Luis VILLEGAS-FORERO, Janusz MACIASZEK

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122 LUIS VILLEGAS-FORERO, JANUSZ MACIASZEK constituents of f. FC 1 (f)= IC(f) — OBC 1 (f), is the set of level 1 functional constituents of f. 2. OBC2 (f) = ( gi∈FC 1 ∪ IC(gi)) ∩ OBO is the set of level 2 (f) objectual constituents of f. FC2 (f) = ( gi∈FC 1 ∪ IC(gi)) — OBC (f) 2 (f) is the set of level 2 functional constituents of f. 3. OBC3 (f) = ( gj∈FC 2 ∪ IC(gj)) ∩ OBO is the set of level 3 (f) objectual constituents of f. FC3 (f) = ( gj∈FC 2 ∪ IC(gj)) — OBC (f) 3 (f) is the set of level 3 functional constituents of f. of f. of f. m = Min {k∈N⏐OBC k (f)= ∅ & FC k (f)= ∅}. We define now: m-1 OBC(f) = ∪ OBC i=1 i (f), as the set of basic constituents of f. OBCe(f) = OBC(f) ∩ Ke, F, as the set of individual constituents OBCt(f) = OBC(f) ∩ Kt, F, as the set of truth-value constituents m-1 FC(f) = ∪ FC i=1 i (f), as the set of functional constituents of f. C(f) = OBC(f) ∪ FC(f), as the set of all constituents of f. Constituents of q-intensions. If f: , then ω C(f) = ∪ {pri(s)⏐s∈SEQ}∪Rang(f) ∪ ( ∪ C( )). i=1 ∈Rang(f)
T ARSKI ON LOGICAL ENTITIES 123 ω Obviously, ∪ {pri(s)⏐s∈SEQ} = E. i=1 Proposition. If y ' ∈ ENTO , then ∪ C( ) ⊆ C( '). ∈C( ') 1.5.2. Replacement operation 10 •: ENTO ×{r : ⏐r ∈ BIJEC} is an operation such that: 1. if a : e, then a•r = r(a). 2. if v : t, then v•r = v. 3. if f : < , < ,...,< n, >...> is a n-ary (n ≥1) function and , ,..., n, are non necessarily different types, then for all : , : ,..., n : n, f( )( )...( n) = f•r ( •r)( •r)...( n•r). Thus • is a replacement operation such that, for each bijection of the universe E onto itself, •r is the structured entity of the same type that , which results of replacing down-top each component, ', of by its value '•r, in such a way that C( •r)={ '•r⏐ '∈C( )}. 2. Tarski-like logical entities 2.1. Logicality criterion “In the extreme case, we would consider the class of all oneone transformations of the space, or universe of discourse, or “world”, onto itself. What will be the science which deals with the notions invariant under this widest class of transformations? Here we will have very few notions, all of a very general character. I suggest that they are the logical notions, that we call a notion 'logical' if it is invariant under all possible one-one transformation of the world onto itself” (Tarski [31]: 149). 10 This operation does correspond to the method, informally characterised by Tarski ([29]: 152), of chain transformations, starting with “transformations of the 'world' onto itself”.
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Luis VILLEGAS-FORERO, Janusz MACIASZEK

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