1. On Denoting.Bertrand Russell - 2005 - Mind 114 (456):873 - 887.
    By a `denoting phrase' I mean a phrase such as any one of the following: a man, some man, any man, every man, all men, the present King of England, the present King of France, the center of mass of the solar system at the first instant of the twentieth century, the revolution of the earth round the sun, the revolution of the sun round the earth. Thus a phrase is denoting solely in virtue of its form. We may distinguish (...)
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On the sense of variables in propositional functions
In "On Denoting" (1905), Russell presents a theory of denotation which relies on the notion of a variable.  Russell says very little about variables in this paper.  He says only that they are "fundamental," and that they are "essentially and wholly undetermined" constituents of propositional functions.  I think I understand the role of this notion in Russell's theory, and why Russell says what he does about it,  He appeals to non-denoting elements in propositions in order to avoid having to interpret "a=b" as "a=a."  By using variables, he can claim that no elements in a propositional function serve the role of the denoting phrase.  For example, in the fully explicit presentation of "Scott is the author of Waverley," we do not find anything for which we could substitute the phrase "the author of Waverley."  The meaning of the denoting phrase is only found when we interpret the proposition as a whole, and cannot be found in any of its parts. 

My problem is, I don't know what it means to say that a proposition contains undetermined elements.  Russellian propositions are regarded either as structured sequences of objects and properties, or as possible worlds.  Fregean propositions are thoughts about objects and properties, or ways of denoting the True and the False.  None of these views seems welcoming of undetermined elements, unless we say that some thoughts and/or objects are undetermined, or that some thoughts and/or objects can have undetermined properties, or that some worlds have undetermined components.  The notion of variables looks problematic to me, though I am admittedly new to Russell's paper and the topic in general.

I would appreciate any thoughts, or pointers to where this issue has been discussed.


March 22, 2010

On the sense of variables in propositional functions
I don't have the passage before me. I would guess he means this.

'x is a philosopher.' is a propositional function. You can make it into a proposition by replacing the variable with a name, e.g. 'Jim' 'Jason.'
So the variable has no determinate meaning or reference. It's a grammatical place holder in a propositional function that can be replaced by various
names to form various propositions. Hence it is a variable. So far it has no semantic content. By contrast, the predicate is determinate in that it denotes a particular property.
It isn't a variable.

On the sense of variables in propositional functions
Reply to Jim Stone
Hi Jim,

Thanks for the post.  I do tend to think of variables as placeholders without semantic content.  Unfortunately, this does not alleviate my concern about Russell's theory of descriptions.

Russell uses the term "proposition" to refer to propositional functions, which suggests he was not making a strong distinction between the two.  He suggests that propositional functions (at least of the sort he is considering) express or indicate hypothetical propositions, which suggests that their constituents are real objects and properties.  So for Russell, variables must have some kind of existence, perhaps as abstract objects.  And I don't see how there can be objects (abstract or not) that form parts of propositions without having any determinate properties.

Also, I don't think a propositional-functional reading of "Scott is the author of Waverley" makes much sense, if we must fill in the variables in order to complete the semantic content. Consider that, for Russell, "Scott is the author of Waverley" means "it is not always false of x that x wrote Waverley, that it is always true of y that if y wrote Waverley, y is identical with x, and that Scott is identical with x."  If we plug in names for x and y, we get, "It is not always false of Jim that Jim wrote Waverley, that it is always true of Tom that if Tom wrote Waverley, Tom is identical with Jim, and that Scott is identical with Jim." It seems obvious that we never, under any circumstances, indicate or express such a proposition when we claim that Scott is the author of Waverley.  Nor is it plausible that such a substitution is required for our meaning to be complete.   If Russell's theory suggests that we only understand that Scott is the author of Waverley when we make such a substitution, then his theory seems obviously wrong,  But I do not think this is how Russell should be read.  I think he wants these propositional functions to have complete semantic content.  He claims the meaning of the denoting phrase is found in the fully expressed propositional function.  He must be talking about the semantic content, and I suspect it should be complete for his theory to work.  Though I admit, I am groping in the dark a little here.

