The tasks of logic Why we need more versatile tools

1. The tasks oflogicWhyweneedmore versatile tools Philosophy and logic 2013 Kyiv 25 May 2013

2. The task oflogic – the classicalview • To analyse reasoning or arguments. Requirestwothings: • thatlogiccancorrectlyrepresent the componentsof reasoning (thoughts) and arguments (sentences), and • that it canalsocorrectlyrepresentcorrectdynamicflowsofsuchcomponents

3. The task oflogic – in common practice • A tool for the analysisofconcepts and definition of terms Requirestwothings: • thatlogiccancorrectlyrepresententities at the levelofconcepts and words, and • that it canalsocorrectlyrepresenthowthesecombineto form entities at the nextlevel (thoughts and sentences)

4. A useful table sphere elementstatics dynamics language word sentencetext worldobject fact process mind ideajudgem. thinking content concept propos. reasoning General problem: what sort ofmorphismsexistbetween the spheres?

5. Inessential limitations oflogic • Cannotrepresentmodalities (add operator) • Cannot express finitude (generalisequantifier) • Cannotrepresentfinitude (go second order) • Cannotrepresentdependencies (branchquantifiers) • Cannotrepresent X (add new symbol)

6. Essential limitation oflogic There is no formal language where a categorical set of arithmetical sentences can be formulated and the syntactic and semantic concepts of consequence coincide. In other words: either content cannot be represented in its entirety, or reasoning cannot. This is a also conflict between the two most fundamental concepts of epistemology, knowing and thinking, the static and the dynamic aspect of knowledge.

7. Lack ofisomorphismbetweenlanguage and the mind for ”belief” • If a belief is acquired by seeing or otherwise experiencing something directly, then it has a fullness that cannot be exhaustively described by any number of sentences because there are no words for most of the concepts involved.

8. Lack ofisomorphismbetweenlanguage and the mind for ”belief” • If a belief is acquired by reading or being told something, it is already given in a linguistic form and therefore presumably propositional in character. It has been depleted in relation to the original experience (if there was one).

9. Is a logicofbeliefpossible? • Beliefs (like thoughts) no doubthavecontent • Some (like Isaac Levi) thinkthatthesecontents form a Booleanstructure • If so, theycan be represented in ”propositionallogic”, but it doesn’tfollowthattheyarepropositional • Morelikely, though, negation is problematic and weneedsomeother representation

10. Is a logicofbeliefpossible? • The simplest solution would be to use a Heyting algebra instead, but it would not really be informative • Preferably, salient aspects of beliefs should be taken into account

11. Beliefshavepropertiesalongmany dimensions • Certainty (of content) • Justification (logical relation to other beliefs) • History of coming into being (pedigree) • Robustness (sensitivity to further inquiry) • Satisfactoriness (relation to holder, determining willingness to further inquiry) • Etc.

12. Fundamental question of the general theory of knowledge Is knowledge is just a species of belief (whether those beliefs that happen to be true and justified, or singled out in any other way); can proper knowledge be studied without regard to beliefs in general? Suspected answer: Knowledge is a limiting case, only the surface of a deep sea of beliefs (or a hyperplane in the space of beliefs, for those with a more formal mind), and one cannot well understand the ripples on the surface unless one studies what goes on beneath.

13. Problem to be studied Which is the nature and structure of belief states? (We know fairly well that they cannot be sets of propositions as modelled in a formal language.) And how can we model them if we want to study them more precisely? Suspicions Many problems in belief revision are spurious and self-generated, resulting from inappropriate modelling. Probability is too one-dimensional for modelling the partiality of beliefs; it cannot, for example, render the disposition to react to new information, i.e. robustness.

14. Tacticalquestion Shall one concentrate on states of full belief, as Isaac Levi does, rather than on belief states in general? Pro: full beliefs are a special case and it is therefore easier to start with them. Contra: one may be tempted to seek ad hoc solutions to problems about full belief, whose drawbacks do not become obvious unless one studies the general case. Solomoniananswer Do both, and see how they interact.

15. The measurementofpower • Power consists in exerting influence over other people • But it counts more if you exert influence over powerful people, in proportion to their power • This is a circular definition, but that is not an argument for its inadmissability • Use a vector/matrix equation!

16. Powers as a vector • pi = powerofindividual i • cij = influenceof i over j • P = n x 1 matrix ofpowers • C = n x n matrix ofinfluences Thensolve the equation: CP = P The same ideaapplicabletootherconceptswithapparentlycircular definition, like coherence.

17. Voting, or preference aggregation • Traditional: function from sequence of individual preferences to group preference • Alternative: look at decisive groups, i.e. groups that can force their will if united • They form an ultrafilter

18. Ultrafilters (”largesubsets”) Set of subsets of a given set satisfying: • Empty set does not belong • Closed under intersection • Closed under supersets • Either a set or its complement belongs Ultrafilters are either principal (= all subsets containing a given element) or free.

19. Ultrafilters The ultrafilter technique is applicable to all sorts of aggregation problems, and also for the construction of infinitesimals in mathematics.

20. The philosopher’stoolbox • Logics • Set theory and transfinite numbers • Lattices • Algebraic structures • Vectors and linear spaces • Topology, metric spaces • Graph theory • Category theory??