Tarski's error & Arrow's imbecility

 Ken Arrow was Tarski's student. He proofread Tarski's 'introduction to Logic' which states

a general and very important principle, which must be retained if a language is to be usefully employed. According to this principle, whenever we wish to state a sentence about a certain object, we have to use in this sentence not the object itself, but its name or designation.

This is nonsense. We can use the object or its name or point to it or leave it to the other party to work out what is being referred to. 

However, for a logical operation, what is important is that the name, or intension, has a fixed, well defined, 'extension'- i.e. the set of things which have all the properties corresponding to the named object is easily specifiable and is not subject to change as more becomes known. 

The application of this principle does not give rise to any problems as long as the object in question is not a word or a symbol or, more generally, an expression of a language.

That does not matter. It is fine to say that £ is the same thing as 'pound sterling'  

Let us imagine, for example, that we have a small blue stone in front of us, and that we write the following sentence: 'this stone is blue'.  

You need to put the scrap of paper with that inscription together with the stone. That way, at a later point, it would be know which stone we thought was blue or how our notion of blue was different from other people's. 

To no one, presumably, would it occur to replace in this sentence the words "this stone" (which together constitute the designation of the object) by the object itself, that is to say, to blot or cut these words out and to place in their stead the stone.

Tarski was clearly wrong.  

For, in doing so, we would arrive at a whole consisting partly of a stone and partly of words, and therefore at something which would not be a linguistic expression, and all the more, would not be a true sentence.

Fuck off! Museums are full of things with descriptions of them attached- e.g. 'This is a Sumerian seal' 

This principle, however, is frequently violated if the object which is talked about happens to be a word or a symbol.

There is no principle here. There is only an arbitrary and wholly foolish assertion. 

And yet the application of the principle is indispensable also in this case; for otherwise we would arrive at a whole which, though being a linguistic expression, would fail to convey the thought intended by us, and might even be a meaningless aggregate of words.

The sentence 'this is blue' is meaningless if we don't have the stone it was referring to.  

Let us consider, for example, the following two words: good, Mary.

Both are highly meaningful, in context.  

Clearly, the first consists of four letters,

That is not clear at all. This stupid cunt forgets that shorthand can be used to transcribe an English sentence. But so can Arabic or Russian script.

and the second is a proper name.

It may be. It may not. In Australia, at that time,  'mary' was a derogatory word signifying an aboriginal woman. 

But let us suppose that we would express these thoughts, which are quite correct, in the following manner: (1) good consists of four letters;

this is quite wrong. Tarski wasn't a native English speaker. I suppose, as a refugee, he needed to earn money by teaching or writing any old shite he could get away with. But Arrow was a native English speaker. He could have got a job as an Accountant or an Actuary. Why did he waste his time and destroy his intelligence on this stripe of shite?  

(2) Mary is a proper name;

sometimes, yes. Sometimes, no.  

then, in talking about words, we would be using the words themselves and not their names.

Nothing wrong with that. Very few 'intensions' have well defined 'extensions'. Set theory and Logic can't be used in a naive manner or they quickly degenerate into nonsense. The proper way to proceed is to look at things where we have some assurance that we have a well defined set or graph of a function. Then we can do set theory and relation algebra and use logical calculi. But, almost all of the time, we can't and even when we can, as our knowledge base grows, we will find that we have to 'restrict comprehension' of any given eligible intension or else employ a type theory etc. This is because epistemic or impredicative intensions can't have clear cut extensions. They don't obey Liebniz's laws of identity. Tarski is introducing Arrow not to Logic but to Stupidity and an algorithmic method for generating mischievous nonsense. 

And if we examine expressions (1) and (2) more closely, we must admit that the first is not a sentence at all, since the subject of a sentence can only be a noun and not an adjective;

No. The subject of a sentence is supplied by its context. I draw a picture. Teechur says 'Good'. The meaning is my drawing is good.  

the second might be considered a meaningful sentence, but, at any rate, would be a false

It is true of some uses of the word and false of others. The law of the excluded middle does not apply to epistemic intensions- e.g. words in natural language where both semantics and pragmatics are knowledge based and change as that base changes.  

one since no woman is a proper name.

Plenty are- e.g. imaginary ones whom a novelist hasn't yet fleshed out in his mind.  

In order to clarify these situations, we have to realize that when the words "good" and "Mary" occur in such contexts as those of (1) and (2), then their meanings differ from the usual ones, and that here these words function as their own names.

We realize much more than this. That's why we aren't impressed by logical aporias or paradoxes. Indeed, our instinct is to consider logic a type of stupidity. Verification matters. Deduction does not.  

In generalizing this viewpoint, we should have to admit that any word may, at times, function as its own name;

but any element of a naive set may be impredicative or dependent on other elements such that we can't assume the set has a unique extension. 

to use the terminology of medieval logic, we may say that in such a case the word is used in SUPPOSITIO MATERIALIS, as opposed to its use in SUPPOSITIO FORMALIS, that is, in its ordinary meaning.

