What Wittgenstein tries to do in the
theses (2.01..) at least in part amounts to articulating the
foundations of what is now known as **model-theory**. This was
effectively started by Gödel, Tarski and Mal'cev in the 1930ies, and
became a specialism in mathematical logic in the 1950ies.

In this summary of the theses
(2.01..) I'll say something about

**1. What Wittgenstein was trying to do
**

2. The approach of model-theory

3. Some general theses about language and reality

**1. What Wittgenstein was trying to do**

What Wittgenstein was trying to do in
(2.01..) and indeed in a considerable part of the
Tractatus was to
give a better explanation for what logic is and does than was given by
Whitehead and Russell in the **Principia Mathematica** and probably also
than by Frege in **Grundgesetze der Arithmetik**.

The first was especially needed, for
close readers of the PM (Principia Mathematica) know that its
introduction is not clear about quite a number of things it should
have addresssed and explained in natural language - propositional
functions, variables, truth, the theory of types, classes, paradoxes -
than happens there.

It is not clear to me how much
Wittgenstein knew of Frege's writings, that were all published before
Russell's "**Principles of Mathematics**" and Whitehead and Russell's in
the **Principia Mathematica**, and that were in the 20th Century
discovered to be more clear and more useful than what Russell and
Whitehead published about the foundations of logic and the relation of
language and thought to reality and to mathematics.

It seems probable to me that in fact
W. had not read most of Frege's writings about semantics, and also
that he did not get far in the Grundgesetze der Arithmetik, that
indeed is written in a quite forbidding symbolism.

In any case, what W. was trying to do
can be outlined briefly as follows:

The PM, indeed following Frege for
the most part, had analysed statements a consisting of (i)
logical
terms, like "not", "and", "all" and "is equal to" (ii)
subject-terms,
like "Aristotle", "Greeks", and "Zeus" (iii)
predicate terms, like
"human", "loves", "is greater than", and had schematized this by
introducing abbreviating terms and
logical rules and
assumptions for
these and some related terms.

Most of the rest of natural language
and grammatical categories were abstracted from, which is to say that,
if they occured were forced into the above schematism - which in
Whitehead and Russell's use of it, as also in Frege's, had **the great
merit of articulating fairly to very precisely what the rules of
inference, statements and terms of the surrected formal systems are
and what is a proof for a statement in the language of the formal
theory**, and besides consisted of
axioms that were or seemed adequate
to state all or nearly all
mathematics in, in the sense that it
allowed the derivation of mathematics in the formal and logical
systems that Frege and Whitehead and Russell had created for that
purpose, namely to show that mathematics can be derived from
logic, in
the sense Frege and Whitehead and Russell believed they had codified
it formally in their systems.

As it happens, Frege's system was
shown to be inconsistent by Russell (but much later shown to be
repairable and then to do what Frege had claimed for his system - see
Boolos); Whitehead and Russell's system was shown to be much more
complicated than necessary (see
Ramsey); and in any case few
mathematicians or philosophers agreed with Frege, Russell and
Whitehead that mathematics was in fact nothing but logic.

The reason for the last disagreement
was mostly that both Frege and Russel and Whitehead had found it
necessary to assume at least one axiom that was **not** properly logical
in any clear sense at all, such as an axiom of
infinity: While it
seems as if mathematics needs the concept of infinity to make sense of
differential and integral calculus and of infinite series (such as
1/2+1/4+1/8+ ..+1/2^{n} i+ ...) it also seems
intuitively quite evident that the statement that there is an infinity
of things, however much it may be needed for mathematics, is not
really a **logical** assumption or axiom. And if it is not, then no
proof that most or all of mathematics can be restated in the terms of
the systems of Frege or Whitehead and Russell, which is true, and derived with the
help of its axioms and principles of inference, which is also true, can be a proof that
mathematics is nothing but
logic.

To return to what Wittgenstein was
trying to do:

To articulate clearly how statements
involving logical operators, subjects and predicates may be able to
represent anything, whether thoughts, facts or things, and indeed what
are the assumptions involved in the logically and mathematically
presumed apparatus of
logical operators,
subjects,
predicates and
formal proofs.

**2. The approach of model-theory**

In fact, Wittgenstein did not get
far, and his explanations, that are mostly in terms of (i) the picture
theory of meaning (statements are like pictures in
representing
whatever they represent) and (ii)
tautologies and
identities (the
statements of logic and mathematics are true because they cannot be
false and hold in any possible case, which is also what distinguishes
them from other kinds of statements) were mostly not correct.

