Sunday, August 25, 2013
Coursera: Introduction to Mathematical Philosophy
Hannes Leitgeb and Stephan Hartmann
Personal notes from lectures of Week 2: Truth
(These are personal summaries and paraphrasings of some of the major points of the lectures that I felt to be important. They are not meant to be comprehensive records nor intended to be reproductions of copyright materials. I encourage you to participate in the course for better understanding.)
The vocabulary of L_simple does not contain the word ‘true’ but English does contain the term 'true' already in its vocabulary. So, a definition of truth for English as a whole should allow us to derive sentences that talk about the truth of sentences. This can, however, lead to the famous Liar Paradox, which can arise when a sentence refers to itself.
* The sentence that is introduced by a star symbol is not true.
Applying the truth predicate:
'The sentence that is introduced by a star symbol is not true' is true if and only if 'The sentence that is introduced by a star symbol is not true' is not true.
This conclusion is of the form 'A if and only if not A', but that is a logical contradiction.
The Tarskian Hierarchy
Kurt Gödel famously proved in his so-called incompleteness theorems that there will always be sentences which in some sense talk about themselves.
In our definition from before, we defined the truth predicate 'true' for all the sentences of L_simple, but the truth predicate itself was not a member of the vocabulary of L_simple.
The truth predicate and its definition do not belong to L_simple, but to a different language. The language in which we stated our definition of truth for L_simple.
When we define truth, we are actually dealing with two languages at the same time:
1) The object language. That's the language for which one defines truth.
2) The metalanguage, the language in which one defines truth for the object language.
L_Simple doesn't contain the word 'true' but meta-language of L_Simple does. Would this lead to Liar Paradox?
To state the truths of this meta-language, we'd have to create another meta-meta-language, which would use a new truth predicate 'true_1' which doesn't occur in the meta-language. And so on and so forth.
We can construct such a hierarchy of meta meta and so on languages and Tarskian truth definitions on top of it. While none of these languages is expressive enough to speak about the truth or falsity of all of its own sentences, they can speak about the truth or falsity of all the sentences at the previous stages in the Tarskian hierarchy.
Consider, however, a universal language, a language in which one can express everything that is meaningful at all. It is not possible to state a formally correct and material adequate definition of truth for such a universal language. Such a definition would have to be carried out in some language again. If the definition is formulated in the universal language itself, it would lead to the contradictory conclusion of the liar paradox. So there is no universal language for which one could state a satisfactory definition of truth in the Tarskian sense. If a natural language such as English is universal, then we cannot state a satisfactory definition of truth for it as a whole, only for fragments of it.