Why Tarski cannot handle propositions? Because there are infinitely many primitive of them

Intensional sentences such as:

(1) Georgio believes that Rome is a beautiful city

are sentences whose truth or falsity (i.e. logical) value depends not only the denotation their constituent parts, but on something else, like in the example above: the content of the sentence “Rome is a beautiful city” and (probably) on how Georgio understands this sentence (or its translation or synonym).

From a certain angle there seems to be no intensionality paradox. Sentence (1) is true if Georgio believes that Rome is a beautiful city. Why should the truth of (1) depend on whether the sentence

(2) Rome is a beautiful city
true or false

[Here we can immediately see the relationship between the intensional sentences and intentional objects, the contents of thoughts. The inexistence of the unicorn does not make my thinking of it any less true. The existence of a turbojet does not make my thinking of it true, if I do not realise its existence]

But of course there is a problem. You simply cannot write a Tarski-style definition for each belief sentence like the one

(3) The sentence “Georgio believes that Rome is a beautiful city is true” if and only if Georgio believes that Rome is a beautiful city is true

The reason is that one would have to write infinitely many of such definitions. This is due to the fact that there are recursively infinitely many sentences like:

(1) „Rome is a beautiful city”
(2) „the city in which there lives his only uncle is a beautiful city”
(3) „the city in which there lives the only uncle of his uncle is a beautiful city”
(4) „the city in which there lives the only uncle of his uncle’s uncle is a beautiful city”
(5) „the city in which there lives the only uncle of his uncle’s uncle’s uncle is a beautiful city”


The subjects of these sentences are coreferential with Rome

And now comes the important part. These sentences may be generated recursively. We may even understand them recursively from bottom up. For instance:

(6) the city in which there lives his only uncle

can be understood in such a way

But as far as truth conditional semantics goes, phrases like (6) are absolutely non-compositional, vocabulary items. We must take them as whole. And you cannot formally write a grammar or truth conditional semantics which has an infinite vocabulary, an infinite number of building blocks.

That the vocabulary (in terms of referential semantics) is indeed can easily be seen if we replace the relevant descriptions with coreferential proper names

(7) Rome is a beautiful city”
(8) „the city in which there lives Silvio Berlusconi is a beautiful city”
(9) „the city in which there lives Luca Barbone is a beautiful city”
(10) „the city in which there lives Giordano Bruno is a beautiful city”
(11) „the city in which there lives Giorgio Pier Georgi is a beautiful city”, etc.

We could quickly run out of telephone directory names

These proper names are primitive (non-decomposable) both in syntactic and semantic terms. But from the point of view of intensional semantics the phrases:

(12) “that Rome is a beautiful city”
(13) „that the city in which there lives Silvio Berlusconi is a beautiful city”
(14) „that the city in which there lives Luca Barbone is a beautiful city”
(15) „that the city in which there lives Giordano Bruno is a beautiful city”
(16) „that the city in which there lives Giorgio Pier Georgi is a beautiful city”

are also primitive. Why? Because nothing inside them will help us establish the truth of sentences like:

(17) “Georgio thinks that Rome is a beautiful city”
(18) „Georgio thinks that the city in which there lives Silvio Berlusconi is a beautiful city”
(19) „Georgio thinks that the city in which there lives Luca Barbone is a beautiful city”
(20) „Georgio thinks that the city in which there lives Giordano Bruno is a beautiful city”
(21) „Georgio thinks that the city in which there lives Giorgio Pier Georgi is a beautiful city”, etc.


So this is the real problem of intensional sentences. The truth-conditional semantics of intensional sentences is not formalisable. Why? Because the formalisation would need to use an infinite number of definitions, an infinite list of them. This again is because what counts as a recursive phrase inside a complement sentence, would need be reanalised as primitive, non-analysable element in intensional semantics.

Ryle Revisited

I had a cursory glance at the previous note after two weeks of absence from this blog during which I pursued other projects (like earning money for my family).

