INVARIANCE CRITERION FROM ABSTRACT LOGICS TO LOGICS OF

INVARIANCE CRITERION FROM ABSTRACT LOGICS TO LOGICS OF ABSTRACT OBJECTS Elena Dragalina – Chernaya National Research University Higher School of Economics 25. 01. 2011 Bridge to Logic II Workshop: Logic and Philosophy This study comprises research findings from the “Formal ontology: from phenomenology to logic” Project № 1001 -0005 carried out within The Higher School of Economics’ 2011 Academic Fund Program

Abstract systems and abstract objects p p Abstract logics are theories of abstract systems Any two isomorphic structures represent the same abstract system. A system is considered to be abstract, if we do not know anything of its objects except the relations existing between them in the system. Formal ontologies as abstract logics are formal theories of relations. Abstract logics are theories of abstract objects Classes of isomorphism are abstract individuals of higher order. Abstract logics are not ‘empty’ in Kant’s sense. They deal with individuals of higher order, classes of isomorphic structures. Categorical objects as classes (types) of isomorphism are quite similar to indivisible species (automon eide) of Aristotle’s ontology.

Generalized quantifiers p p Classes of structures closed under isomorphism are generalized quantifiers. For example, Mostowski’s generalized quantifiers (1957) interpreted by classes of subsets of the universe attribute cardinality properties to the extensions of first-level unary predicates. Generalized quantifiers express Husserl’s mental properties and relations which, unlike physical, do not influence on other properties and relations, but exist because of them.

G. Frege: Existence as a property of concepts “ In the sentence 'There are men' we seem to be speaking of individuals that fall under the concept 'man', whereas it is only the concept 'man' we are talking about” Gottlob Frege - Dialogue with Punjer on Existence (written before 1884) p “Existence is a property of concepts” Gottlob Frege - Foundations of Arithmetic (1884)

G. Frege: Quantifiers as properties of concepts p p p F (µ) is a name of a unary first-level predicate. The object variable µ is not a name; it is just a place-holder. х F (х) is obtained by inserting the name F (µ) into the secondlevel predicate name х ψ (х) is a name denoting the universal quantifier, i. e. the unary second-level predicate which is true for precisely those first-level unary predicates which are true for every object. Again the first-level variable ψ is just a placeholder. х F (х) denotes the value of the universal quantifier applied to predicate F (µ). х F (х) is true iff F (µ) is true for every object.

Mostowski’s quantifiers as second-order properties Mostowski’s generalized quantifiers also attribute second-order cardinality properties. More precisely, a Mostowski’s quantifier is a function Q associating with every structure A a family Q (A) of subsets of the universe of A closed under permutations of the universe of A. p Thus Mostowski’s quantifiers perfectly satisfy the permutation invariance criterion by Alfred Tarski. p

Logical quantifiers A unary monadic quantifier is higher-order predicate witch yield truth values when applied to predicate extension. It is interpreted on a domain M by an operation Qm: µ (M)→ {T, F}. Qm is said to be invariant under permutation π iff for all permutations π, for all A from M, Qm (π (A)) = Qm (A). For example, we can interpret on a domain M by an operation Q defined by Q (A ) = T if A ≠ Ø, and Q (A ) = F if A =Ø (where A is a subset of M). The image π (A) of a non-empty subset A will always be another non-empty subset, and the empty set will always be mapped to the empty set. So is invariant under permutation, hence logical.

Nonlogical quantifiers Barwise and Cooper “Generalized quantifiers and Natural Language” (1981): Noun phrase are natural language quantifiers (1) Einstein (x) [x is among the ten greatest physicists of all time] (2) Einstein (x) [x is among the ten greatest novelists of all time] Sher, G. The Bounds of Logic. A Generalized Viewpoint. Cambridge, 1991 Barwise-Cooper’s quantifiers are nonlogical, because they are sensitive to the difference between elements in the domain.

Tarski’s thesis: “Our logic is logic of cardinality” Tarski’s examples of logical notions. Among individuals there are no such examples, among classes the logical notions are the universal class and the empty class. The only properties of classes of individuals which we can call ‘logical’ are “properties concerning the number of elements in these classes” Alfred Tarski “What are Logical Notions? ” What does cardinality have to do with logicality? Tarski’s thesis: “our logic is logic of cardinality” Alfred Tarski “What are Logical Notions? ”

Polyadic quantifiers Lindström’s quantifiers (1966) interpreted as second -order relations between first-order relations on the universe are polyadic. Binary examples of Lindström’s quantifiers are p syllogistics quantifiers, e. g. «all … are…» = {<X, Y>: X, Y U and X Y}, p Resher’s quantifiers QR = {<X, Y>: X, Y U and card(X) ‹ card (Y)}, p Hartig’s quantifiers QH = {<X, Y>: X, Y U and card (X)= card (Y)}.

