ATOMIC AND MOLECULAR PROPOSITIONS I did not quite finish last time the syllabus that I intended for

Lecture 2, so I must first do that.

I had been speaking at the end of my last lecture on the subject of the self-subsistence of particulars, how each particular has its being independently of any other and does not depend upon anything else for the logical possibility of its existence. I compared particulars with the old conception of substance, that is to say, they have the quality of self-subsistence that used to belong to substance, but not the quality of persistence through time. A particular, as a rule, is apt to last for a very short time indeed, not an instant but a very short time. In that respect particulars differ from the old substances but in their logical position they do not. There is, as you know, a logical theory which is quite opposed to that view, a logical theory according to which, if you really understood any one thing, you would understand everything. I think that rests upon a certain confu-sion of ideas. When you have acquaintance with a particular, you understand that particular itself quite fully, independently of the fact that there are a great many propositions about it that you do not know, but propositions concerning the particular are not necessary to be known in order that you may know what the

particular itself is. It is rather the other way round. In order to understand a proposition in which the name of a particular occurs, you must already be acquainted with that particular. The acquaintance with the simpler is presupposed in the understand-ing of the more complex, but the logic that I should wish to combat maintains that in order thoroughly to know any one thing, you must know all its relations and all its qualities, all the propositions in fact in which that thing is mentioned; and you deduce of course from that that the world is an interdependent whole. It is on a basis of that sort that the logic of monism develops. Generally one supports this theory by talking about the

“nature” of a thing, assuming that a thing has something which you call its “nature” which is generally elaborately confounded and distinguished from the thing, so that you can get a comfort-able see-saw which encomfort-ables you to deduce whichever results suit the moment. The “nature” of the thing would come to mean all the true propositions in which the thing is mentioned. Of course it is clear that since everything has relations to everything else, you cannot know all the facts of which a thing is a constituent without having some knowledge of everything in the universe.

When you realize that what one calls “knowing a particular”

merely means acquaintance with that particular and is presup-posed in the understanding of any proposition in which that particular is mentioned, I think you also realize that you cannot take the view that the understanding of the name of the particu-lar presupposes knowledge of all the propositions concerning that particular.

I should like to say about understanding, that that phrase is often used mistakenly. People speak of “understanding the universe” and so on. But, of course, the only thing you can really understand (in the strict sense of the word) is a symbol, and to understand a symbol is to know what it stands for.

I pass on from particulars to predicates and relations and what we mean by understanding the words that we use for predicates

and relations. A very great deal of what I am saying in this course of lectures consists of ideas which I derived from my friend Wittgenstein. But I have had no opportunity of knowing how far his ideas have changed since August 1914, nor whether he is alive or dead, so I cannot make anyone but myself responsible for them.

Understanding a predicate is quite a different thing from understanding a name. By a predicate, as you know, I mean the word that is used to designate a quality such as red, white, square, round, and the understanding of a word like that involves a different kind of act of mind from that which is involved in understanding a name. To understand a name you must be acquainted with the particular of which it is a name, and you must know that it is the name of that particular. You do not, that is to say, have any suggestion of the form of a proposition, whereas in understanding a predicate you do. To understand

“red”, for instance, is to understand what is meant by saying that a thing is red. You have to bring in the form of a proposition.

You do not have to know, concerning any particular “this”, that

“This is red” but you have to know what is the meaning of saying that anything is red. You have to understand what one would call “being red”. The importance of that is in connection with the theory of types, which I shall come to later on. It is in the fact that a predicate can never occur except as a predicate.

When it seems to occur as a subject, the phrase wants amplifying and explaining, unless, of course, you are talking about the word itself. You must say “ ‘Red’ is a predicate”, but then you must have “red” in inverted commas because you are talking about the word “red”. When you understand “red” it means that you understand propositions of the form that “x is red”. So that the understanding of a predicate is something a little more compli-cated than the understanding of a name, just because of that.

Exactly the same applies to relations, and in fact all those things that are not particulars. Take, e.g., “before” in “x is before y”: you

understand “before” when you understand what that would mean if x and y were given. I do not mean you know whether it is true, but you understand the proposition. Here again the same thing applies. A relation can never occur except as a relation, never as a subject. You will always have to put in hypothetical terms, if not real ones, such as “If I say that x is before y, I assert a relation between x and y”. It is in this way that you will have to expand such a statement as “ ‘Before’ is a relation” in order to get its meaning.

The different sorts of words, in fact, have different sorts of uses and must be kept always to the right use and not to the wrong use, and it is fallacies arising from putting symbols to wrong uses that lead to the contradictions concerned with types.

