Alice C. Linsley
Ancient logic
Plato’s view of reality was based on his belief that pattern can be seen in the order of nature and that we recognize pattern because it reflects the pattern intuitively grasped by the soul or inner person. The observable patterns in nature include the binary sets uniformly applied in the logic of the ancient world.
In mathematical terms, a binary set would be 1a+1b = {1} (binary set) in which the two entities are not equal. An example is male + female = human, but the male is larger and stronger than the female. The paradoxical nature of this is evident. This paradox was rejected as logical by modern pioneers in logic who, like Frege, insisted that 1+1 = a set of 2 and 2 = 1+1, a set of equals.
The anthropologist Levi-Strauss has demonstrated how binary logic is reflected in the thought of primitive peoples, and Jacques Derrida has shown how binary thinking is inescapable even in modern logic. This is because in a binary set we have more than one thing. Through disclosure or deconstruction, we find there are sets within sets.
Derrida saw the middle as a function by which binary opposites acquire extended meaning as merisms and reversals. These thinkers discovered in analysis of primitive myths that there are some universal ideas related to the most fundamental of human experiences such as birth, the Sun, and nourishment. This suggests weakness in nominalism, the view that there are no universal essences; that no abstract entities, essences, classes, or propositions have real existence.
Syllogistic reasoning
Aristotle (384-322 BC) is often credited with being the "father" of logic. He taught his students syllogistic logic. A syllogism attempts inference of one proposition from two premises. Each premise has one term in common with the conclusion. If (1)A = B, and (2)B = C, then (3)A = C. A standard-form categorical syllogism meets the following four conditions:
- All three statements are categorical propositions.
- The two occurrences of each term are identical.
- Each term is used in the same sense throughout the argument.
- The major premise is listed first, the minor premise second, and the conclusion is last.
Aristotle also deserves credit for modal logic because his evaluation of involves concepts like possibility, necessity, belief and doubt. He also identified several common fallacies
Aristotle dealt with the logical constants and, or, if ... then ..., not, and some and all. This is referred to as Naïve Set Theory because it relies on natural language to describe sets and the words and, or, if ... then, not, for some, for every are not subject to rigorous definition. Syllogisms may be useful in training young minds to reason according to a pattern, but they do not lead to discovery of anything new.
Naive sets are common in informal logic, but the 19th and 20th centuries saw the rise and refinement of formal logic, initiated by John Stuart Mill (1806-1873) and the brilliant mathematician Gottlob Frege (1848–1925).
John Stuart Mill
Mill was a British empiricist who formulated five principles of inductive reasoning in his A System of Logic, Ratiocinative and Inductive (1843). These are 1 Direct method of agreement; 2 Method of difference; 3 Joint method of agreement and difference; 4 Method of residue and 5 Method of concomitant variations.
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| John Stuart Mill |
Mill's Nominalism
Mill believed that the early versions of nominalism held that "there is nothing general except names." He was mistaken in this, as anthropologists subsequently demonstrated.
The earliest nominalism, as reflected in ancient lexemes, involved general names and complex related experiences including attributes and related ideas. Lexemes are the basis for early scripts such as Thamudic. A solar lexeme such as Y or T or O represented a deified or divinely appointed ruler, his territory, his people, and all his resources such as water and gold. The Horite rulers are identified with the solar symbol Y: Yaqtan, Yishmael, Yitzak, Yisbak, Yacob, Yosef, Yeshua, etc. Other ancient lexemes include V, W and X.
Mill was correct that names of entities (nouns) have levels of meaning, or connotative variables. He showed that names denote either individuals or the attributes of individuals. A general name - white, for example - connotes an attribute and denotes all individuals that have that attribute. White connotes the attribute whiteness, and denotes all things that have that attribute.
Mill explored the variability of names through analysis of syllogisms. Here is one he used to illustrate that there is a connection between attributes connoted by terms, but the attributes are logically independent.
Consider the syllogism:
Man is mortal.In the syllogism "man" and "mortal" are attributes. Given Mill's claim that attributes are logically independent, the major premise - Man is mortal - adds nothing to the truth of the propositions concerning Socrates. What matters is the particulars - Socrates and the attributes man and mortal. In other words, deductive inference cannot advance knowledge. There is nothing new discovered by the syllogism.
Socrates is a man.
Ergo, Socrates is mortal.
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| George Boole |
Boolean logic is easy to understand. Begin with the idea that some statement (P) is either true or false. (He applies the law of the excluded middle). Then other statements can be formed, which are true or false, by combining these initial statements together using the fundamental operators And, Or and Not.
