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Quantitative Epistemology

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Quantitative Epistemology & the Structure ofLogical KnowledgeMichael BaranySubmitted 9 April, 20091 Logical KnowledgeThe Pythagorean theorem states that in any right triangle the legs of lengtha and b and hypotenuse of length c are related by the equationa2+ b2= c2. (1)Equation 1 represents an element of mathematical knowledge. More gen-erally, the Pythagorean theorem is an element of logical knowledge. But theequation itself does not make this so.The Pythagorean theorem is an element of logical knowledge because ofthe way it is established and the way it is used. It is established througha step-by-step logical derivation from a collection of foundational principles.It is used in combination with other established elements of knowledge ina particular way to establish further elements of logical knowledge. Logicalknowledge is recognizable less by the contents of its assertions than by thelogical way in which they are connected.How, then, is the Pythagorean theorem connected to other statements?In Euclid’s Elements, it appears as the 47thproposition of Book I, after a longsequence of definitions, postulates, common notions, and other propositions.Euclid does not hold a monopoly on proofs of the theorem, however. Loomis(1968) compiles over 250 pages of proofs and demonstrations of the theorem.When taught to school children, the theorem appears after explanations ofsuch things as right triangles, side lengths, and square numbers. In more1advanced mathematical settings, it is often stated without proof, and is usedas a foundational idea with which to establish other statements.In this essay, we shall seek an account of logical knowledge which appliesto all these different settings for the Pythagorean theorem, and, indeed, toall works which might be called logical.1In so doing, we shall treat logicalknowledge as a system, and ask what features of the system are propertiesof the system itself, as opposed to the isolated meanings and referents ofthe statements therein. Roughly speaking, we shall enquire after structurerather than content.2A starting point for our analysis of systems of logical knowledge comesfrom the field of proof theory.3Accounts typically begin with a logical systemdivided into individual statements or elements.4How a work is so dividedcan depend on one’s analytic framework and goals. Statements can be logicalsentences, theorems, proofs, or other units of an argument. At a minimum,statements should contain some indicative assertion.5Proof theorists thenstudy the relationships between these statements. Such relationships takea variety of forms. They could involve inference, explanation, definition,dependency, or some other form of logical connection. Many include logicalprinciples such as implication and negation in their structural analyses.6Structural accounts of logical knowledge are relevant to many consider-ations. In mathematics, they give a means of understanding and compar-ing different proof arguments. One might ask, for instance, how differentproofs of the Pythagorean theorem using different techniques are nonethe-less related. Conceptually, they give a means of analysing the relationshipsbetween different abstract concepts and conclusions.7Thus, one can relatepoints, angles, lines, triangles, and the theorems about them in a single net-work of knowledge and understanding. Pedagogically, one might ask which1Importantly, we do not make the logicist claim that such works are reducible to logicalprinciples, only that they carry a logical structure.2On this distinction, see Suppes, 1969, 373.3On proof theory, see Hunter, 1971, 7–8; Avigad and Reck, 2001; Paggiolesi, 2009.4cf Carnap, 1937, 1–2; Tarski, 1956, 30. We shall prefer the term statements.5Lepore,2000, 6. Our use of ‘statement’ is slightly broader than Lepore’s.6cf Tarski, 1956, 30–37.7Field, 1984, 512, makes these relationships a central feature of mathematical knowl-edge. Van Bendegem and Van Kerkhove, 2009, frame these relational concepts in termsof argumentation.2presentational approaches are simplest, or which concepts are most centralto an area of study.8Structural analyses also play an important role in boththe philosophy and practice of proof mechanization, proof languages, and thefoundations of mathematics.9The account in this essay will focus on two closely tied logical relations.The first is that of dependency. Dependency can take many forms in differentlogical frameworks. One might be primarily interested in the mathematicalclaims needed to establish the Pythagorean theorem, or the logical principlesused in its proof. One might also be interested in the linguistic, semantic, orconceptual prerequisites for understanding the theorem or its proof.10Ouraccount will not distinguish between these different forms of dependency, andmay be applied to whichever suits one’s particular analytic goals. A logicalwork is logical, in part, because of the network of logical interdependenciesamong its statements. These dependencies are a direct consequence of thestructurally inferential character of logical knowledge.11However one parsesa logical work, one finds that certain statements depend on other logicallyanterior statements.The second logical relationship we shall consider is that of citation. Thepresentational process of citation connects the statements of a logical workto their dependencies. Citations are the explicit connections which showthe implicit relations of dependency permeating a logical work. We say astatement A cites another statement B if B is cited in order to justify A. Asbefore, different analytic contexts may call for different notions of dependencyand citation.In this essay, we understand the structure of logical knowledge in terms ofits networks of citations and dependencies. Each element of logical knowledgefits into a logical work—a collection of statements, each themselves elementsof logical knowledge. To each statement we associate a collection of otherstatements upon which it depends and another collection of statements whichit cites.The citation and dependency relations between statements form struc-8cf studies of definitional systems in Kieffer, 2007, 49–52, 59–63; Friedman and Flagg,1990; and Kieffer, et al. 2008.9See Azzouni, 2009; Bridges and Reeves, 1999; Eder, 1992; Friedman, 2005; Kino, etal., 1970; Longo, 2003; MacKenzie, 2001; Pelc, 2009; Rav, 1999, 2007.10On semantic and logical relations, see Tarski, 1956, 401–420.11Audi, 1998, 157.3tures of


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