plays no real role in the Tractatus. There are
indeed brief references to "rules of logical syntax", but the upshot is that these are
consequent upon, rather than constitutive of, the proper functioning of signs.32 Prior to
writing the Tractatus, Wittgenstein surely understood the possibility of attempting to
frame rules for a language such as Russell's type restrictions. His considered response
was that such rules are useless pseudo-propositions. By the same token, the sentences of
the Tractatus itself are not formulations of rules. They are, as Wittgenstein tells us at
6.54, "elucidations" (erläuterungen). Their function is in one sense akin to the
elucidations given above for expressions such as “gp1” in L0. Elucidations are not a
matter of laying down rules governing the application of signs, because propositional
symbols like “gp1” do not appear as such until their use in picturing is grasped (compare
3.263). This grasping requires understanding what is pictured by “gp1”. Yet once “gp1” is
understood, any attempt to formulate rules for its use is rendered redundant. The
31 As Floyd has pointed out, this leads to important differences between Wittgenstein and the contemporary
understanding of propositional functions like “(x)(y)(fxy & fyx) ⊃ (x)fxx”. For Wittgenstein this is not a
truth of logic but is a significant proposition telling us something about the relation f. Cf. Floyd 2002, 325-
6.
32 Cf. 3.334, quoted above. Compare also 3.325 and 6.124, which convey the sense that the rules of logical
syntax are determined by the "nature of the essentially necessary signs" of a language.
27
sentences of the Tractatus are equally redundant for any one who has grasped the proper
use of the expressions of their language.
There is a tension here, however. As we have seen, many of Wittgenstein's
Satzvariablen emerge from the use of propositions as pictures by "giving a function fx,
whose values for all values of x are the propositions to be described" (5.501). The
function in this case is a propositional function, a Satzvariable formed from actual, used
propositions. How is this model to be applied to a proposition such as, "All men die
before they are 200 years old"? This proposition involves a generalization over
collections that are neither surveyable in the way that L0's world is, nor constructible by
means of formal series. This proposition therefore must be the logical product of all
values of the Satzvariable "x died before he was 200 years old", where this in turn is
formed from statements such as "Socrates died before he was 200 years old", "Plato died
before he was 200 years old", and so on. Yet the totality of the possible values of this
variable are nothing that we can plausibly be said to see, as we can be said to see all the
possible values of "xp1" in L0. Wittgenstein must say that (as he later put it), "though its
terms aren't enumerated here, they are capable of being enumerated (from the dictionary
and the grammar of language)".33
Wittgenstein indirectly gave expression to this idea in his Tractatus claim that the
world can be completely described by completely general propositions without
coordinating any name with a definite object (5.526). Such a set of propositions would
delimit all and only the possible states of affairs of the world (5.5262). But without
knowing all the terms of the logical sums and products that the Tractatus holds that such
propositions are equivalent with, our understanding of such a set of propositions is left a
mystery.

It is no use to promise here that we could provide an enumeration of the relevant
instances were we to perform a complete analysis. For Wittgenstein's whole account of
the Satzvariablen and the truth-operations formed with them gets its grip from genuine
propositions used to describe possible facts.
It is possible that Wittgenstein was misled by the use of simple, finite examples.
In his "Criticism of my former view of generality", he reported that:
it is correct that (∃x).ϕx behaves in some ways like a logical sum and (x).ϕx like a
product; indeed for one use of words "all" and "some" my old explanation is
correct, -- for instance for "all the primary colours occur in this picture" or "all the
notes of the C major scale occur in this theme". But for cases like "all men die
before they are 200 years old" my explanation is not correct.34
In his Notebooks of 1914, Wittgenstein provided a small example illustrating the
claim, later made at Tractatus 5.526, that the world could be described by completely
general propositions without coordinating signs with names. He first described the world
as one that "consisted of the things A and B and the property F, and that F(A) were the
case and not F(B)" and then described it again by means of the general propositions:
“(∃x,y).(∃φ).x≠y.φx.~φy:φu.φz.⊃u,z . u = z”, “(∃φ).(ψ).ψ = φ”, and “(∃x,y).(z).z = x v z =
y.”35 One here sees that the general propositions describe the world given the prior
description of it as consisting of A, B, F, and so on. Given that we have, as it were, a
God's-eye view that allows us to see before us all of the objects, along with all of their
combinatorial possibilities, we assent to the general descriptions as complete, because we
see how to construct the Satzvariablen and thereby form the required logical sums and