March 26, 2010

On the sense of variables in propositional functions

I wrote, "[Russell] suggests that propositional functions (at least of the sort he is considering) express or indicate hypothetical propositions, which suggests that their constituents are real objects and properties."

I do not think that is accurate.  Perhaps Russell regards propositional functions as a class of propositions, or perhaps they express or indicate propositions.  I do not think Russell is clear on this point, at least not in "On Denoting."  Either way, to be consistent, I think Russell would have to regard their constituents as real objects and properties.

Also, Russell is not only considering hypothetical propositions in "On Denoting," so I was wrong to suggest that he was.  The fully expressed version of "Scott is the author of Waverley" does contain a hypothetical proposition, however, so we might say the proposition as a whole is hypothetical in some sense.  I do not think this is relevant to my general concern, though.  I only mention it to correct the remark in my previous post.

March 27, 2010

On the sense of variables in propositional functions
Russell writes:

‘I use `C(x)' to mean a proposition in which x is a constituent, where x, the variable, is essentially and wholly undetermined.’

However there is a footnote–click on ‘proposition.’

‘More exactly, a propositional function. BR’

Propositions, not propositional functions, have a truth value.  ‘x is mortal’ has no truth value.
I don’t think there is a deep issue here. 

On the sense of variables in propositional functions
Reply to Jim Stone

Mustn't some impredicative propositional functions have truth value? Maybe being a function or not isn't indicative of differential truth valuehood. A minor point.


On the sense of variables in propositional functions
Reply to Jim Stone

You say "x is mortal" has no truth value.  Yet, Russell claims that "if x is human, then x is mortal" can be true--in fact, he says the truth of this function is what we assert when we say, "all men are mortal."

Let's take another example.  Russell says (1) means (2):

     (1)  I met a man.

     (2)  "I met x, and x is human" is not always false.

Now, surely, (1) can be either true or false, so (1) is true iff (2) is true, and (1) is false iff (2) is false.

Russell also says that (1) does not denote a definite man.  It does not mean (3):

     (3) I met some definite man.

He says (1) entails (3), but when we affirm (1), we are not affirming (3).  They are distinct propositions.  So (1) cannot mean something like (4):

    (4)  "I met Jim and Jim is human" is not always false.

It follows that (2) can be either true or false, but this does not depend on our substituting a name for the variable.

Furthermore, as I indicated in an earlier post, concerning definite descriptions, Russell takes "Scott was the author of Waverley" to mean a complex proposition which includes variables, and it is not plausible that this proposition requires substitution to be true or false.

In conclusion, It seems that Russell does regard propositional functions as having truth values, and that this does not depend on our substituting names for variables.

P.S. I'm looking at Russell's paper as it appears in Martinich, A. The Philosophy of Language, 3rd edition, pp.  199-207, and not an online version.

On the sense of variables in propositional functions
Reply to Jim Stone
It looks like I'm not the only one who thinks there is a deep issue here. 

I just took a look at Peter Hylton's Russell, Idealism, and the Emergence of Analytic Philosophy (1990).  On page 256, he relates the following exchange between Moore and Russell:

"What I should like explained is this.  You say 'all the constituents of propositions we apprehend are entities with which we have immediate acquaintance.'  Have we, then, immediate acquaintance with the variable?  And what sort of entity is it?" (Moore to Russell, 23 October, 1905)

Russell responded two days later:

"I admit that the question you raise about the variable is puzzling, as are all questions about it.  The view I usually incline to is that we have immediate acquaintance with the variable, but it is not an entity.  Then at other times I think it is an entity, but an indeterminate one.  In the former view there is still a problem of meaning and denotation as regards the variable itself.  I only profess to reduce the problem of denoting to the problem of the variable.  This latter is horribly difficult, and there seems equally strong objections to all the views I have been able to think of."

Hylton observes that Russell never arrives at an explanation or account of variables, and in a later paper ("On Fundamentals") takes them as fundamental and unexplained.  Hylton also traces the problem of the the variable to the obscurity of Russell's notion that propositional functions are fundamental yet essentially ambiguous entities.

