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.