I still think that the basic idea presented there is quite plausible. “Rome” and “the capital of twentieth century Italy” are phrases which are replacable in:

(3) Marek believes that Rome is a beautiful city
(4) Marek believes that the capital of twentieth century Italy is a beautiful city

but not in

(5) Cicero believes that Rome is a beautiful city (true perhaps)
(6) Cicero believes that the capital of twentieth century Italy is a beautiful city (false or nonsensical)

just because Marek but not Cicero believes that “Rome” and “the capital of twentieth century Italy” are the same thing.

But this is where things start to go wrong. “Believes” reappears in the explanation and we have again intensional sentences which need explanation. So we have an infinite regress We can only be saved by converting to behaviorism and not using any psychological language. But to become a behaviorist at my age?

When I cogitated at my predicament, I just remembered that this is not the first infinite regress connected with the behaviorism/mentalism dispute. Remember Gilbert Ryle and his campaign against a “Ghost in the Machine”.

According to the legend, whenever an agent does anything intelligently, his act is preceded and steered by another internal act of considering a regulative proposition appropriate to his practical problem. [...] Must we then say that for the hero's reflections how to act to be intelligent he must first reflect how best to reflect how to act? The endlessness of this implied regress shows that the application of the criterion of appropriateness does not entail the occurrence of a process of considering this criterion.(The Concept of Mind (1949), p. 31.)
The crucial objection to the intellectualist legend is this. The consideration of propositions is itself an operation the execution of which can be more or less intelligent, less or more stupid. But if, for any operation to be intelligently executed, a prior theoretical operation had first to be performed and performed intelligently, it would be a logical impossibility for anyone ever to break into the circle.(The Concept of Mind (1949), p. 30.)

There seem to be similarities between the two regresses. But I am not sure whether they have so much in common. I will need to think about this.

Farewell to Semantics (part two)

So when are words replaceable in sentences starting with "I believe"?

Not when they denote the same thing, as shown by many examples of the "Morning Star/Evening Star" type. Not even when they have the same meaning, because the speaker (believer) many not know that they have the same meaning.

I suggest that replaceablity of words in intensional contexts must always taken relative to the subject of the sentence (= believer). The relevant relation would look more less like this"

(1) aD(B)a = a has for B the same denotation as b

where a and b are names such "Morning Star" and "Evening Star" respectively, "Rome" and "capital of Italy" respectively, "Cicero" and "Tully" respectively or "Superman" and "Clark Kent respectively. Let me suggest the following definition

(2) a is replaceable by b in sentences about B believing something iff aD(B)a

(this should be properly formalised, but I am now only testing my intuitions)



This definitely explains why the sentences

(3) Marek believes that Rome is a beautiful city
(4) Marek believes that the capital of twentieth century Italy is a beautiful city

are both true

but not sentences

(5) Cicero believes that Rome is a beautiful city (true perhaps)
(6) Cicero believes that the capital of twentieth century Italy is a beautiful city (false or nonsensical)

Simply, the two relevant descriptions have the same denotations for Marek but not for Cicero

There are four things to be said about this account of belief sentences. Two not so good and two quite good.

Not so good

(A) This solution is a capitulation. D(B) is not a general relation of meaning. It is not about meaning in a language. It is relativised to a particular believer. So it is not very general. So it may not be very interesting.

(B) You must be very careful how you formulate even informal paraphrases of D(B). For instance, you cannot say that aD(B)b means that B believes that "a" has the same denotation is "b". This would smack of a vicious circle, but this may not be the biggest problem. The problem is that the using of "believe" in the definition of aD(B)b could invite the same paradoxes as those which we tried to explain in the first place. Wherever we have belief we have intensionality

The solution is that "a has for B the same denotation as b" would be explained in behavioristic terms which would not include "believe". This is the philosophy of mind bit. I am very unwilling to recourse to behaviorism, because I like mental concepts and do not want to explain them away, but what can I do? I am just forced to accept it in this case, to explain the semantic puzzle.

