9. Logical Form

This section focuses on discussing some remarks on logical forms, a key issue in naturalizing the philosophy of language.

In his (1914) Russell made the following remarkable claim about logical forms:

“Some kind of knowledge of logical forms, though with most people it is not explicit, it involved in all understanding of discourse. It is the business of philosophical logic to extract this knowledge from its concrete integuments, and to render it explicit and pure.”^[1]

According to Wittgenstein:

“Every proposition has a content and form.”^[2]

We get the picture of the pure form if we abstract from the meaning of the single words, or symbols (so far as they have independent meanings). That is to say, if we substitute variables for the constants of the proposition. The rules of syntax which applied to the constants must apply to the variables also. By syntax in this general sense of the word I mean the rules which tell us in which connections only a word gives sense, thus excluding nonsensical structures. The syntax of ordinary language, as is well known, is not quite adequate for this purpose. It does not in all cases prevent the construction of non-sensical pseudopropositions (constructions such as “red is higher than green” or “the Real, though it is an in itself, must also be able to become a for myself,” and so on).

As observed by Wittgenstein

“If, now, we try to get at an actual analysis, we find logical forms which have very little similarity with the norms of ordinary language. We meet with the forms of space and time with the whole manifold of special and temporal objects, as colours, sounds, so on and so forth, with their gradations, continuous transitions, and combinations in various proportions, all of which we cannot seize by our ordinary means of expression.”

And he wishes to make first definite remark on the logical analysis of actual phenomena:

“It is this, that for their representation numbers (rational and irrational) must enter into the structure of the atomic propositions themselves.”^[3]

Wittgensteins maintains that the statement which attributes a degree to a quality cannot further be analyzed, and, moreover, that the relation of difference of degree is an internal relation and that it is therefore represented by an internal relation between the statements which attribute the different degrees. That is to say, the atomic statement must have the same multi-plicity as the degree which it attributes, whence it follows that numbers must enter the forms of atomic propositions. The mutual exclusion of unanalyzable statements of degree contradicts an opinion which was published by him several years ago and which necessitated that atomic propositions could not exclude one another. He here deliberately says “exclude” and not “contradict,” for there is a difference between these two notions, and atomic propositions, although they cannot contradict, may exclude one another. For proving, he tries to explain this. There are functions which can give a true proposition only for one value of their argument because there is only room in them for one. Take, for instance, a proposition which asserts the existence of a colour R at a certain time T in a certain place P of our visual field. And he writes this proposition “R P T,” and abstract for the moment from any consideration of how such a statement is to be further analyzed. “B P T,” then, says that the colour B is in the place P at the time T, and it will be clear to most of us here, and to all of us in ordinary life, that “R P T & B P T” is some sort of contradiction (and not merely a false proposition). Now if statements of degree were analyzable we could explain this contradiction by saying that the colour R contains all degrees of R and none of B and that the colour B contains all degrees of B and none of R. But from the above it follows that no analysis can eliminate statements of degree. How, then, does the mutual exclusion of R P T and B P T operate? He believes it consists in the fact that R P T as well as B P T are in a certain sense complete. That which corresponds in reality to the function “() PT” leaves room only for one entity -in the same sense, in fact, in which we say that there is room for one person only in a chair. Our symbolism, which allows us to form the sign of the logical product of “R P T” and “B P T” gives here no correct picture of reality. He has said that “a proposition “reaches up to reality,” and by this I meant that the forms of the entities are contained in the form of the proposition which is about these entities.” (see Peter Ludlow 1997: 213). For the sentence, together with the mode of projection which projects reality into the sentence, determines the logical form of the entities. This remark, he believes, gives us the key for the explanation of mutual exclusion of R PT and B PT. For if the proposition contains form of an entity which it is about, then it is possible that two propositions should collide in this very form. The propositions, “Brown now in this chair” and “Jones now sits in this chair” each, in a sense, try to their subject term on the chair. But the logical product of these propositions will put them both there at once, and this leads to a collision, a mutual exclusion of these terms. How does this exclusion represent itself in symbolism? We can write the logical product of the two propositions, p and q, in this way:

<table border="0" cellpadding="0" cellspacing="0" class="TableGrid" style="border-collapse:collapse; width:114.1pt"> P Q   T T T T F F F T F F F F

What happens if these two propositions are R P T and B PT? In this the top line “T T T” must disappear, as it represents an impossible combination. The true possibilities here are:

RPT	BPT
T	T
F	T
F	F
F	F

That is to say, there is no logical product of R P T and B P T in the first sense, and herein lies the exclusion as opposed to contradiction. The contradiction, if it existed, would have to be written:

RPT	BPT
T	T	F
T	F	F
F	T	F
F	F	F

but this, according to Wittgenstein, is nonsense, as the top line, “TTF,” gives the proposition a greater logical multiplicity than that of the actual possibilities. It is, of course, a deficiency of our notation that it does not prevent the formation of such nonsensical constructions, and a perfect notation will have to exclude such structures by definite rules of syntax. These will have to tell us that in the case of certain kinds of atomic propositions described in terms of definite symbolic features certain combinations of the T’s and F’s must be left out. Such rules, however, cannot be laid down until we have actually reached the ultimate analysis of the phenomena in question. This, as we all know, has not yet been achieved.

In the logic, sentences were constructed from predicates using a small number of operators corresponding to traditional forms of judgment, such as universal affirmative judgments, which are of the form ‘All Fs are Gs’. Proper names, such as ‘Socrates’, were regarded as predicates; that is, as being of the same logical type as expressions like ‘is mortal’. Thus, the famous argument:

All humans are mortal
Socrates is a human
Therefore, Socrates is mortal
Might have been represented as:
All H are M
All S are H
All S are M

The correctness of the argument then follows from the validity of the form of syllogism known as Barbara.

Footnotes and references:

[1]:

B. Russell. 1914. Logic as the Essence of philosophy. In Our Knowledge of the External World, London: Geoege Allen and Unwin.

[2]:

Ludwig Wittgenstein. 1929. ‘Some Remakers on Logical Form’. In Copi and Beard, (1966), 31-37; orig. in Proc. Aris. Soc. Supp. (1929), 162-167.

[3]:

Ludwig Wittgenstein. 1997. Some Remakrs on Logical Form. In Readings in the Philosophy of Language. ed. Peter Ludlow, 211. Massachusetts/London: The MIT Press.