p
q
T
F
KRITERION
Nr. 17 (2003), pp. 1-6
Contentless Syntax, Ineffable
Semantics, and Transcendental
Ontology. Reflections on
Wittgenstein’s Tractatus
Arkadiusz Chrudzimski
p
q
T
F
q
T
F
Abstract
1
Syntax does not represent
One of the most influential ideas of Wittgenstein’s
Tractatus is that the logical constants do not represent anything. Logical constants combine sentences truth-functionally, so that if we are to construct a model for a language L, we only need to
assign the semantical values to the atomic formulas of L. The semantic values of the molecular
formulas will be generated automatically, according to the well known functions that could be represented as truth-value tables. (5, 5.101) For the
conjunction ‘p ∧ q’ we obtain the following table:
I should like to thank the Austrian Foundation for the
Promotion of Scientific Research (FWF) for the financial
support.
∗
T
F
F
F
T
F
T
T
T
F
And for the material implication ‘p ⊃ q’ we have:
p
Wittgenstein’s Tractatus [6] contains some
very striking theses. We read, e.g., that “in
a sense” we could not be wrong in logic, and
that the whole subject matter of the theory
of modalities could be reconstructed on the
ground of the insights in the mechanism of
the linguistic reference. Yet in the light of
the last sentences of Tractatus the whole semantics turns out to be principally ineffable.
In our paper we will try to clarify these matters. We show how these theses could be
made plausible in the context of the transcendental method of Wittgenstein’s Tractatus.
F
For the disjunction ‘p ∨ q’ the table looks like this:
∗
Universities of Zielona Góra, Poland
and Salzburg, Austria
T
T
F
T
F
T
T
Accordingly, we can say that the logical forms of
composition, represented by ‘∧’, ‘∨’, ‘⊃’ and, for
that matter, also by the one-place conjunctive ‘¬’,
do not represent. They need no separate semantical correlates over and above the semantical correlates of the flanking sentences. Consequently, if we
want to formulate a semantical truth-definition à
la Tarski [5], we only need truth-makers for atomic
formulas.
All that is well known. Note, however, that
nearly the same procedure could be applied to the
under-atomic logical forms. Consider the simplest
syntactical structure ‘F a’. Its semantical value depends exclusively on the question, whether or not
the semantical value correlated with the name ‘a’
belongs to the set which is the semantical value of
the predicate ‘F ’. We can thus construct the analogical (possibly infinite) table for the predicationform:
a
F
∅
{x}
{y}
{z}
{x, y}
{x, z}
..
.
x
y
z
...
F
T
F
F
T
T
F
F
T
F
T
F
F
F
F
T
F
T
...
...
...
...
...
...
KRITERION, Nr. 17 (2003), pp. 1-6
Indeed, Wittgenstein’s view was not just that
the logical constants do not represent. His idea
was that the entire logical syntax does not represent. And Tarski’s concept of satisfaction explores
in point of fact, the same idea.
If we accept all that, we have ipso facto something like the principle of extensionality. If the
semantic values of compound sentences are functions of the semantic values of their constituents
and the semantic values of atomic sentences are
functions of the semantic values of their ultimate
(sub-sentential) constituents, then it is clear that,
for any constituent expression ‘x’ involved in the
compound sentence ‘A’, we can substitute any
other expression ‘y’ without any change of the
semantic value of ‘A’, providing that ‘x’ has the
same semantic value as ‘y’. To have the standard
principle of extensionality we need only to assume,
that the semantic values in question are the same
entities that are commonly called extensions. And
so far, as we have seen, they were extensions.
2
We have two unknown values. Whatever is actual, is possible, and whatever is not actual, is
not necessary, but not all non-actual is impossible,
and not all actual is necessary. To extensionalize
modal contexts we need some more fine-grained
semantic values than the Fregean the True and
the False.
The standard procedure is to move to the ontology of possible worlds. The propositions are
treated no longer as entities that simply have
truth-values. They become functions from possible worlds to truth-values. In the extensional semantics operating in the frame of one world, sentences could be construed as 0-ary predicates (i.e.
as 0-place functions). In the frame of the possibleworlds semantics, they become 1-ary predicates
(i.e. 1-place functions from the possible worlds to
the truth values).1
The world-theoretic explanation of modalities
is then as follows: The sentence ‘p’ is true simpliciter, if ‘p’ is true at the actual world (i.e. at the
“distinguished” world w ∗ ). The sentence ‘p’ is possibly true, if there is at least one possible world, at
which ‘p’ is true, and the sentence ‘p’ is necessarily true, if ‘p’ is true at all possible worlds. 2 The
modal discourse has been thus “extensionalized”.
The box and the diamond have been replaced by
the quantifying over the possible worlds.