March 31, 2010

On the sense of variables in propositional functions

‘x is mortal,’ that propositional function has no truth value. Which is intuitive, as it doesn’t predicate mortality of anything. It can be turned into a proposition in two ways. First, by replacing ‘x’ with a singular term, e.g. ‘Jim.’ Then you get a proposition.

Second, by attaching a quantifier. (x) (x is mortal). This says that for anything there is (or for anything in a certain predetermined domain), it is mortal. This quantified sentence does express a proposition, it does have a truth value. The propositional function ‘x is a mortal’ is merely part of the proposition
and it still has no truth value, just as ‘is mortal’ has no truth value in ‘Jim is mortal.’
Same goes for (Ex) (x is mortal). The whole sentence expresses a proposition, not the propositional function which is a part of it.

There is some question as to how to construe the quantifier. One way of doing it is this.
Suppose we have domain D which contains all men, and a set of names such that each one names just one man in D.

(x) (x is mortal) is true just in case whatever name with which one replaces ‘x’ in the propositional function (‘Jim,’ ‘Jason,’...) produces a true proposition.
(Ex) (x is mortal) is true just in case there is at least one name such that if one substitutes it for the ‘x’ in the propositional function, it produces a true proposition.

‘I met a man’ means “‘x is a man and I met x’ is not always false.” Now suppose I met a man. Then this second claim (within the double quotes) is true. But what does it assert? It is, in fact,
the assertion (Ex) (x is a man and I met x). That’s true and you will note that it works fine
on the semantics I gave–it’s the assertion that for at least one denoting term, substituting that name for ‘x’ is the propositional function ‘x is a man and I met x’ produces a true proposition. “‘x is a man and I met x’ is not always false” should be construed this way, I submit. It isn’t saying that ‘x is a man and I met x’ is sometimes true simpliciter (which really would be nonsense), but that there is a name such that ‘x is a man and I met x’ produces a true proposition if you substitute that name.

Alternatively we might take "'x is a man and I met x' is not always false" to say that there is at least one thing such that, if we substitute a singular term that denotes it for ‘x’, this produces a true proposition. I submit this is what Russell has in mind. 

On the sense of variables in propositional functions
I wrote:

'Propositions, not propositional functions, have a truth value.  ‘x is mortal’ has no truth value.
I don’t think there is a deep issue here.'

That  is, propositional functions aren't propositions, the latter alone have truth values.
Also the indeteminacy of the variable in a propositional function amounts to this:
it doesn't determine a particular subject. This is because it doesn't denote anything.
That's WHY propositional functions don't have truth values. They aren't about anything.
So I don't see these
as deep issues. 'x is mortal' plainly has no truth value and it's because the
variable doesn't determine a subject.

Russell in addition maintained that we are immediately acquainted with every constituent of a proposition we can understand.
Variables are constituents of some propositions, e.g. (x) (x is  mortal). So he has to give an account of
what variables really are ontologically. There may or may not be deep issues about this..

On the sense of variables in propositional functions
Reply to Jim Stone
I appreciate your wanting to focus on the distinction between propositions and propositional functions, and I do think Russell did, at one point, make the distinction you are making.  In fact, in The Principles of Mathematics (1903), Russell defines the distinction between propositions and propositional functions by appeal to Peano's distinction between real and apparent variables.  Real variables are like placeholders, so that an expression like "x is mortal" has a real variable, and thus no truth value.  To get a proposition, you have to replace the variable with a value.  On the other hand, apparent variables occur in propositions and thus cannot be replaced with values.  When a variable appears in a proposition, the proposition does not depend upon it being replaced with a name.  But, of course, the variable is still there.  It's just not "real."  So what happened to the variable?  According to Russell, it somehow got "absorbed" in the proposition.  That leaves us (or me, at least) skeptical, when the most explicit statement of the proposition still contains the variable.  If it had been absorbed, then it would not be in the most explicit statement of the proposition.  So Russell's initial account cannot be right.  We still do not know what the variable means, or how we could be acquainted with it, or how we could recognize different instantiations of the same variable, or how we could distinguish one variable from another.  These are deep problems, I think.

