1. LOGIC А ND PHENOMENOLOGY OF INVARIANCE Elena Dragalina – Chernaya National Research University Higher School of Economics National Research University Higher School of Economics22.06.2011 This study comprises research findings from the “Formal ontology: from phenomenology to logic” Project № 10- 01-0005 carried out within The Higher School of Economics’ 2011 Academic Fund Program carried out within The Higher School of Economics’ 2011 Academic Fund Program

2. Invariance Conditions for Logicality 1.“[general logic] treats of understanding without any regard to difference in the objects to which the understanding may be directed” I. Kant “[p]ure logic... disregard[s] the particular characteristics of objects” G.Frege 2. For Quine, logic cannot assume any special entities as existing ones. 3. A.Tarski ( “What are Logical Notions?”, 1966) proposed to call a notion logical if and only if “it is invariant under all possible one-one transformations of the world onto itself”

3. Permutation Invariance Permutation Invariance Criterion Permutations Invariance criterion: A notion is logical iff it is invariant under all permutations of the individuals in the “world” (or universe of discourse) A.Tarski (1966) Invariance under Isomorphism: An operator O is logical iff it is invariant under all isomorphisms of its argument- structures. Lindström’s (1966) generalization of Mostowski’s approach (1957)

4. vs Permutation Invariance Geometric Invariance vs Permutation Invariance Geometric Invariance: Geometric notion O is invariant under all one-one transformations of the geometrical space onto itself which preserve X F.Klein’s Erlanger Program (1872) A.Tarski: What would happen if we weakened X as much as possible, i.e., if we set no requirements on the transformations taken into account? Permutations Invariance: Notion O is invariant under all one-one transformations of space (the universe of discourse, the “world”) onto itself (or under all permutations of the “world”). Logic deals with our most general notions, notions which are invariant under all 1-1 transformations of the world onto itself.

5. “Overgeneration” Problem An operation is logical according to Tarski’s criterion if and only if it is definable in the infinitary language L∞, ∞ which allows conjunctions and disjunctions of any cardinality together with universal and existential quantification over sequences of variables of any cardinality ( V.McGee, 1996 ). L∞, ∞ is powerful enough to express substantial set-theoretic claims

6. E. Husserl considered formal ontology as an a priori science of objects in general. Logic is formal apophantic---domain of judgment ---uses linguistic categories Logic is formal ontology---domain of formal objects ---uses object categories E. Husserl believed that the transcendental justification of logic is possible only if we postulate a special region of abstract categorical objects. What is the nature of these objects? Categorical objects are classes (types) of isomorphism Formal ontologies

7. Husserl vs Tarski Formal ontology is “an a priori discipline that investigates all truths belonging to the essence of objectivity in general in formal universality” Husserl. Introduction to Logic and Theory of Knowledge. Lectures 1906/07 Logic as formal ontology studies “higher-level object formations” like set, cardinal number, quantity, ordinal number, etc. “People are”, in his view, “in the habit (a habit thousands of years old) of keeping the two bodies of knowledge in drawers far apart from one another. For thousands of years, mathematics has been considered a unique, special science, self-contained and independent like natural science and psychology, but logic, on the other hand, an art of thinking related to all special sciences in equal measure, or even as a science of forms of thinking not related any differently to mathematics than to other special sciences and not having any more to do with it than they” Husserl. Introduction to Logic and Theory of Knowledge. Lectures 1906/07

8. Generality vs Logicality  "The mark of a logical proposition is not general validity. To be general means no more than to be accidentally valid for all things. An ungeneralized proposition can be tautological just as well as generalized one» [Tractatus, 6.1231].  “The general validity of logic might be called essential, in contrast with the accidental general validity of such propositions as “All men are mortal”” [Tractatus, 6.1232]. What kind of validity is essential or necessary for Wittgenstein?

9. Internal properties and relations A property is internal if it is unthinkable that its object should not posses it. (This shade of blue and that one stand, eo ipso, in the internal relation of lighter to darker. It is unthinkable that these two objects should not stand in this relation.)” [Tractatus, 4.123].

10. Color exclusionproblem Color exclusion problem “Just as the only necessity that exists is logical necessity, so too the only impossibility that exists is logical impossibility. [Tractatus, 6.375]. For example, the simultaneous presence of two colours at the same place in the visual field is impossible, in fact logically impossible, since it is ruled out by the logical structure of colour (It is clear that the logical product of two elementary propositions can neither be a tautology nor a contradiction. The statement that a point in the visual field has two different colours at the same time is a contradiction.)” [Tractatus, 6.3751].

11. Chair analogy “For if the proposition contains the form of an entity which it is about, then it is possible that two propositions should collide in this very form. The propositions, "Brown now sits in this chair " and " Jones now sits in this chair" each, in a sense, try to set their subject term on the chair. But the logical product of these propositions will put them both there at once, and this leads to a collision, a mutual exclusion of these terms” Some Remarks on Logical Form (1929)

12. Phenomenology as Grammar “The proposition "at one place at one time there is only room for one color" is of course a masked proposition of grammar. Its negation is not a contradiction; rather it speaks against a rule of our accepted grammar. "Red and green don't go together at the same place" does not mean, they are never actually together, rather it means that it is nonsense to say that they are at the same place at the same time and therefore also nonsense to say they are never at the same place at the same time” (Wittgenstein, 1929)

13. Chair analogy-2 “We say three people can’t sit side by side on this bench; they have no room. This is a grammatical rule and states a logical impossibility. The propositions "three men can’t sit side by side on a bench a yard long" states a physical impossibility; and this example shows clearly why the two impossibilities are confused. (Compare the proposition "He is 6 inches taller than I" with "6 foot is 6 inches longer than 5 foot 6." These propositions are of utterly different kinds, but look exactly alike.)” (Wittgenstein, 1929)

14. Geometry as Grammar "the axioms-e.g.-of Euclidean geometry are disguised rules of syntax" "the geometry of visual space is the syntax of the propositions about objects in visual space“ (Wittgenstein, 1929)

15. Geometry as Phenomenology “The genuine criterion for the structure is precisely witch propositions make sense for it – not, witch are true. To look for these is the method of philosophy” “Physics differs from phenomenology in that it is concerned to establish laws. Phenomenology only establishes the possibilities” (Wittgenstein, 1929)

16. Logic as Geometry From the Klein’s ideology point of view a logic may be considered as a member of a family of logics whose notions are invariant under one- one transformations respecting some additional structures. If our logic takes into account a spectrum of invariance preserving some addition structures we may get various kinds of formal objects. Including in this spectrum invariance preserving «structure of color» in Wittgenstein’s sense, we can speak, for example, about «logic of color», considering "color" as a formal abstract object, a type of invariance of a special kind.