That is, more precisely: The picture
theory of meaning as an explanation of how
language succeeds in
representing ideas and
things was soon rejected by nearly everyone
including Wittgenstein, and while W. had hit upon an important insight
about mathematics and logic with his notion that these were - at
least: the logiical statements - characterized by being true in all
possible circumstances, unlike statements of fact, that are
true only
if the facts
exist, and
false otherwise, he did not clearly work out
this insight.

In fact, something like it was worked
out by the founders of
model theory, that did succeed in providing a
fairly clear semantics for formal languages, mostly in the 1950ies and
1960ies, after Wittgenstein's death, and with little or no inspiration
from the Tractatus, and also in a somewhat peculiar fashion, that
probably would not have found approval in Wittgenstein's eyes.

The fashion in which this happened
was in fact that a formal language was supposed to be about
sets or
classes of things, and that one could elucidate what a formal language
and its axioms amounted to by showing how the
terms,
axioms and
statements of the language could be articulated in
set theory, that
then also could be used to
prove statements about the statements and
properties of the formal language (thus practising what was often
called "metamathematics" or "metalogic", since it consisted of
mathematical or logical statements and proofs about systems of
mathematics and logic).

Formally and mathematically, model
theory goes quite far and is able to shed considerable light on what a
formal system in fact states and cannot state.

The two conceptual difficulties that
remain are that (1) this in fact explains mathematics and logic in
terms of set theory - which seems to need
explanations themselves, and
that (2) in many texts of
model theory it is not clear what is really
happening in terms of logic.

Both points involve a certain amount
of circularity and circular reasoning, that indeed probable is
unavoidable in principle (it is rather similar to one's needing the
idea of meaning to explain the idea of meaning, and the idea of truth
to explain truth).

The first point, that model theory
does little more than reduce or explain logic and mathematics in terms
of set theory is normally met by the claim that
set theory is
mathematics or embodies it, and cannot be avoided in clarifying what
mathematics is - which is OK as far as it goes, but doesn't go very
far since in the end it must assume set theory as fundamentally clear
and given, which it isn't, and besides there are **other** foundations of
mathematics than set theory, such as combinatory logic or systems of
lambda calculus, that are as powerful as settheory mathematically
speaking, while suggesting quite other approaches to semantics.

The second point is a major
shortcoming that is rarely avoided, and notably it often is not clear
what "the models" of model theory really are: Sets of real things;
sets of ideas; sets of functions relating strings to some set of
things called a domain again
represented as a term in the set theory
that is said to be a model for the formal system it is interpreting by
mapping it to set-theoretical constructs...

In brief, things can get quite
confusing, also because often the verbal introductions to the
mathematics or logic of model theory are far from clear or
unambiguous. ("**Beginning Model Theory**" by **Jane Bridges** is clear in
most respects, but still is confusing if one is not aware of what has
been said here.)

**3. Some general theses about language
and reality**

As I said, W. tried in (2.01..) and
indeed all of (2...) to explain how the logically and mathematically
presumed apparatus of logical operators, subjects, predicates and
formal proofs such as one can find in Whitehead and Russell's
Principia Mathematica are capable of saying anything about reality,
and about how language, ideas and the facts and processes of the real
world are related.

Unfortunately, most of W.'s
explanations involve his picture theory of meaning, and the idea that
somehow reality has the same sort of structure as predicate and
propositional logic: It consists of facts, which are whatever makes
statements true, and it consists of objects as referred to by subject
terms, and of forms as referred to by predicate terms, and that is
about it, as far as W. was concerned, besides rather a lot more about
this somehow "showing itself" in and through language without being
articulable in language, which in turn propped up Wittgenstein's
peculariar Wittgensteinian "mysticism". (Both will crop up later in
the Tractatus.)

Here I will repeat and extend a
number of points I made in my notes to (2.01..)

2. The **correspondence theory of truth**
is presumed:

Statements represent because the
subject-terms in the statements represent individual things or classes
of things; the predicate-terms in the statement represent stuctures or
classes of things; while - and here is a statement of the
correspondence theory of
truth - a statement is true precisely if the
things represented by its subject-terms are related as or have the
properties stated by the predicate term(s) in the statement: "Snow is
white" is true precisely if indeed whatever is referred to by "Snow"
indeed is one of the (sets of) things that is referred to by "white",
and is false otherwise (namely in particular if something correctly
called by the term "snow" is not something that is correctly called by
the term "white").