There is just one more point before I leave the subjects I meant to have dealt with last time, and that is a point which came up in discussion at the conclusion of the last lecture, namely, that if you like you can get a formal reduction of (say) monadic relations to dyadic, or of dyadic to triadic, or of all the relations below a certain order to all above that order, but the converse reduction is not possible. Suppose one takes, for example, “red”. One says, “This is red”, “That is red”, and so forth. Now, if anyone is of the opinion that there is reason to try to get on without subject-predicate propositions, all that is necessary is to take some standard red thing and have a relation which one might call “colour-likeness”, sameness of colour, which would be a direct relation, not consisting in having a certain colour. You can then define the things which are red, as all the things that have colour-likeness to this standard thing.

That is practically the treatment that Berkeley and Hume recommended, except that they did not recognize that they were reducing qualities to relations, but thought they were getting rid of “abstract ideas” altogether. You can perfectly well do in that way a formal reduction of predicates to relations. There is no objection to that either empirically or logically. If you think it is

worth while you can proceed in exactly the same way with dyadic relations, which you can reduce to triadic. Royce used to have a great affection for that process. For some reason he always liked triadic relations better than dyadic ones; he illustrated his preference in his contributions to mathematical logic and the principles of geometry.

All that is possible. I do not myself see any particular point in doing it as soon as you have realized that it is possible. I see no particular reason to suppose that the simplest relations that occur in the world are (say) of order n, but there is no a priori reason against it. The converse reduction, on the other hand, is quite impossible except in certain special cases where the relation has some special properties. For example, dyadic relations can be reduced to sameness of predicate when they are symmetrical and transitive. Thus, e.g., the relation of colour-likeness will have the property that if A has exact colour-likeness with B and B with C, then A has exact colour-likeness with C; and if A has it with B, B has it with A. But the case is otherwise with asymmetrical relations.

Take for example “A is greater than B”. It is obvious that “A is greater than B” does not consist in A and B having a common predicate, for if it did it would require that B should also be greater than A. It is also obvious that it does not consist merely in their having different predicates, because if A has a different predicate from B, B has a different predicate from A, so that in either case, whether of sameness or difference of predicate, you get a symmetrical relation. For instance, if A is of a different colour from B, B is of a different colour from A. Therefore when you get symmetrical relations, you have relations which it is formally possible to reduce to either sameness of predicate or difference of predicate, but when you come to asymmetrical relations there is no such possibility. This impossibility of reducing dyadic relations to sameness or difference of predicate is a matter of a good deal of importance in connection with

traditional philosophy, because a great deal of traditional phil-osophy depends upon the assumption that every proposition really is of the subject-predicate form, and that is certainly not the case. That theory dominates a great part of traditional meta-physics and the old idea of substance and a good deal of the theory of the Absolute, so that that sort of logical outlook which had its imagination dominated by the theory that you could always express a proposition in a subject-predicate form has had a very great deal of influence upon traditional metaphysics.

That is the end of what I ought to have said last time, and I come on now to the proper topic of today’s lecture, that is molecular propositions. I call them molecular propositions because they contain other propositions which you may call their atoms, and by molecular propositions I mean propositions having such words as “or”, “if”, “and”, and so forth. If I say, “Either today is Tuesday, or we have all made a mistake in being here”, that is the sort of proposition that I mean that is molecular. Or if I say,

“If it rains, I shall bring my umbrella”, that again is a molecular proposition because it contains the two parts “It rains” and “I shall bring my umbrella”. If I say, “It did rain and I did bring my umbrella”, that again is a molecular proposition. Or if I say,

“The supposition of its raining is incompatible with the suppos-ition of my not bringing my umbrella”, that again is a molecular proposition. There are various propositions of that sort, which you can complicate ad infinitum. They are built up out of proposi-tions related by such words as “or”, “if”, “and”, and so on. You remember that I defined an atomic proposition as one which contains a single verb. Now there are two different lines of complication in proceeding from these to more complex pro-positions. There is the line that I have just been talking about, where you proceed to molecular propositions, and there is another line which I shall come to in a later lecture, where you have not two related propositions, but one proposition containing two or more verbs. Examples are got from believing,

wishing, and so forth. “I believe Socrates is mortal.” You have there two verbs, “believe” and “is”. Or “I wish I were immortal”.