Polyadic quantifiers vs. Tarski’s thesis In standard logical notation polyadic quantifier are not to be regarded as having an independent value, but interpreted as iterated unary quantifiers. p On the other hand, any iterated quantifier prefix may be viewed as a polyadic quantifier. Heterogeneous quantifier prefixes expressing properties of classes of pairs of individuals (binary relations) distinguish equicardinal relations. p

Z. Mikeladze’s Model Let us consider a simple model with the universe U= {a, b, c} Let set two binary relations on U, F 1= {(a, a), (a, b), (a, c)} and F 2= {(a, a), (b, b), (c, c)}. p p These relations have an identical number of elements. However x y. F 1(x, y) is not equivalent to x y. F 2(x, y), and х у. F 1(x, y) is not equivalent to х у. F 2 (x, y). In other words, binary quantifiers x y and х у distinguish equicardinal relations F 1 and F 2. Зураб Микеладзе. Об одном классе логических понятий» // Логический вывод, Москва: Наука, 1979, с. 296

Tarski’s thesis is not correct Tarski’s thesis of ‘our logic’ as ‘logic of cardinality’ may be fair for theory of monadic quantification (logic of properties of classes of individuals). But this thesis is not correct for theory of binary quantification (logic of properties of classes of pairs of individuals).

Ontology of cardinality vs. ontology of structures Polyadic quantifiers take into account not only cardinalities, but more refined formal features of the universe. Not only cardinalities, but also patterns of ordering of the universe have to be taken into account by logical conceptualization. p Logic with polyadic quantifiers is not an ontology of cardinality but a formal ontology of structures, types of ordering of the universe. p

J. van Benthem’s proposal p The permutation invariance criterion may be viewed as “only one extreme in a spectrum of invariance, involving various kinds of automorphisms on the individual domain” van Benthem, Johan 1989 “Logical Constants Across Varying Types”, Notre Dame Journal of Formal Logic 30, p. 320 p The invariance criterion generalized this way is wide enough to include logics of abstract objects. Thus abstract logics become logics of abstract objects quite similar to domain ontologies of ontological engineering.

Formal objects “Speaking in terms of objects we can say that formal objects are not just elements of formal structures, they are themselves formal structures” G. Y. Sher. Did Tarski Commit "Tarski's Fallacy"? The Journal of Symbolic Logic, Vol. 61, No. 2 (Jun. , 1996), p. 678 What kinds of abstract objects are formal? Why, for instance, Ludwig Wittgenstein considered a relation between “this shade of blue and that one” as structural or internal relation? In other words, why he considered a «structure of color» as a formal or logical structure?

Tractatus : ‘logical structure of colour’ «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]. “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].

Some Remarks on Logical Form (1929): “An atomic form cannot be foreseen” “we can only arrive at a correct analysis by, what might be called, the logical investigation of the phenomena themselves, i. e. , in a certain sense a posteriori, and not by conjecturing about a priori possibilities” RLF p “A proposition "reaches up to reality", and by this I meant that the forms of the entities are contained in the form of the proposition which is about these entities. For the sentence, together with the mode of projection which projects reality into the sentence, determines the logical form of the entities. . This remark, I believe, gives us the key for the explanation of the mutual exclusion of RPT and BPT [Red and blue are at the same place-time]” RLF "RPT" says that the color R is in the place P at the time T "BPT" says that the color B is in the place P at the time T

RLF: color-incompatibility claims as tautologies a posteriori p p “it will be clear to most of us here, and to all of us in ordinary life, that "RPT and BPT" is some sort of contradiction (and not merely a false proposition)” RLF “It is a characteristic of these properties [e. g. , the brightness or redness of a shade of color] that one degree of them excludes any other. One shade of colour cannot simultaneously have two different degrees of brightness or redness, a tone not two different strengths, etc. And the important point here is that these remarks do not express an experience but are in some sense tautologies” RLF

Conversation recorded by Waismann (1929) Wittgenstein remarks that statements about colour can be represented in geometrical terms— assigning them a position along certain colour axes. “Every statement about colours can be represented by means of such symbols. If we say that four elementary colours would suffice, I call such symbols of equal status elements of representation. These elements of representation are the ‘objects’ ”

Geometry of a colour space

Jaakko Hintikka’s interpretation p “the conceptual incompatibility of color terms can be turned into a logical truth simply by conceptualizing the concept of color as a function mapping points in a visual space into color space. This is an instructive example of how nonlogical but “analytic” truths can be interpreted as logical ones. Then they are uninformative (“tautological”) by the same token and in the same sense as logical truths” p Thus “nonlogical analytical truths sometimes turn out to be logical ones when their structure is analyzed properly” Hintikka, Jaakko 2009 “Logical vs. nonlogical concepts: an untenable dualism? ” in: Logic, Epistemology, and the Unity of Science, Springer Science+Business Media B. V. , p. 52

Game-theoretical semantics “the distinction between logical and non-logical constants is put to a new light by the approach known as game-theoretical semantics (GTS). The meaning of each logical constant is defined by the game rule that applies to a logical constant when it occurs in a given sentence”… “from the vantage point of GTS there is no difference between logical and nonlogical constants as long as their semantics can be captured game-theoretically” Hintikka, Jaakko 2009 “Logical vs. nonlogical concepts: an untenable dualism? ”

Tarski’s skeptical conclusion “Further research will doubtless greatly clarify the problem which interests us. Perhaps it will be possible to find important objective arguments which will enable us to justify the traditional boundary between logical and extra-logical expressions. But I also consider it to be quite possible that investigations will bring no positive results in this direction, so that we shall be compelled to regard such concept as ‘logical consequence’ and. . . ‘tautology’ as relative concepts which must, on each occasion, be related to a definite, although in greater or less degree arbitrary, division of terms into logical and extra-logical” Tarski “What are Logical Notions? ”