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.