Modalities
But we know very well that the principle of extensionality holds only for a very rudimental part of
our language. Consider modal contexts, ‘It is possible/necessary that . . .’. If we try to construct the
truth-functional tables for the modal operators,
we see that the truth-values of modal sentences are
The modal
The “extensionalized”
not always determined by the mere truth-values of
discourse
modal discourse
their constituents and therefore are not functions
p
p(w∗)
of the truth-values of their constituents. Syntac3p
∃w[p(w)]
tically, the box (2) and the diamond (3) behave
2p
∀w[p(w)]
like the negation (¬). They are sentential operators building sentences out of sentences. But while
Not all philosophers, however, are happy with
for the negation we have the well known simple ta- the abundant ontology of possible worlds. 3 And
ble:
Wittgenstein has shown, how such an ontology
could be simulated. The possible worlds need not
p ¬p
T
F
F
T
1
Every n-ary predicate becomes analogically n + 1-ary.
Alternatively, propositions could be construed not as
functions from possible worlds to the truth-values but as
sets of possible worlds at which they are true.
3
It is far from clear what the possible worlds are. Carnap
construes worlds linguistically as “state descriptions” [1, cf.
p. 9]. For Chisholm and Plantinga worlds are “abstract
entities” (maximal states of affairs) [2, cf. p. 43] and [4,
cf. p. 44f]. David Lewis [3] proposes to treat worlds as
concrete individual entities.
2
for the box and diamond we obtain positions at
which the truth values are indeterminate:
p
T
F
3p
T
?
2p
?
F
2
Chrudzimski, A.: Contentless Syntax, Ineffable Semantics, and Transcendental Ontology.
be construed as primitive individuals of the sys- illustrated by the following simple picture:
tem, as in the orthodox possible worlds semantics.
If we had a set of basic building blocks of the universe and a set of rules specifying how these blocks
could be combined, we could develop a combinatorial theory of possibilities to the effect that possible is, as it were, all that could be “constructed
from the given material”. In effect Wittgenstein
proposes such a theory. The world is composed
of simple objects which prescribe all possible configurations in which they could appear. (2.011,
2.012)
3
Semantics and ontology
But why should all that be true? What guarantee
could we have that the combinatorial ontology of
possibility that we could formulate on the ground
of our analysis of our world, is really valid. To
mention just a few problems: Why could there be
no “alien” objects that exist in some possible, but
not in the actual world? How could we be sure
that we know all the ways in which the objects
could be combined etc.?
The answer lies in the philosophical method of
the Tractatus. The ontological theses formulated
in this work are for the most part deduced from
semantical considerations. The way the world is,
appears thus, in a sense, as a consequence of the
way the world is represented. In this sense the
ontology of Tractatus is “transcendental” in the
Kantian sense. Wittgenstein assumes that our
thoughts refer, and seeks the “conditions of possibility” of this semantic relation. He claims that
such a reference is only possible, if the thought
has a structure that is able to “mirror” a structure of the (fragment of the) world. (2.151) But
the very “mirroring” assumes that it must be “in
advance” some correlation between the elements
of the structure of the thought and the elements
of the structure of the world. The main device of
the tractarian semantics is thus an a priori correlation between the logical form of the language
and the ontological form of the world that is explained by the metaphor of the correlation of the
syntactically simple names with the ontologically
simple objects. (2.1514, 2.1515)
The ontology and semantics of Tractatus can be
The world consists of facts that are configurations of ontologically simple objects (A, B, C, D,
E, F). Conscious subjects think of the world by
means of sentences that are configurations of syntactically simple names. Sentences are meaningful because of the semantical relation correlating
every simple name with an exactly one simple object. The sentence is true, if the configuration of
the involved names corresponds to the configuration of the objects in the world (cf. the left side
of our diagram). Otherwise the sentence is false
(cf. the right side of our diagram). (2.222) The
simple objects are called the logical form of the
world, the simple names – the logical form of the
language.
A moment’s reflection shows, however, that the
Wittgensteinian concepts of a name and an object
are technical concepts that have very little to do
with their commonsensical ancestors. As we have
said, Wittgenstein suggests that the set of the ontologically simple objects prescribe somehow all
possible configurations in which the objects could
appear and by means of that, generates the set of
possible worlds. Indeed, this is the crucial feature
of Wittgensteinian objects. They are just a label for “something” that generates the set of possible worlds, whatever the nature of this “something” might be. The talk of “simple objects” is
just a picture. So, Wittgenstein is not persuading
us that there are really Wittgensteinian objects –
a kind of Democritean atoms of which the world
consists. Instead, he gives us a picture that is in3
KRITERION, Nr. 17 (2003), pp. 1-6
tended to help us to understand what the logical
form of the world could be like, but the metaphor
used should not be pressed behind its intended
meaning. The logical form is just any thing (or
better any aspect of the world) that would generate all possible worlds in an analogue way, as the
Wittgensteinian objects would generate Wittgensteinian worlds, if there really were such Wittgensteinian objects and worlds at all.
