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# Note for Soc 320 at WSU

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This 8 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Washington State University taught by a professor in Fall. Since its upload, it has received 13 views.

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Date Created: 02/06/15

Deductive Theory L N Gray August 29 2004 GENERAL PRINCIPLES OF AXIOMATIC THEORY 1 Introduce Primitive Terms undefined terms Primitive Terms are those that are not themselves defined but are understood by our audience Ideally these are terms in our language for which we share definitions and which are not controversial If they are to be measured directly at some point additional clarification may be required These are our starting points and should be considered carefully Care in the choice of primitive terms is not always evident however For example a term like quotattitudequot should probably not be chosen as a primitive term since there is considerable disagreement about what it stands for 2 Develop Derived Terms defined terms Derived terms are those defined through the use of primitive terms and logical or linguistic connectives Ideally they have no empirical content though in practice derived terms and definitions often function in a way similar to axioms or postulates In some cases there is considerable disagreement over whether a given statement in a theory is a definition or derived term or something more One of the most famous examples is Newton s definition of force here abbreviated as f ma force mass X acceleration Theorists have argued about the status of this statement since its original appearance around 1687 3 Present Axioms or Postulates The axioms or postulates are statements often of relation linking primitive terms and such derived or defined terms as may have been developed Ordinarily but not always these relations are assumed to represent knowledge on which we currently agree andor take as true Often these are considered untestable for the sake of theoretical development Any derivations that follow depend on the truth of these axioms so if any are false derivations are likely to be faulty 4 Theorems or derived relations Based on the primitive terms derived terms and the axioms additional derivations using an appropriate calculus eg mathematics symbolic logic etc are developed When all aspects have been clearly specified it should be possible to follow the logic of theoretical development clearly That makes this approach useful for teaching since the structures of theoretical argument can be made apparent so that mistakes can be shown Of course in practice this approach is rarely used as generally described here EXAMPLE I Classical Mechanics Physics Primitive undefined Terms 1 Distance 2 Mass 3 Time Derived defined Terms l velocity dX Vil dt Where X represents distance velocity is measured as a unit of distance per unit of time eg meter per second or msquot 2 Acceleration dv dZX a 447 2 dt dt Acceleration is measured as a unit of distance per unit of time per unit of time eg meter per second per second or msquotamp 3 Force dv F m 47 ma dt Force is measured as a unit of mass times a unit of distance per unit of time per unit of time eg kilogram meter per second per second or kgnrs Axiom postulate Gmlm2 F 44777 Newton s Law of GraVity r Where Rh and m2 stand for the masses of objects 1 and 2 r stands for the distance between the two objects and G is some constant number Note that the units of measurement for both sides of this equation must be the same so the units of G must be mikg39LsJ Substitutions Let Rh M the mass of Earth a constant Let m m2 the mass of an object on Earth Let r the distance from an object on the surface of Earth to the center of the planet a constant Deductions With substitutions definition 3 can be restated as F mg and the single axiom as Since equals can always be substituted for equals we have The n5 values cancel ie divide each side of the equation by m2 leaving Conclusion Each component of the right side of this equation is a constant term Thus the acceleration of gravity on Earth is a constant Since the same derivation could be used for any object on Earth quotEverything on Earth falls at the same ratequot EXAMPLE II A Sociological Example liberally adapted from Bruce H Mayhew and Roger L Levinger 1976 quotSize and the Density of Interaction in Human Aggregatesquot American Journal of Sociology 8286 110 Primitive undefined Terms 1 Behavior 2 Person 3 Interaction Derived defined Terms 1 AF count of instances of a behavior within some political unit over a known period of time For example a count of instances of behavior X within a state in a given year 2 S count of persons within the same political unit over the same period of time For example the population of a state in a given year 3 D SS 1 z SA The interaction potential of a count of persons is the number of permutations of pairs of persons self excluded 4 RD interaction realization The count of pairs of persons that meet in the defined political unit during the defined period of time 5 b Intensity A function of temporal economic and social conditions that together produce increased interaction in a population Axioms postulates 1 AF p R n The count of a behavior is a constant proportion p of a function of interaction realization 2 Run z Db Interaction realization can be approximated by interaction potential exponentiated by intensity Derivations 1 Under the assumption that the interaction potential represents an upper limit for interaction realization 5 s b s l 2 Employing a statistical approach b Ami pDieiI or lnA lnpblnDilnei Fi where eiis a residual or error term Many macro studies of behavior use rates rather than frequencies or counts A rate is defined as the count of the events divided by the population size for the geographic unit and time period This number is usually multiplied by a constant c so that we can talk about the number of events per 100000 persons or any other base we desire eg C t fE Rate of EventE w x 100 000 S Given Ami prei psiz bei39 we can divide both sides of the equation by Sidivide equals by equals A ch R 1 Si 5 D1 Si A Dquot5 x c Dbe Dquot5 x c so Fi 1 P 1 1 1 I Ding Ami CpDi 6139 or ln A Ri lncp b 5 ln Di ln e1 3 If intensity increases with the count of persons at an increasing rate b gt 5 then the rate of a behavior will increase with the count of persons 4 If intensity cannot be distinguished from the count of persons b 5 then rate is independent of the count of persons If an application is made to quotcriminal behaviorquot defined as appearing on the list of Federal quotindexquot offenses and some crimes eg quotviolentquot are identified as quotsocialquot and others eg quotpropertyquot are identified as quotnon socialquot then H1 k gt bp and H0 bV k5 Thus the rate of violent behavior should depend on the count of persons Since violent behavior is assumed to be social we cannot adjust completely for population size by dividing through by it ie taking rates The following figures suggest that this approach may have some value Crime Frequency Rate Crime Frequency by Population O FViolent I FProperty Power FOIiolent Power FProperty 10000000 1000000 100000 10000 1000 100 100000 1000000 10000000 100000000 Population Crime Rate by Population O ROIiolent I RProperty Power RVioent Power RProperty 10000 1000 100 3 O 10 100000 1000000 10000000 100000000 Population

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