Luis VILLEGAS-FORERO, Janusz MACIASZEK

More documents

Recommendations

Info

124 LUIS VILLEGAS-FORERO, JANUSZ MACIASZEK is a Tarski logical entity in the ontological system O iff. for all bijection r: , •r = . 2.2. Categories of logical entities (Klog, τ,F ) τ∈Τ , is the family of categories of logical entities such that for all τ ∈ Τ, Klog, τ, F , is the set of logical entities of this type, relatively to the frame F. This set is to be determined relatively to each frame. As we said in the introduction, if changes of frames or universes are permitted, each logical notion is to be the same, but the specific entity which falls under the notion can vary. So, the abstract basis for building each logical category is the same in all possible frames, but its inhabitants may be different. Moreover, if a quasi-intensional ontology is given, then also quasi-intensional logical entities appear, which were not supposed directly by Tarski. First we list some categories of logical entities in Tarski's paper: 1. Individuals “... we can start with individuals, with objects of the lowest type,... There are no logical notions of this type, simply because we can always find a transformation of the world onto itself where one individual is transformed into a different individual.” (Tarski [31]: 150): Klog,e , F = ∅. 2. Truth-value based entities “If we proceed to the next level, to classes of individuals,..., there are exactly two classes of individuals which are logical: the universal class and the empty class. Only these two classes are invariant under every transformation of the universe onto itself.” (Tarski [31]: 150) 11 . 11 The editor of Tarski's paper, J. Corcoran, remarks on page 150, note 6: “In his Buffalo lecture, Tarski indicated that the present remarks apply to 'notions' taken in the narrow sense of sets, classes of sets, etc., but that the truth-functions, quantifiers, relation-operators, etc. of Principia Mathematica can be construed as notions in the narrow sense and, so construed, the present remarks apply equally to them. For example, construing the truth-values T and F as the universe of discourse and the null set leads immediately to construing truth-functions as (higherorder) notions”. (Tarski [11]: 150, note 6. from J. Corcoran (ed.) ).
T ARSKI ON LOGICAL ENTITIES 125 Let Ttv, the set of types of truth-value based entities, be the smallest set such that: (1) t ∈ Ttv. (2) If , ∈ Ttv, then < , > ∈ Ttv 12 . Then, for all t ∈ Ttv, Klog,t, F = Kt, F . 3. Relations between individuals and, in general, between entities of the same type that: “If we... consider binary relations, a simple argument shows that there are only four binary relations which are logical in this sense: the universal relation..., the empty relation..., the identity relation..., and its opposite, the diversity relation... If you consider ternary relations, quaternary relations, and so on, the situation is similar: for each of these you will have a small finite number of logical relations.” (Tarski [31]: 150). f ∈ Klog, < < ,...,< , t>...>, F if f is a n-ary (n≥2) relation such i) for each 1, 2, ... , n: , f( 1)( 2)...( n) = T iff 1= 2= ... = n (n-ary Identity) or ii) for each 1, 2, ... , n: , f( 1)( 2)...( n) = T iff for each i, j, i ≠ j (n-ary Difference ) or iii) for each 1, 2, ... , n: , f( 1)( 2)...( n) =T (n-ary universal relation) or iv) for each 1, 2, ... , n: , f( 1)( 2)...( n) = F(n-ary empty relation). 4. Cardinality properties “It turns out that the only properties of classes (of individuals) that are logical are properties concerning the number of elements in these classes. That a class consists of three elements, or four elements... that is finite , or infinite - these are logical notions and are essentially the only logical notions on this level.” (Tarski [31]: 151). 12 Obviously, Τtv = {t} ∪ Τtff ∪ ΤtFF where Τtff, ΤtFF are, as above, the set of types of truth-functions and truth-functionals, respectively.
Page 1 and 2: Logica Trianguli, 1, 1997, 115-141
Page 3 and 4: T ARSKI ON LOGICAL ENTITIES 117 tie
Page 5 and 6: T ARSKI ON LOGICAL ENTITIES 119 the
Page 7 and 8: T ARSKI ON LOGICAL ENTITIES 121 〈
Page 9: T ARSKI ON LOGICAL ENTITIES 123 ω
Page 13 and 14: T ARSKI ON LOGICAL ENTITIES 127 6.
Page 15 and 16: T ARSKI ON LOGICAL ENTITIES 129 12)
Page 17 and 18: T ARSKI ON LOGICAL ENTITIES 131 eac
Page 19 and 20: T ARSKI ON LOGICAL ENTITIES 133 tra
Page 21 and 22: T ARSKI ON LOGICAL ENTITIES 135 “
Page 23 and 24: T ARSKI ON LOGICAL ENTITIES 137 pre
Page 25 and 26: T ARSKI ON LOGICAL ENTITIES 139 the
Page 27: T ARSKI ON LOGICAL ENTITIES 141 [34

Luis VILLEGAS-FORERO, Janusz MACIASZEK

Create successful ePaper yourself

Delete template?

Save as template?