Boole's syllogistic reasoning did not place importance on the existential question, as happened with Aristotle. For Aristotle any proposition involving unicorns or satyrs, for example, would render an argument invalid. In Boole's view, anything named in a proposition does not imply that it in fact exists.
Consider the difference in their views:
Aristotle
All men are humans. This implies the existence of men.
No maple trees are oaks. This implies the existence of maple trees.
All unicorns are grand creatures. This does not imply the existence of unicorns.
Boole
All men are humans. This does not imply the existence of men. (Heidegger would love this!)
No maple trees are oaks. This does not imply the existence of maple trees.
All unicorns/satyrs are grand creatures. This does not imply the existence of unicorns/satyrs.
Gottlob Frege
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| Gottlob Frege |
His first major work was Begriffsschrift, published in 1879. In 1884, The Foundations of Arithmetic appeared, and this was followed by two volumes known by the German title Grundgesetze.
Frege's writings were largely ignored when first published, partly because the English philosophers were not capable of reading the original German. That would change with Bertrand Russell (1872–1970), whose knowledge of German made it possible for him to understand Frege and even to refute him.
Russell was introduced to the significance of Frege's work in logic through Giuseppe Peano (1858–1932), who he met at a conference in Paris in and both of whom saw the merit of his work.
In 1903, Russell wrote an appendix to The Principles of Mathematics in which he presented deficiencies in the assumptions that Frege made in Grundgesetze, which Russell recognized led to paradox or contradiction. Russell communicated this to Frege by letter and Frege's response was to accuse Russell of undermining the whole of mathematics.
Others influenced by Frege's work include Alfred North Whitehead (1861–1947), Ludwig Wittgenstein (1889–1951), Willard van Orman Quine (1908-2000), and the Logical Positivist Rudolf Carnap (1891–1970). In his Der logische Aufbau der Welt, Carnap attempted to apply the concepts of Principia Mathematica to his discourse about sense data, the external world and logic. Quine would go beyond the
Proper Names and Cognitive Value
Leibniz developed an approach to questions of necessity, possibility, contingency that served an important function within his metaphysics and epistemology. This is called modal metaphysics and it has important implications for logic. Carnap thought one could give a possible world semantics for the modalities of necessity and possibility by giving the valuation function a parameter that ranges over Leibniz's possible worlds. We will explore this under "Modal Logic."
Frege’s term for such a language -“Begriffsschrift” - may have been borrowed from a paper on Leibniz written by Adolf Trendelenburg, considered by Søren Kierkegaard "one of the most sober philosophical philologists I know."
Leibniz’s Problem: Why doesn’t ‘2+3 = 5’ reduce to ‘5 = 5’? Frege does not address this problem, but instead recasts it. Frege’s Puzzle is about the semantics of proper names.
(1) Hesperus is Hesperus. (Hesperus is a proper name for the planet Venus.)
(2) Hesperus is Phosphorus. (Phosphorus is a personification of the planet Venus.)
Each of these sentences is true. 'Hesperus' refers to the same object as 'Phosphorus' (the planet Venus). Nonetheless, (1) and (2), though synonymous, differ in what Frege called "cognitive value." Frege thus rejects John Stuart Mill's view that a proper name has no meaning above and beyond the object to which it refers. Frege develops this in his book Sense and Reference.
"I am David Kaplan", spoken by David Kaplan.
"He is David Kaplan", spoken by someone pointing at David Kaplan.
"David Kaplan is David Kaplan", spoken by anyone.
All express the same content and refer to the same individual. Yet each has a different cognitive value. Kaplan explains this by associating cognitive value with character rather than content, thus providing what seems a remedy to Frege's problem.
Frege's Concept-Writing
Frege's most famous work is Begriffsschrift (Concept-Writing: A Formal Language for Pure Thought Modeled on that of Arithmetic). It marked a turning point in logic. Here Frege demonstrates that true contents do not follow directly from other true contents. This requires a mediating system or what has come to be called axiomatic predicate logic. The book includes a rigorous treatment of functions and variables.
Frege declared nine of his propositions to be axioms, and justified them by arguing informally that, given their intended meanings, they express self-evident truths. In contemporary notation, these axioms are:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Frege wanted to show that mathematics grows out of logic, and devised techniques that took him beyond syllogistic and propositional logic to symbolic or formal logic. He accomplished this through his invention of quantified variables, which solved the problem of multiple generality.