But to continue with my point.  As I was saying, Russell did draw a sharp distinction between propositions and propositional functions, much like you are doing here.  However, later, he changed his mind.  In his 1927 introduction to the second edition of the Principia, Russell rejects Peano's distinction between real and apparent variables, and seems to suggest that there is no difference at all between (1) and (2):

     (1) If x is human, then x is mortal.
     (2) For all values of x, (1) is true.

That is, Russell seems to take so-called "propositional functions" as propositions which range over all values of their variables.  So, for an expression like (1), or like "x is mortal", we should say it can be true or false, depending on whether or not it is true or false for all values of x.  And I think this makes sense--at least, it makes a lot more sense than claiming that only (2), and not (1), could be true.

In any case, I think it is clear that the problem of the variable is there from the beginning, and it does not disappear when Russell disposes of Peano's distinction between real and apparent variables.

April 12, 2010

On the sense of variables in propositional functions

Dear Jason

I think you may be right. In the applied use of logic [in which words are mixed with symbols] we often seem to use free (‘real’) variables as an indefinite mode of reference, and this may have been the usage Russell had in mind. Imagine a group of people that we want to keep anonymous [eg in a court of law] then we may agree to call them ‘Mr X’, ‘Mr Y’, etc. In each case the variable denotes a particular person without identifying that person; we may call this ‘indefinite reference’. Thus in Russell’s example, ‘I met x’ there would be a unique person denoted by ‘x’ which, for reasons of concealment or poor memory the speaker has decided not to identify by his singular term. Of course it is only the act of reference that is indefinite, not the person; the person is a definite individual.

Thus the real meaning of ‘I met x’ would seem to be

(1) I met the person referred to by ‘x’ .

The variable in ‘I met x’ is used here to indicate that we want to refer to a person only as being the referent of the symbol ‘x’. It is intended by this means to create anonymity.

Consequently, ‘I met x’ does not mean the same as ‘I met y’ since currently ‘y’ may be used to designate a different person. Free variables used in this way may be combined in a shared narrative: thus I may have been to a party and met a father and son, whom I cannot now identify, in which case I could assert truly,

(2) ‘I met x and I met y and x is the father of y’

Here it is the speaker who decides who to designate as x, and who as y. Isn’t this a formalisation of ‘I met a father and his son’ ?

I believe this provides a theory for what Russell referred to as ‘indefinite descriptions’ and, as Russell required, it does not appear to assert the existence of anything, ie it uses no quantifiers. However, if I am right, since ‘I met x’ means (1) “ I met the person referred to by ‘x’ ” then this translation is itself none other than a definite description, and according to Russell’s own theory of definite descriptions it says,

(3)     There is one and only one person referred to by ‘x’ and I met that person.

Similarly for each of the clauses in the narrative (2).

So it turns out, ironically, that indefinite descriptions seem to require definite descriptions [in the metalanguage] for their analysis. If ‘I met x’ is a propositional function then clearly, with this full expansion (3) the propositional function is indeed true or false as you claimed because, implicitly, the variable itself is being referred to, as well as the object.

I’d be most interested to hear your comments on this theory of indefinite descriptions,


On the sense of variables in propositional functions
I believe the best solution is to think of two layers of booleans. The lower layer consists of predicates and the upper layer consists of propositions. Variables in each of the layers can take on one of two values, {0,1} BUT the interpretations of the Boolean values are different in each layer.
In the lower predicate layer, 0= 'present' or 'exists' and 1= 'absent' or 'nonexistent'. So consider the sentence 'John has a blue shirt'. The subject 'John' we denote by y. The predicate 'has a blue shirt' we denote by x. Don't get stuck on whether we include the verb 'has' because that can be attached to 'John' so that 'John has' becomes a subject capability. The argument is the same. 

In the upper proposition layer 0= 'false' and 1= 'true'. Now we are in a position to determine what can be truth. Let the proposition variable be z. Then z=(y=x). That is, truth (whether z= 1, or 0) is determined by two things. 1. Is y=x a possible attribution! ie a WFF 2. Is the computation (y=x) true?

