Tarski and Truth, Part 1

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.)

Truth and falsity are ascribed to descriptive or declarative sentences, or to what is expressed by descriptive sentences. Descriptive sentences express propositions. What does the sentence 'Snow is white' express? It expresses that snow is white.

Traditionally truth has been defined as correspondence with reality. The meaning of ‘corresponds to’, however, is far from clear. 

The Polish philosopher Alfred Tarski was the first person to state a precise definition of truth in familiar terms.

The Truth Scheme

Tarski suggests two requirements for a satisfactory definition of truth: It should be formally correct and materially adequate. 
By 'formally correct' Tarski means that the definition should be precise, free of contradictions, and it should have the right form. 
By 'materially adequate' Tarski refers to an equivalence sentence. This equivalence sentence is of the form (T): A is true if and only if A. 
'Snow is white' is true if and only if snow is white.

Let us call (T) the truth scheme, and let us call each of its instances a truth equivalence. 

A definition of truth for the descriptive sentences of a language L is materially adequate if and only if the definition implies all truth equivalences for language L. 

Truth scheme is not itself the intended definition of truth. 

It is important to note that we are dealing with a definition of truth for the descriptive sentences of some given language L, not a definition of truth for all sentences of all languages. 

Let's create a simple toy language 'L_simple' that contains only descriptive sentences, and define truth for that.

Truth for L_Simple

(Predicates are terms that express properties or relations). 

Here is the complete vocabulary of L_simple. 
Names: 'Socrates', 'Plato' 
Predicates: 'is a teacher of'. 
Logical symbols: 'and', 'or'. 

These are the grammatical rules of L_simple: 
If we put a name before 'is a teacher of' and another one after it, we get a sentence of L_simple. 
Finally, for every two sentences of L_simple, if we put an 'and' or an 'or' between them, then we get sentences of L_simple. 

We can now define 'truth for L_simple' in a Tarskian manner. 

For all sentences x of L_simple: 

if x is the result of putting together the name 'Socrates' with the predicate 'is a teacher of' and with the name 'Socrates' again, then x is true if and only if Socrates is a teacher of Socrates; (same case for Plato)

if x is the result of putting together the name 'Socrates' with the predicate 'is a teacher of' and with the name 'Plato', then x is true if and only if Socrates is a teacher of Plato; (and vice versa)

if there is a sentence y of L_simple and a sentence z of L_simple such that x is the result of putting together y with the logical symbol 'and' and with z, then x is true if and only if y is true and z is true. 

if there is a sentence y of L_simple and a sentence of z of L_simple such that x is the result of putting together y with the logical symbol 'or' and with z, then x is true if and only if y is true or z is true.

All truth equivalences for sentences in L_simple are derivable from the definition above. In other words: The definition of truth for L_simple is materially adequate.

Recursive Definitions

You would notice that in the definition of truth the term 'true' does occur on the right-hand sides of some of the various parts of our definition. 
Does this mean the definition is not formally correct? No, that is not the case.
The truth condition for the complex sentence is determined by definition from the truth conditions for the simpler sentences and the truth condition for the latter of these is determined ultimately by maximally simple sentences, which are formulated without invoking the term 'true' anymore. 

The definition of truth given above is therefore a recursive definition. Languages that are defined from a given vocabulary in such a precise recursive manner are called formal languages. (Linguists and philosophers of language also believe that recursion is true of natural language in general.)

Explicit Definitions

Every recursive definition can be transformed into a standard definition without any kind of circularity, using concepts from set theory. 
This second version of the definition of truth would look like this: 

For all sentences x of L_simple: x is true if and only if x is a member of all sets capital Y of sentences of L_simple for which the following holds: 

For all x': if x' is the result of putting together the name 'Socrates' with the predicate 'is a teacher of' and with the name 'Plato', then x' is a member of capital Y if and only if Socrates is a teacher of Plato.

And so on with other parts of the definition. 

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