products. The general propositions then delimit the range of possibility only because we
see from the initial description the range of Satzvariablen like "φx". From such a
perspective, it is indeed possible to describe the world without coordinating any name
with an object, as Wittgenstein says. We might then imagine that the case of ordinary
language must be similar, and thus that while we cannot survey all of the values of the
Satzvariable beforehand, some such totality of values must nonetheless be present
(compare 5.5562).
For anyone in the world however, logic must, as Wittgenstein put it, "have contact
with its application" (5.557). This "contact" in the context of general propositions
includes the specification of the actual propositions from which the Satzvariable required
for the quantified proposition is constructed. I think that Wittgenstein's use of Russell's
notation for generality conceals this; nothing about the Principia's use of the quantifiers
seems to require of us that we be able to survey the possible substitution instances of a
variable beforehand. Our understanding of “(x).φx” seems to be exhausted by saying,
with Russell, that it denotes "all values of φx". But by the Tractatus' lights it is not so
exhausted; we must know how the Satzvariable “φx” was formed before we can
understand what is indicated by "all values".
Generality and Logical Form
That Wittgenstein's conception of generality cannot be fully squared with the
existence of general propositions for which we cannot provide the required analysis does
not mean that his conception has nothing to offer us. For there are different senses of
generality in logical formulas, and the differences are akin to those Wittgenstein was
30
trying to elucidate with his separation of the generality of logical form from
quantificational generality.36
Consider a quantified formula of contemporary model-theoretic logic, such as
“(∀x)(Fx⊃Fx)”. It is general in two ways. On the one hand, there is the general
applicability of the formula to a collection of independently specifiable instances. On the
other hand, there is the generality of the form of a logical construction, which is
expressed by arbitrary instances or schematic variables. In the first type of generality, the
formula “(∀x)(Fx⊃Fx)” is general by being generally applicable to the objects of its
domain, which are specifiable independently of the formula itself. This is the type of
generality that expressed by the quantifier “(∀x)”. 37 The generality of the form of the
logical construction on the other hand, is not expressed but is rather shown by its being a
form of possible formulae that all of its instances, such as:
(*) Fa⊃Fa, Fb⊃Fb, Fc⊃Fc, etc.,
have in common. This form is not expressed by the quantifier but rather is presupposed
by the entire quantified formula, as we see when we compare the following two
sentences:
(1) All the equations (*) have the form Fx⊃Fx.
(2) All the equations (*) are validities.
Statement (1) is vacuous in a way that I think is akin to the way that Wittgenstein had
thought the sentences of the Tractatus were. The vacuity of (1) lies in the fact that one
36 The following paragraph is inspired by Sören Stenlund's analysis of the equation "a + b = b + a", and the
forms of generality contained in it, in Stenlund 1990, 158-9.
37 I'm here assuming an "objectual" interpretation of the quantifier. On a substitutional interpretation, the
quantifier expresses that all of the substitution instances of the formula are valid. In this case, all of the
substitution instances must be understood independently of what the quantifier says, similarly to what I say
here about the objectual case.
31
cannot understand what is meant by the expression “All the equations (*)” without
understanding that sentence (1) is true. On the other hand, recognizing the validity of all
the equations (*) is not a precondition of understanding what is meant by “All the
equations (*)” in statement (2). It would make sense, for instance, to ask of someone that
they prove the truth of (2) by showing that “(∀x)(Fx⊃Fx)” holds in all models. But there
could be no such proof of the truth of (1); to understand the “etc.” in (*) is to be able to
produce or recognize equations of the appropriate form. Understanding the generality of
the form of ‘(∀x)(Fx⊃Fx)” presupposes a recognition of (*) as a partial list of its
instances. This generality of the form is not stated by the quantifier, but is instead a
condition of its general applicability.
Wittgenstein's Tractatus articulated this distinction and brought it to the fore. By
separating the concept all from the quantifier, the Tractatus worked to elucidate the fact
that apprehending a quantified sentence presupposes apprehending a form of use of
expressions that is already general. Wittgenstein's observation of the distinction was to
survive the demise of the Tractatus project, although it would have to find a new
rationale.38 Finding that rationale would involve a return to the notion of elucidations,
and would lead Wittgenstein to the idea that, contrary to what the Tractatus had
maintained, explanations, such as ostensive definitions, are in fact meaningful rules for
the use of expressions.39
38 Consider for instance Wittgenstein's later claim that the infinity of a number series is given by the rules
for a number system rather than by classes in his 1975, 161f., his continued dissatisfaction with the Frege-
Russell notation for generality when applied to ordinary language (cf. 1978, 265-7; 1979b, 227-9), or his
claim that a proposition about all propositions or functions is "a priori an impossibility: what such a
proposition is intended to express would have to be shown by an induction" (cf. 1975, 150).
39 Thus for instance, the ostensive definition of something comes to be treated as "a rule for translating
from a gesture language into a word language." Wittgenstein 1978, 88.
32


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