Inquire about purchasing the book | Table of Contents | Annotated Bibliography | Class on Nature of Mind
These are excerpts and elaborations from my book "The Nature of Consciousness"
|
Truth and Meaning One more notion was
necessary to complete the picture: meaning. What did all this mean in the end? Aristotle had realized the importance of
"truth" for logical reasoning and had offered his definition: a
proposition is true, if and only if it
corresponds with the facts. This is the "correspondence theory of
truth". Frege had founded Logic on truth: the
laws of logic are the laws of truth. Truth is Frege's unit of meaning. In fact,
it was Frege who introduced "true" and "false", the so
called "truth values". Frege regarded logical propositions as
expressing the application of a concept to an object, as in "author(piero)"
that states that Piero is an author. Indirectly, he partitioned the universe
into concepts and objects, equated concepts with mathematical functions and
objects with mathematical terms. The proposition "Piero is an author"
has a concept "author" that is applied to a term "Piero".
All of this made sense because, ultimately, a proposition was either true or
false, and that could be used to think logically. According to Aristotle, if this proposition is true, then its meaning is that the person
referred to as Piero is an author; and viceversa. Hilbert had taken this course of action
to its extreme consequences. Hilbert had emancipated Logic from reality, by
dealing purely with abstractions. In 1935 the Polish mathematician
Alfred Tarski grounded Logic back into reality. He gave "meaning" to
the correspondence theory of truth. Logic is ultimately about
truth: how to prove if something is true or false. But what is truth? Tarski was looking for a definition of
truth that would satisfy two requirements, one practical and one formal: he
wanted truth to be grounded in the facts, and he wanted truth to be reliable
for reasoning. The second requirement was easily expressed: true statements
must not lead to contradictions. The first requirement was more complicated.
How does one express the fact that "Snow is white" is true if and
only if snow is white? Tarski realized that "snow is white" is two
different things in that sentence. They are used at different levels. A
proposition p such as "snow is white" means what it states. But it
can also be mentioned in another sentence, which is exactly the case when we
say that "p is true". The fact that "p" is true and the
sentence "p is true" are actually two different things. The latter is
a "meta-sentence", expressed in a meta-language. In the meta-language
one can talk about elements of the language. The liar's paradox, for example,
is solved because "I am lying" is a sentence at one level and the
fact that I am telling the truth when I am lying is a sentence at a different
level; the contradiction is avoided by considering them at two different levels
(language and meta-language). Tarski realized that truth within a
theory can be defined only relative to another theory, the meta-theory. In the
meta-theory one can define (one can list) all the statements that are true in
the theory. Tarski introduced the concepts of
interpretation and model of a theory. A theory is a set of formulas. An
interpretation of a theory is a function that assigns a meaning (a reference in
the real world) to each of its formulas. Every interpretation that satisfies
all formulas of the theory is a model for that theory. For example, the formulas
of Physics are interpreted as laws of nature. The universe of physical objects
becomes a model for Physics. Ultimately, Tarskis trick was to build models
of the world which yield interpretations of sentences in that world. The
important fact is that all semantic concepts (i.e., meaning) are defined in
terms of truth, and truth is defined in terms of satisfaction, and satisfaction
is defined in terms of physical
concepts (i.e., reality). The meaning of a proposition turns out to be the set
of situations in which it is true. What Tarski realized is that truth can only
be relative to something. A concept of truth for a theory (i.e., all the
propositions that are true in that theory) can be defined only in another
theory, its meta-theory, a theory of that theory. All paradoxes, including
Goedels, can then be overcome, if not solved. Life goes on. Tarski grounded meaning in truth and
in reference, a stance that would set the stage for a debate for the rest of
the century. Back to the beginning of the chapter "Machine Intelligence" | Back to the index of all chapters |
| 
| 
| 
|
