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An Introduction to the Theory of Knowledge
would hold that ‘‘red’’ is a simple concept. A simple concept is one that is indefinable or unanalyzable, one that cannot be broken down into logically independent concepts.15 According to some, we cannot analyze the concept of ‘‘red’’ into logically independent concepts the way we can analyze ‘‘square’’ into the logically independent concepts of ‘‘equilateral’’ and ‘‘rectangle.’’ Other allegedly simple concepts would be perceptual qualities such as ‘‘sweet,’’ ‘‘sour,’’ ‘‘salty,’’ ‘‘C#,’’ as well as other color concepts, such as ‘‘blue,’’ ‘‘yellow,’’ ‘‘green,’’ etc. If ‘‘red’’ is a simple concept, then (7) is not an analytic proposition. If so, then our a priori knowledge and justification is not confined to what is, according to A1, analytic. There are, however, other views about what it is for a proposition to be analytic. The German philosopher and logician Gottlob Frege suggested another approach. Roughly, Frege proposed the following: A2
Proposition p is analytic ¼ Df. Either (a) p is a logical truth or (b) p is reducible to a truth of logic by substituting synonyms for synonyms.
In order to understand this account we need to know what it is for something to be a ‘‘logical truth.’’ Consider the following propositions: (8) If all men are mortal, then all men are mortal. (6) Either all men are mortal or it is not the case that all men are mortal.
Each of these propositions has a certain ‘‘logical form.’’ The logical form of (8) is ‘‘If p, then p’’ and the logical form of (6) is ‘‘Either p or not-p.’’ Both of these logical forms are such that if we substitute any proposition for p, the result will be a true proposition. In other words, any of the ‘‘substitution instances’’ of these logical forms will result in a true proposition. What is a logical truth? Let us say that a proposition is a logical truth just in case its logical form is such that all of its substitution instances are true. As we have seen, A1 implies that the following propositions are analytic: (3) All squares are rectangles. (4) All bachelors are unmarried. 15
A and B are logically independent concepts just in case it is possible for something to have A without anything having B and vice versa. ‘‘Equilateral’’ and ‘‘rectangle’’ are logically independent concepts since it is possible for something to be equilateral without anything being a rectangle and vice versa.