There are certain combinations of subject y and predicate x which are not legal, or 'well formed'. For example 'John has a blue horizon' is not computable. In the Pierceian triad ( the semiotic triangle consisting of <symbol, representation, object>) , there is no possible realistic representation. 

If you believe the mind is a computer, as I do, then this result is a no-brainer, because all programs (eg the sentence 'John has a blue shirt') must be 'compiled' into a binary state machine that can be executed. The first step of compilation is lexical analysis, which ideally traps syntactic and semantic (ie typing) errors.

The test of any such theory is to see how it handles the 'liars paradox' usually written as 'This statement is false'. Since falsehood is not a predicate value, it cannot legally appear inside the computation. It is as if you have written z=(z=z). You have conflated the predicate and propositional layers. In other words truth value  must arise from comparing two existential variables.

On the sense of variables in propositional functions
Reply to Jim Stone
I use the mnemonic z=(y=x) to help me negotiate the confusing aspects of propositional calculus.The statement has two logical (Boolean) levels, a lower, predicate level where 0=Nonexistant and 1=exists, and an upper propositional level where 0=false and 1=true. Here is the 'kicker'- propositions only result when two predicate variables (eg a subject and predicate) are compared for sameness.

Consider 'John has a blue shirt'. This is a legal combination of predicates. 'John' and 'has a blue shirt' (or equivalently, the affordance 'John has' and 'a blue shirt') can both exist in a possible world. therefore 1. You can put y = x together legally so the Boolean computation (y=x) resolves to either true or false.

In the possible micro-world 'John has a blue shirt' , the statement is either true (yep, I can see him on the surveillance video, running out the door to work in the morning, suitcase in hand, pecking the wife on the cheek) or false (no, he's late again, his favourite shirt is dirty, he has a brown shirt on today).

But the micro-world 'John has a blue horizon' DOES NOT COMPUTE, not in a reality processor like our mind, which consist of a conceptual part and an embodied part. We CAN form the statement above, and if we were a magic realist author, or creative poet, we might even get a literature prize, but as soon as the statement (which exists in the conceptual mind) is compared on the basis for similarity with the knowledge of predicate constructs in the 'realistic, possible' embodied mind, we obtain a '0' which doesn't mean 'false' (it isn't a proposition) it means 'cannot exist' because it is a predicate.

The problem arises because our minds ( and all computers) are ultimately binary machines, but the world is not. It has a predicate layer which uses unary 'base 1' in which there is arity=1. If something isn't possible in the real world, it simply doesn't exist, because it was never created. It was evolution which created brains and computers, in which binary images of unary reality are possible. The main reason for this was to model propositional falsity (past and future) , not modelling predicate falsity.

On the sense of variables in propositional functions
A propositional function is an expression that contains at least one occurrence of a free variable; whereas a proposition is an expression that does not contain any free occurrence of any variable.

Now, (1) contains two occurrences of free variables (namely two occurrences of "x"). So, (1)* is NOT a proposition, but a propositional function. Since (1) is a propositional function, it does not have a truth-value. On the other hand, (2)** contains no free occurrence of any variable; the two occurrences of "x" in (2) [I read (2) as "For all values of x, if x is human, then x is mortal"] are bounded by the quantifier 'for all values of x'; and hence, they are not free variables, they are bound variables. Since, there is no free variable in (2), it is a proposition, NOT a propositional function. That is why, (2) has a truth-value.

Note that, a proposition may contain bound variables, but cannot contain any free variable. And, a propositional function may contain both- bound and free variables, and must contain at least one free variable.

Does it help, Jason?


* (1) If x is human, then x is mortal.
** (2) For all values of x, (1) is true.

On the sense of variables in propositional functions
Hi Mustafa.  Thanks for contributing.  Whie the free/bound distintion is a coherent one, I don't think it alleviates my concern.  The discussion of the distinction between propositions and propositional functions was a bit of a side issue.  My concern is with how we can be acquainted, in the Russellian sense, with variables as such.  Regards, Jason