Piero Scaruffi(Copyright © 2013 Piero Scaruffi | Legal restrictions )
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, Tarski’s 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 Goedel’s, 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.
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