040507 truththeories1

Truth theories.: 

Truth theories. Philosophy of Language V83.0085 Summer 2004

What is a theory?: 

What is a theory? A language is a set of sentences. A theory is a consistent set of sentences of a language, closed under logical deduction. A theory T is finitely axiomatizable if there is a finite set of sentences S such that all sentences of T are deducible from S. A theorem of a theory T is a sentence of T (sometimes: that aren’t axioms).

What is a theory of truth?: 

What is a theory of truth? A truth theory is a theory. So it’s a set of sentences. It’s theorems say something about another language. A theory of truth for a language L is a theory satisfying the following conditions: (i) For each sentence S of L, it has a theorem of the form “S* is true-in-L iff p”. (i) All its theorems are true.

T-theorems: S* is true-in-L iff p.: 

T-theorems: S* is true-in-L iff p. This theorem is a sentence of L*. “S*” is a name or description in L*. It refers to a sentence S in L. “p” is a sentence in L*. “is true-in-L” is a predicate in L*. “iff” is a sentence connective in L*. Example: “neige est blanc” is true-in-French iff snow is white. Example: Ronald is true-in-French iff snow is white. Example: The French sentence beginning with an en, then an ee, then… is true-in-French iff snow is white.

When is a sentence of the form “S* is true-in-L iff p” true?: 

When is a sentence of the form “S* is true-in-L iff p” true?

What is a theory of truth?: 

What is a theory of truth? A truth theory is a theory. So it’s a set of sentences of a language. It’s theorems say something about another language. A theory of truth for a language L is a theory satisfying the following conditions: (i) For each sentence S of L, it has a theorem of the form “S* is true-in-L iff p”. (i) All its theorems are true.

Examples.: 

Examples. “neige est blanc” is true-in-French iff snow is white. Ronald is true-in-French iff snow is white. “snow is white” is true-in-English iff snow is white. “snow is white” is true-in-English iff grass is green. “snow is white” is true-in-English iff grass is white.

More examples.: 

More examples. “grass is green” is true-in-English iff snow is white. “grass is green” is true-in-English iff snow is green. “grass is white” is true-in-English iff snow is white. “grass is white” is true-in-English iff snow is green.

Axiomatization.: 

Axiomatization. Many languages have an infinite number of sentences. If L has an infinite number of sentences, a truth-theory for L has an infinite number of theorems. Some theories are finitely axiomatizable. Question: Can a language with an infinite number of sentences have a truth-theory that’s finitely axiomatizable?

Example.: 

Example. The language: Two sentences: “Snow is white” and “Grass is green”, and one sentence-connective “&”. The result of flanking “&” by two sentences is a sentence. Axioms: (A1) “Snow is white” is true-in-L iff snow is white. (A2) “Grass is green” is true-in-L iff grass is green. (A3) For all sentences S and T, “S & T” is true-in-L iff “S” is true-in-L and “T” is true-in-L. Note: I’ve simplified (A3) at the expense of accuracy.

Sample Derivation.: 

Sample Derivation. “Snow is white and grass is green”. “Snow is white and grass is green” is true-in-L iff “snow is white” is true-in-L and “grass is green” is true-in-L (A3, UI). “Snow is white” is true-in-L iff snow is white. (A1) “Grass is green” is true-in-L iff grass is green. (A2) “Snow is white and grass is green” is true-in-L iff snow is white and “grass is green” is true-in-L. (lines 1, 2, subst). “Snow is white and grass is green” is true-in-L iff snow is white and grass is green. (lines 3, 4, subst.)

Another language.: 

Another language. The terms: Contains three names: “Venus”, “Tony Blair”, and “Mt. Rushmore”. Contains three predicates: “is a planet”, “is a man”, and “is a landmark”. Contains one sentence-connective: “&”. The sentences: The result of writing a name followed by a predicate is a sentence. The result of flanking the connective by two sentences is a sentence.

The truth-theory.: 

The truth-theory. (A1) The referent of “Venus” in L is Venus. (A2) The referent of “Tony Blair” in L is Tony Blair. (A3) The referent of “Mt. Rushmore” in L is Mt. Rushmore. (A4) For all names N, “N is a planet” is true-in-L iff the referent of N is a planet. (A5) For all names N, “N is a man” is true-in-L iff the referent of N is a man. (A6) For all names N, “N is a landmark” is true-in-L iff the referent of N is a landmark. (A7) For all sentences S and T, “S&T” is true-in-L iff S is true-in-L and T is true-in-L. Note: Again, in (A4) – (A7) I simplify at the expense of accuracy.

Sample derivation.: 

Sample derivation. “Venus is a planet”. For all names N, “N is a planet” is true-in-L iff the referent of N is a planet. (A4) “Venus is a planet” is true-in-L iff the referent of “Venus” is a planet. (line 1, UI). The referent of “Venus” is Venus. (A1). “Venus is a planet” is true-in-L iff Venus is a planet. (lines 2 and 3, subst.).

Truth and meaning.: 

Truth and meaning. Davidson notes in “Truth and Meaning” that speakers of a language have knowledge of the meanings of any of an infinite number of sentences. How can this be? Finitely axiomatizable truth-theories are such that knowledge of a finite number of axioms yields knowledge of any of an infinite number of theorems. Davidson suggests that what speakers of a language know is a finitely axiomatizable truth-theory for that language.