Tractatus Logico-Philosophicus by Ludwig Wittgenstein + comments  by Maarten Maartensz          



In the following text I use some conventions to refer to Wittgenstein's text or my remarks. These conventions have most to do with Wittgenstein's peculiar format: The Tractatus consists of theses numbered in such a way as to show their logical dependencies. To refer to a thesis in the Tracatus I enclose its number in square brackets, as in "[2.01]". To refer to a thesis in the Tractatus together with the theses Wittgenstein numbered in such as a way to indicate that they are dependent on it, I enclose its number in square brackets followed by two dots, as in "[2.01..]". And to refer to my comments on a thesis or group of dependent theses I enclose the number (possibly followed by two dots) in normal brackets, as in "(2.01)" or "(2.01..)".

To set off the theses of Wittgenstein from my comments they are indented. Also, to most of the times I will write "W." instead of Wittgenstein.


The propositions [1..] provide an ontology of facts in analogy with propositional logic. The basic ideas involved seem to be that

1. whatever may be is a possible fact
2. whatever is, is a real fact
3. possible facts are represented by statements
4. real facts are represented by true statements

The analogy with propositional logic may be made clear by presenting some of the foundations of propositional logic in a non-standard way, namely by assuming that statements represent ideas and ideas represent possible facts.

The standard way is to leave out ideas and be vague about possible facts, but neither can be easily combined with W.'s remarks in the Tractatus, while in fact the following non-standard way, whatever its ultimate merits, makes perfectly good intuitive sense.

It is convenient to have a handy and short notation, and there is an obvious one available: we shall write statements between double quotes; the ideas they express between single quotes; and the possible facts they refer to without quotes. Since all of this in fact is done within language, sometimes clarifying phrases are added.

Accordingly, for example, the English statement "it rains" expresses the idea that 'it rains', which happens also to be expressed by the German statement "es regnet", and both refer to the possible fact that it rains.

That is, we shall assume semantical assumptions that may be formulated as

S1. "p" represents 'p' or nothing (and if nothing "p" is meaningless)
S2. 'p' represents p or nothing (and if nothing 'p' is false)
S3. "p" is true iff 'p' is true iff p is not nothing

which may be read as

S1. a statement "p" represents an idea 'p'
S2. an idea 'p' represents a possible fact p
S3. A statement is true iff the idea the statement represents is true iff the idea represents a (real) fact

in which "(real)" merely serves to emphasize that this fact is not just possible but exists (in some sense, depending on some criterion).

One may well ask what the use of this non-standard and simple-minded propositional semantics is, especially since it doesn't change anything fundamental about propositional logic.

There are three pertinent differences with the standard way.

1. In the standard way each proposition has, it is assumed, a truth-value T or F, somehow mysteriously depending on what the facts are. The assignment of truth-values to propositions without propositional operators in them is not a matter of logic, and anyhow logic needs to consider both possible outcomes.

In the non-standard way the reason why a proposition gets a T rather than an F is explicated as: Because the idea the proposition expresses represents a fact.

So the first difference is that the non-standard way reduces truth-values to a notation for succeeding or failing to represent a fact rather than construing truth-values as fundamental - and rather mysterious - entities somehow given to logic to work with. (The metaphysical troubles to which basic truth-values lead can be found out through studying Frege.)

In the non-standard way the basis of propositional logic is the assumption that propositions represent ideas and ideas represent or fail to represent facts, and truth-values are mere notational devices to indicate whether or not a proposition represents the fact it expresses; in the standard way the basis of propositional logic is the assumption that propositions somehow have truth-values, that are assigned by some extra-logical process.

2. The standard way does not consider ideas at all. A proposition either represents a fact or not, and whichever is the case does not depend on logic, and that is all that is relevant for propositional logic.

The non-standard way explicitly assumes that propositions do not represent or fail to represent the facts directly, but only indirectly, in virtue of whether or not the idea the proposition expresses represents a fact or not.

The non-standard way, accordingly, presumes an explananation for the meaning of a proposition: This appears as the idea a proposition expresses; the standard way does not provide a clear possibility to explain what the meaning is or could be.

This may seem a trivial difference, but it is a fact that the absence of an assumption that coordinates ideas and statements has caused a lot of muddle in the presentations of propositional logic. Indeed, the term "proposition" is often ambiguously used with the senses of "idea expressed by a statement" and "statement expressing an idea" and even "fact represented by the idea expressed by a statement".

Furthermore, on an intuitive level it seems absolutely self-evident that propositions express ideas and have meanings apart from whatever the facts may be, and that when we understand a proposition we usually understand the idea it expresses rather than the fact that it represents or fails to represent. (Indeed, the interesting cases are where we understand the idea a proposition expresses and do not know whether it represents a fact.)

And finally, the fact that - seen from the point of view of the nonstandard way - meanings and facts are confused has led to other confusions, such as the verificationist theory of meaning, and the confusion of meaning and truth.

3. The standard way cannot give an obvious, clear and direct account of when a proposition is meaningful rather than true. The non-standard way does suggest an obvious, clear and direct account of when a proposition is meaningful, namely that the terms of which a proposition is composed represent ideas which in turn represent or fail to represent possible things, and that a proposition a proposition is meaningful if its terms are understood.