If we remember that the entire conception of the
Wittgensteinian objects in point of fact flows from
the semantical considerations – in fact, from the
assumption that the language refers at all – and if
we remember that the aspect of the language that
is a priori correlated with the objects are names,
that in an analogue way generate all that could
be said in a given language, we see that we had
better think of Wittgenstein’s names and objects
as something resembling rather a grammar than
a set of building blocks.
Wittgenstein’s thesis is that the logical form of
the language is somehow a priori correlated with
the logical form of the world, and we understand
why he claims this, if we remember that the logical
form of the language is its logical syntax, and that
this syntax does not represent anything. Syntax
is essentially something that could not be false,
because it is not semantically correlated with any
correlate in the contingent world. Indeed, the logical syntax is perfectly contentless.
The metaphor by which Wittgenstein tries to
express this fact is the picture according to which
the syntax-aspect of the language (i.e. the set of
the simple names) is correlated with the eternal
and unchangeable form of reality (i.e. with the set
of simple objects), while the descriptive aspect of
the language (i.e. the configurations in which the
names could be placed) represent the contingent
content of the world (i.e. the way the simple objects are arranged in facts). (2.024, 2.0271) But
in point of fact the eternality and unchangeability
of the semantical correlate of the syntax, consists
in the fact that there is no such correlate. It is, as
it were, the eternality and unchangeability of the
nothingness.
We can now clearly see the structure of
Wittgenstein’s transcendental argumentation.
The point of departure is language by means
of which we represent the world.
The first
assumption is that such a language must have
something like a syntax.4 The syntax is further
claimed to be “contentless”. It does not represent
anything. Consequently whatever is true on the
ground of syntax alone, is necessarily true. The
last thesis is that such “syntactic truths” are the
only necessary truths.
4
Logical omniscience and intentionality
This idea of the “contentless” logical syntax makes
it understandable why Wittgenstein could claim
that “in a sense” we could not be wrong in
the logic. (5.473) The logical omniscience of a
Wittgensteinian subject consists precisely in the
fact that the logic is contentless (6.1, 6.11), and
so there is no point at which one could be wrong.
As long as a subject uses a given language, he
must respect its grammar. If he, e.g., uses the
conjunctive ‘∧’ in a way that does not conform to
the table for conjunction, then he is actually interpreting the sign ‘∧’ not as conjunction. Consequently, he is not speaking a language in which ‘∧’
means ‘and’. This is the sense in which the meaning of our words lie in their use. To speak of the
“logical omniscience” of the Wittgensteinian users
of language could be thus very misleading. Note
that, as there are no separate “logical objects”,
there is also no “logical knowledge” involved. To
be logically omniscient and infallible means therefore not “to know” something. It means merely to
speak a given language.
The thesis of the logical omniscience is the reason why in the frame of the tractarian philosophy
there are no problems of the ontology of intentionality in addition to the problems of the ontology of modalities. The Wittgensteinian analysis of
modalities automatically gives us a solution of the
problem of intentionality.
But first of all we ask, why it is commonly
believed that the intentional contexts pose such
additional problems. The reason is simple. If
we move from the modal structures (‘2 . . .’ and
‘3 . . .’) to the constructions involving the so called
4
It seems that this assumption is plausible for any finite
mind.
4
Chrudzimski, A.: Contentless Syntax, Ineffable Semantics, and Transcendental Ontology.
intentional operators, i.e., to the constructions
like ‘John believes that . . .’, ‘Mary thinks of . . .’,
‘Jack wonders, if . . .’ etc., we seem to have an
additional source of non-extensionality. The nonextensionality of the modal contexts consists in
the fact that we have to take into account not
only how the world actually is, but also how it
could be. And it is an initially plausible idea that
in the case of the intentional contexts we have to
take into account not only how the world actually
is, and not only how it could be, but also what the
subject in question thinks the world is and could
be. Accordingly, if we want to use a possible world
semantics for the intentional contexts, we will expect that the “doxastically possible” worlds will be
in general much more fine grained then the “alethically possible” worlds.
The intentional contexts are thus nonextensional and they seem to be essentially
“more subjective” than the modal contexts. The
question, whether two given constituents could
be substituted salva extensione, depends not
on the question what their semantic values are,
but instead on the question, what the involved
subject believes these semantical values are. It
seems therefore that the logical form of the
belief-context must involve at least an additional
reference to the subject in question. It does not
have the simple form: ‘B(p)’ where ‘B’ is an
intentional operator like ‘S believes that’ but
takes the form: ‘B(S, p)’ where ‘B’ stands for a
two-place operator ‘believes that’ and ‘S’ refers to
the believing subject.