In fact, this is not articulated
clearly by Wittgenstein at all, but he probably did have a fair
inkling of it, and indeed it was articulated first in the 1920ies,
quite probably in part because some mathematicians tried to formally
restate what is involved in the correspondence theory of truth. (See
Tarski.)

2.011. To try to explain how human
beings can learn anything at all with the help of language, the the **Finite
Characteristic Properties** thesis is useful:

- every (kind of) thing has some
characteristic
properties, and
- the number of characteristic
properties of a thing is finite, and
- a (kind of) thing's characteristic
properties are those properties which determine when

anything is that (kind of) thing, and
- a (kind of) thing's characteristic
properties determine which
structures it may be part of.

Indeed, this seems to be embodied in
natural language, for this is used normally as if the FCP is true:

- All
reasoning must start from
assumptions, and all
learning must involve the principle that one must
try out what seems sensible and give up what doesn't work, and many
things do seem to come in
**
kinds**,
- in language we proceed as if FCP is
true, and that FCP, if true, explains at least to some extent why
we can know what the world is like, and
- it seems that our perceptive system
involves a similar principle, i.e. we recognize and

reidentify perceptions as being identical or similar to other
perceptions on the basis of a finite set of properties (perceptual
features), and
- it can be proved - see e.g. Keynes
and Broad - that if the FCP is true for real things, then some
inductive generalizations may be supported, i.e. if the FCP is
true, then we can learn from experience.
- The FCP goes some way in the
direction of explaining and supporting kinds of things.

2.0121: An important principle
involved in theorizing and talking of almost all kinds is the
principle of adequacy:

- The world may be and indeed seems
to be such as to enable us to acquire
**sufficient **true information
about at least some of its constituents and their ways of behaviour so
as to be **adequate** to
**some **of our
purposes,
namely in the sense that we more often than not can realize our ends
on the basis of very partial knowledge of all the (possibly)
relevant fact - and indeed, after all, everybody who survived in the world
such as it is must have guessed rightly often

2.0122: One further reason or
assumption that enters into learning what the world is like is this:

**Relevancy-postulate**: As a
matter of fact, all contingent facts are
relevant to some facts and
not relevant to other facts, in the probabilistic sense of
"relevant": That learning that P is true, alters the probability
that Q is true (making it larger or smaller than it was, which is
why P is said to be relevant to Q), while learning that R is true
does not alter the the probability that Q is true (which is why R
then is said to be irrelevant to Q).

In fact, both relevance and
irrelevance are necessary to make any test at all of a statement,
since one can only test any particular
prediction by assuming it is relevant to some
statements, and specifically some statements the truth or falsity can
be established empirically, and also one can only test it by assuming
that most that happens while one is testing it is not relevant to its truth
or falsity (for if anything whatsoever is relevant to anything else
whatsoever, there is little chance of making decisive empirical
tests).

2.0123: Three further assumptions
that enter into the relation of language, though and reality are

**Natural kinds of structures:**
A - mathematically and linguistically - useful assumption is that
reality consists of structures that may be sorted in
natural kinds,
where structures are composite things with interrelated parts, and
natural kinds are classes of things with definite
antecedents (that
bring them about or make them more probable if true) and definite
consequents (that they bring about or make more
probable if
true)
**The characteristic properties**,
and also many other properties of things, are such as to be **
invariant in time**: They last as long as the things last, and
therefore allow its being tested and allow the making of
predictions
and explanations (which are pointless if things can unaccountably
change in time), and also is at the basis of making and testing
inductive generalizations (to the effect that what has often been
seen to be correlated will continue to be correlated, unless there
arises a reason it is not).
**There is no reason to assume
that reality is much like simple predicate or propositional logic**
- that indeed seem to be mostly as they are because natural language
and human minds are as they are: **There may be and very probably
are far more complicated entities and structures then can be stated
by only the logical apparatus involved in
First Order Logic**,
namely things that can only be adequately represented by
set theory,
mereology, or
higher
mathematics (fields, differentials, matrices, differential
manifolds etc.).

Note that none of the above was
clearly seen or stated by W., which is the reason to insert this
summary of some of my notes to (2.01..).