Even then, what matters is that the thing being referred to has a well defined extension which stays fixed. But this just means 'be sure you are talking about the same thing. If you are talking about different things, you will get contradictions.'

Consider Blau (1957) on Arrow's Social Welfare Function

The starting point is a set a of (possibly infinitely many) objects of unspecified nature, called alternatives.

This is not a set. Depending on the knowledge base, some things appear to be alternatives though they may be complementary, interdependent, or, indeed, the same thing under a different name.

They may be thought of as candidates, or alternative economic states, or alternative courses of action.

Arbitrarily, a person or group of persons can stipulate for a particular set of alternatives and then stipulate there is an ordering upon them. But, they may find that they were wrong because alternatives are epistemic and thinking about a thing changes it. There is no set. There is no identity (a given alternative isn't even the same thing as itself). There is no relations or orderings or graphs of functions. This is because our arbitrary stipulation is nonsense.  

A weak ordering (hereafter called merely an ordering) of the set a is a reflexive, connected, and transitive relation R defined on (t; that is, (I) For each (not necessarily distinct) pair x, y of alternatives, at least one of the following is true: x R y, y R x. (Connected and reflexive.)  (II) For each (not necessarily distinct) triple x, y, z of alternatives, x R y and y R z imply x R z. (Transitive.)

We will see that xRx is impossible for anything epistemic unless nothing in the knowledge base is affected by anything else in it. But, if that were the case, a knowledge unit would be itself a material object. It would be a chattel which could be transferred. Its production and distribution and extinction would be outside the mind. Thus it couldn't be mental. It couldn't be knowledge. Relations too are mental- i.e. epistemic. They too don't 'coincide with themselves'. Moreover they can be impredicative- e.g. preferred preference or meta-preference. This is a relation which isn't identical to itself because my preferences (which are slobbish) are different from my meta-preferences (I'd prefer to have snobbish preferences). 

Now consider the definition-

 A social welfare function (SWF) is a function whose domain D is a subset of the set of all ordered n-tuples 6R = (R1, R2, * **, R.) of orderings of A(set of Alternatives).

The Domain is not well-defined. It is not a set. There can't be any such function. Now you may say, we can talk about things which can't exist- 'e.g. the cat which is a dog because it isn't a dog'.  Indeed, we could agree to worship or execrate such things. We could say 'there is an impossibility result such that the cat which is a dog because it isn't a dog can't have properties we want it to have- e.g. the ability to be eaten as a hot-dog in a baseball stadium.' But we could equally say the opposite or just fart vociferously and claim that our fart proves, mathematically, that the cat which is a dog just proved the Reimann hypothesis. In some mystical sense, this may be regarded as true. But only because it is also false and true only because it is false and false only because it truly is the fart of the cat which is a dog.

What is going on in the previous paragraph is 'ex falso quodlibet' or the 'principle of explosion'- i.e. the notion that from what is false any and every nonsense can be logically deduced. Tarski throws the baby out with the bath-water when he deals with this problem by declaring erroneously that a language can't contain its own truth-predicate. Consider the law courts. Clearly, they do have a protocol bound, buck stopped, method of doing this for themselves. True, most issues are not justiciable. But 'completeness' is not required. Only utility is. If a particular type of language 'pays its way', that's all we ask of it. 

Still, within Tarski's (or any other mathematical) framework, the first requirement when using set theory or relation algebra is that you ensure that you are actually dealing with well defined sets and relations. If they are epistemic or impredicative then the intensional fallacy arises and you can't assume Liebniz's laws of identity or the principle of excluded middle. Similarly, when dealing with dogs- which is what the good folks at Crufts do- the judges have strict protocols re. what is a dog. That's why my g.f. didn't win a prize though she is a basic enough bitch. 

Ordinary people use natural language and are seldom troubled by intensional fallacies or 'ex falso' because we quickly converge to a good enough 'extension' for any word used. Moreover, in any useful field- e.g. the 'higher faculties' in a University (viz. those that deal with Law, Divinity & Medicine) there are strict protocols which assign well defined 'buck stopped' extensions to intensions. This is not in the case of lower faculties where you get a Doctorate in Philosophy even if you write wholly fallacious Arrowvian or Sen-tentious shite. 

It is a separate matter that Arrow's nonsense could lead you to inaccessible cardinals or other such topics in the foundations of set theory. But those foundations have deeper foundations yet and themselves represent the set of alternatives for a type of Social Choice- viz. that with which mathsy peeps might be genuinely concerned or to which they might themselves contribute. 

Voting theory, auction theory, preference revelations mechanisms etc. can proceed well enough and may even 'pay their way' in certain narrow contexts. What will be lacking is canonicity, naturality, categoricity, universality, completeness etc. Math can be useful. Mathsy masturbation is mischievous shite.