Anything like that where you have a wish or a belief or a doubt involves two verbs. A lot of psychological attitudes involve two verbs, not, as it were, crystallized out, but two verbs within the one unitary proposition. But I am talking today about molecular propositions and you will understand that you can make pro-positions with “or” and “and” and so forth, where the constitu-ent propositions are not atomic, but for the momconstitu-ent we can confine ourselves to the case where the constituent propositions are atomic. When you take an atomic proposition, or one of these propositions like “believing”, when you take any prop-osition of that sort, there is just one fact which is pointed to by the proposition, pointed to either truly or falsely. The essence of a proposition is that it can correspond in two ways with a fact, in what one may call the true way or the false way. You might illustrate it in a picture like this:

True: Prop. Fact

False: Fact Prop.

Supposing you have the proposition “Socrates is mortal”, either there would be the fact that Socrates is mortal or there would be the fact that Socrates is not mortal. In the one case it corresponds in a way that makes the proposition true, in the other case in a way that makes the proposition false. That is one way in which a proposition differs from a name.

There are, of course, two propositions corresponding to every fact, one true and one false. There are no false facts, so you cannot get one fact for every proposition but only for every pair of propositions. All that applies to atomic propositions. But when you take such a proposition as “p or q”, “Socrates is mortal

or Socrates is living still”, there you will have two different facts involved in the truth or the falsehood of your proposition “p or q”. There will be the fact that corresponds to p and there will be the fact that corresponds to q, and both of those facts are relevant in discovering the truth or falsehood of “p or q”. I do not sup-pose there is in the world a single disjunctive fact corresponding to “p or q”. It does not look plausible that in the actual objective world there are facts going about which you could describe as “p or q”, but I would not lay too much stress on what strikes one as plausible: it is not a thing you can rely on altogether. For the present I do not think any difficulties will arise from the suppos-ition that the truth or falsehood of this propossuppos-ition “p or q” does not depend upon a single objective fact which is disjunctive but depends on the two facts one of which corresponds to p and the other to q: p will have a fact corresponding to it and q will have a fact corresponding it. That is to say, the truth or falsehood of this proposition “p or q” depends upon two facts and not upon one, as p does and as q does. Generally speaking, as regards these things that you make up out of two propositions, the whole of what is necessary in order to know their meaning is to know under what circumstances they are true, given the truth or falsehood of p and the truth or falsehood of q. That is perfectly obvious. You have as a schema, for “p or q”, using

“TT” for “p and q both true”

“TF” for “p true and q false”, etc.

TT TF FT FF

T T T F

where the bottom line states the truth or the falsehood of “p or q”. You must not look about the real world for an object which you can call “or”, and say, “Now, look at this. This is ‘or’.” There is no such thing, and if you try to analyse “p or q” in that way

you will get into trouble. But the meaning of disjunction will be entirely explained by the above schema.

I call these things truth-functions of propositions, when the truth or falsehood of the molecular proposition depends only on the truth or falsehood of the propositions that enter into it. The same applies to “p and q” and “if p then q” and “p is incompatible with q”. When I say “p is incompatible with q” I simply mean to say that they are not both true. I do not mean any more. Those sorts of things are called truth-functions, and these molecular propositions that we are dealing with today are instances of truth-functions. If p is a proposition, the statement that “I believe p” does not depend for its truth or falsehood, simply upon the truth or falsehood of p, since I believe some but not all true propositions and some but not all false propositions.

I just want to give you a little talk about the way these truth-functions are built up. You can build up all these different sorts of truth-functions out of one source, namely “p is incompatible with q”, meaning by that that they are not both true, that one at least of them is false.

We will denote “p is incompatible with q” by p/q.

Take for instance p-p, i.e. “p is incompatible with itself ”. In that case clearly p will be false, so that you can take “p/p” as meaning

“p is false”, i.e. p/p = not p. The meaning of molecular proposi-tions is entirely determined by their truth-schema and there is nothing more in it than that, so that when you have got two things of the same truth-schema you can identify them.

Suppose you want “if p and q”, that simply means that you cannot have p without having q, so that p is incompatible with the falsehood of q. Thus,

“If p then q” = p/(q/q).

When you have that, it follows of course at once that if p is true, q is true, because you cannot have p true and q false.

Suppose you want “p or q”, that means that the falsehood of p is incompatible with the falsehood of q. If p is false, q is not false, and vice versa. That will be:

(p/p)/(q/q).

Suppose you want “p and q are both true”. That will mean that p is not incompatible with q. When p and q are both true, it is not the case that at least one of them is false. Thus,

“p and q are both true” = (p/q)/(p/q).

The whole of the logic of deduction is concerned simply with complications and developments of this idea. This idea of incompatibility was first shown to be sufficient for the purpose

The whole of the logic of deduction is concerned simply with complications and developments of this idea. This idea of incompatibility was first shown to be sufficient for the purpose