The tasks of logic Why we need more versatile tools Philosophy and logic 2013 Kyiv 25 May

The task of logic – the classical view To analyse reasoning or arguments. Requires two things: that logic can correctly represent the components of reasoning (thoughts) and arguments (sentences), and that it can also correctly represent correct dynamic flows of such components 2

The task of logic – in common practice A tool for the analysis of concepts and definition of terms Requires two things: that logic can correctly represent entities at the level of concepts and words, and that it can also correctly represent how these combine to form entities at the next level (thoughts and sentences) 3

A useful table sphereelementstaticsdynamics languagewordsentencetext worldobjectfactprocess mindideajudgem.thinking contentconceptpropos.reasoning General problem: what sort of morphisms exist between the spheres? 4

Inessential limitations of logic Cannot represent modalities (add operator) Cannot express finitude (generalise quantifier) Cannot represent finitude (go second order) Cannot represent dependencies (branch quantifiers) Cannot represent X (add new symbol) 5

Essential limitation of logic 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. 6

Lack of isomorphism between language 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. 7

Lack of isomorphism between language 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). 8

Is a logic of belief possible? Beliefs (like thoughts) no doubt have content Some (like Isaac Levi) think that these contents form a Boolean structure If so, they can be represented in ”propositional logic”, but it doesn’t follow that they are propositional More likely, though, negation is problematic and we need some other representation 9

Is a logic of belief possible? 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 10

Beliefs have properties along many 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. 11

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. 12

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. 13

Tactical question 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. Solomonian answer Do both, and see how they interact. 14

The measurement of power 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! 15

Powers as a vector p i = power of individual i c ij = influence of i over j P = n x 1 matrix of powers C = n x n matrix of influences Then solve the equation: CP = P The same idea applicable to other concepts with apparently circular definition, like coherence. 16

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 17

Ultrafilters (”large subsets”) 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. 18

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

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