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.