Good

(C) This is a very simple solution at first glance (but complications lurk)

(D) It seems to save compositional semantics, although a reference is made to metalanguage, on both side of truth conditions. The intuitive sketch would look as follows.

The sentence "John believes Rome is a beautiful city" iff John believes Rome is a beautiful city and "Rome" denotes Rome for John

If it looks again like a vicious circle, but let me remind that the explication of "Rome" denotes for John Rome would be behaviouristic and "Rome" could be treated as a physical sound and denotation of Rome by "Rome" for John perhaps some physical disposition or co-occurance.

Am I satisfied with this solution? I will need to look at it again later

Farewell to Semantics (part one)

Some entries earlier I described the old and famous intensionality paradox my philosophy teachers were fascinated about. (do not forget to click)

Here is another example:


(1) Cicero believed that Rome was a great town

But Rome is also the capital of Italy, so Rome and the capital of Italy are the same thing.

(2) Rome = capital of Italy,

so by way of substitution, the following should be true

(3) Cicero believed that the capital of Italy was a great town.

But this is false. Cicero did not have the faintest idea about Italy, not in the modern sense, and what its capital was.


Why is this paradox disquieting? This is connected with three names

1’) Frege and compositional semantics
2’) Tarski and definition of truth
3’) Davidson and truth-conditional theory of meaning


Re 1') The compositionality principle says that the meaning of a complex expression is determined by the meanings of its constituent expressions and the rules used to combine them. In particular, it depends on what entities are assigned to constituent words. The town of Rome is assigned to the word “Rome”. But the meaning of (1') and (3') does not depend only on the reference of words, but also what words are actually used (“Rome” or “capital of Italy”). And this seems to be unpredicable.

Re 2') This is similar. Once you assign meaning to basic proper names, variables and predicates (or put more simply to basic constituents of language), you can determine by automatic rules the truth of complex sentences which involve logical connectives (and quantifiers). But this apparently cannot be done if we admit verbs expressing propositional attitudes like "believe" or "know". "Rome" and "capital of Italy" refer to the same town. So whether we use the word "Rome" or "capital of Italy" should have no bearing on the truth of the sentences.

(1) Cicero believed that Rome was a great town.
(3) Cicero believed that the capital of Italy was a great town.

But the first is true and the second untrue.

Re 3') Donad Davidson and some other philosophers suggested that the meaning of natural language sentences should be their truth conditions. But for this this you need a Tarski-style semantics, which with belief sentences you apparently are not going to get.

In the next entry I will try to give a solution to the intensionality paradox. I am still a little sketchy about the details and I am sure that it cannot be faultless, because in my experience no one has yet succeeded to solve this problem. But for now my solution seems quite intuitive and, moreover, is rooted in philosophy of mind.

See you soon,

Marco

Banking and semantics

I have a job at a bank. On a bad day it seems to make little sense (other than in pounds and pence). On a good day there is enough sophisticated semantics to pass round, straightforward as it might look at first sight.

Why straightforward? Because banking is about balance sheets. Balance sheets are about numbers. Numbers are mathematics. Mathematics is extensional and does not allow intensional paradoxes.
So banking does not allow intensional paradoxes.

But this reasoning is certainly untrue.

(if you do not know what intensional contexts are click here to see previous entry)


Banking’s basic equation, sort of E = mc2 , is

(1)TOTAL ASSETS = TOTAL LIABILITIES

Hence,

(2) ASSETS = LIABILITIES

Now let us take the assumption which all the bankers recognise as true

(3) DEPOSITS ARE LIABILITIES

Next, from (2) and (3) you can derive by substitution

(4) DEPOSITS ARE ASSETS

which is false, absurd even (because deposits are on the liabilities side of the balance sheet).

The bottom line is that when we talk about the balance sheet the Fregean, compositional semantics does not work, because the talking of assets and liabilities involves intensional contexts. Now, this is puzzling, because it would seem that the balance sheet is pure mathematics, which is extensional.