Wittgenstein believed, however, that a philosophically analyzed language poses no such problems. The analysis revealing the deep logical
structure of a language must therefore lead to the
language that is fully logically transparent. In this
case we could omit the reference to the particular
subject and refer simply to the language. His next
important claim was that in point of fact, every
language functions according to its deep logical
structure. The functioning of a language is thus
logically transparent, whether or not the language
in question is actually analyzed, and to use a language is precisely to let the language in question
properly function. (3.262) This is the meaning of
the thesis that the mere using of a language im-
plies in a sense, a logical omniscience. A subject
who really uses the language L must thus conform to the logical syntax of L and consequently
he could not “think” non-logically. (3.03) Now it
is the logical syntax of L that prescribes what is
possible and, as this syntax must be transparent
for any subject using L, there could be no difference between (i) what is possible and (ii) what
the subject think is possible. There is thus no
need for an additional ontology of intentionality.
(i) and (ii) collapse.
5
An ineffable semantics
Transcendental argumentations are notoriously
difficult to express. The reason for this difficulty
is instead relatively easy to grasp. What we talk
or think about, are by definition objects of our
thought. But what a transcendental argumentation wants to concentrate on, is precisely the conditions of possibility of such objects as they emerge
from the way they are thought about. The conditions that were identified by Wittgenstein were:
(i) the logical form of the language, (ii) the ontological form of the world, and (iii) their a priori
correlation. It is thus no surprise that all Wittgenstein could give us, is a handful of metaphors like
“simple objects”, “the form” or “the substance”
of the world – the metaphors discredited explicitly
in the last sentences of the Tractatus. (6.54) The
conditions of possibility of a semantical reference
– the logical form of representation – could only
be “shown” (or “mirrored”), but not described.
(4.12) A sentence could be “about” the world, but
not “about” its own “aboutness”. Still this “being about” – the logical form of representation –
is “mirrored” in each use of a sentence. (4.121)
In point of fact, the whole semantics is, according to Wittgenstein, principally ineffable. And the
reason is easy to understand, when we think of
Tarski’s analysis of the concept of truth. According to Tarski the condition of the material adequacy of any theory of truth T T for language L is
that for every sentence ‘p’ of L the T T must imply
the so called “Tarski’s biconditional”:
(T) ‘p’ is true if and only if p.
But to the conditions of the formal correctness
5
KRITERION, Nr. 17 (2003), pp. 1-6
of T T it belongs that the possibility of expressing References
the paradoxical “liar sentence”:
[1] R. Carnap. Meaning and necessity. The Uni(*) The sentence (*) is not true
versity of Chicago Press, Chicago, 1960.
should be blocked.
[2] R. Chisholm. On metaphysics. University of
Tarski’s proposal is that the truth-predicate for
Minessota Press, Minneapolis, 1989.
the language L could not belong to the language L
itself. The biconditional (T) must be expressed in [3] D. Lewis. On the plurality of worlds. Basil
a language of higher order (in a meta-language)
Blackwell, Oxford, 1986.
which contains the names for all sentences of L
(that appear at the left side of the (T)) and the [4] A. Plantinga. The nature of necessity. Oxford
University Press, Oxford, 1974.
translations of all sentences of L (that appear at
the right side of the (T)). We consequently obtain
[5] A. Tarski. Pojecie prawdy w jezykach nauk
the following picture:
dedukcyjnych. In A. Tarski, editor, Logic, Semantics, Metamathematics. Papers from 1923
to 1938, pages 152–277. Clarendon Press,
1956. English version. Originally published
1933 in Warszawa.
[6] L. Wittgenstein.
Tractatus logicophilosophicus.
Routledge & Kegan Paul,
London, 1922.
We see that the semantical relations between an
object-language and its universe could be cognitively accessed, only through the language of the
higher order. But according to Wittgenstein there
is only one universal language. The semantical relations are consequently principally ineffable.
This is the next point that allows us to understand the special status of the “logical omniscience” of the Wittgensteinian subject. The logical form of the world and the language is a crucial
element of the mechanism of the intentional reference. We see now that this mechanism principally
could not be intentionally grasped. Now if there is
any object of the logic, it is this mechanism itself.
(Cf. 6.124) The thesis that the logic of the language L could not be described (by any sentence of
L) but only shown (in each use of a sentence of L)
becomes thereby much more understandable. And
we also seem to understand better why to have a
logical omniscience is simply to use a language.
6