Frege's symbolic logic was able to move beyond the logical constants and, or, if ... then ..., not, and some and all to more complex inferences. He prepared the way for the analysis of logical concepts by Bertrand Russell (theory of descriptions), Kurt Gödel's incompleteness theorems, and to Alfred Tarski's (1901–1983) theory of truth. All owe a great debt to Frege's work.
Other philosophers who have responded to Frege's work include Saul Kripe (Naming and Necessity, 1980), David Kaplan, Ruth Barcan Marcus, and Hilary Putnam. (We will look closer at Kripe's contribution when we study Modal Logic.)
Frege's most famous work is Begriffsschrift (Concept-Writing: A Formal Language for Pure Thought Modeled on that of Arithmetic). It marked a turning point in logic. Here Frege demonstrates that true contents do not follow directly from other true contents. This requires a mediating system or what has come to be called axiomatic predicate logic. The book includes a rigorous treatment of functions and variables.
Frege declared nine of his propositions to be axioms, and justified them by arguing informally that, given their intended meanings, they express self-evident truths. In contemporary notation, these axioms are:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Frege wanted to show that mathematics grows out of logic, and devised techniques that took him beyond syllogistic and propositional logic to symbolic or formal logic. He accomplished this through his invention of quantified variables, which solved the problem of multiple generality.
Frege's symbolic logic was able to move beyond the logical constants and, or, if ... then ..., not, and some and all to more complex inferences. He prepared the way for the analysis of logical concepts by Bertrand Russell (theory of descriptions), Kurt Gödel's incompleteness theorems, and to Alfred Tarski's (1901–1983) theory of truth. All owe a great debt to Frege's work.
Other philosophers who have responded to Frege's work include Saul Kripe (Naming and Necessity, 1980), David Kaplan, Ruth Barcan Marcus, and Hilary Putnam. (We will look closer at Kripe's contribution when we study Modal Logic.)
Frege's Intention
In his work Frege intended to set forth a system to isolate logical principles of inference. He perceived the need for this because he saw that only formalized logic could be applied to the sciences. No longer would an intuitive element be permitted as an assumption/premise. It would be isolated and represented separately as an axiom. The proof was to be logical and without gaps.
Frege was an important influence on Russell who influenced Quine. Russell's work inspired Quine to pursue logic. Quine wrote, “Russell’s name is inseparable from mathematical logic, which owes him much, and it was above all Russell that made that subject an inspiration to philosophers.” In 1962, Quine wrote to Russell that “Principia Mathematica was what, of all books, has influenced me the most.”
In his work Frege intended to set forth a system to isolate logical principles of inference. He perceived the need for this because he saw that only formalized logic could be applied to the sciences. No longer would an intuitive element be permitted as an assumption/premise. It would be isolated and represented separately as an axiom. The proof was to be logical and without gaps.
Frege was an important influence on Russell who influenced Quine. Russell's work inspired Quine to pursue logic. Quine wrote, “Russell’s name is inseparable from mathematical logic, which owes him much, and it was above all Russell that made that subject an inspiration to philosophers.” In 1962, Quine wrote to Russell that “Principia Mathematica was what, of all books, has influenced me the most.”
Wittgenstein also studied Frege's work and in 1911, he wrote to Frege concerning his solution to Russell’s paradox (see below). Frege invited him to Jena to discuss his views. The two engaged in a philosophical debate, and Wittgenstein reported that Frege “wiped the floor” with him. However, Frege was impressed with Wittgenstein and suggested that he study with Russell at Cambridge.
Russell's Paradox and Wittgenstein's solution
Russell's paradox Z = {x : x is not a member of x}
If Z is a member of Z how can it not be a member of Z?
Is Z a member of Z? If yes, then by the defining quality of Z, Z is not a member of itself. This forces us to declare that Z is not a member of Z. Then Z is not a member of itself and so, again by definition of Z, Z is a member of Z. What we have here is the following contradiction: Z is a member of Z if and only if Z is not a member of Z.
The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself. For let us suppose that the function F(fx) could be its own argument: in that case there would be a proposition 'F(F(fx))', in which the outer function F and the inner function F must have different meanings, since the inner one has the form O(f(x)) and the outer one has the form Y(O(fx)). Only the letter 'F' is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of 'F(Fu)' we write '(do): F(Ou). Ou = Fu'. That disposes of Russell's paradox. (Tractatus Logico-Philosophicus, 3.333)
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