E. Schroder, Der Operationskreis des Logikkalkuls, B. G. Teubner, Leibzig, 1877.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....# such that, for each a Bval, #(a) 1 if and only if a(x) 1. For # 1 , # 2 Bfun, we write # 1 # 2 to denote the function # such that, for each a Bval, #(a) 1 if and only if both # 1 (a) 1 and # 2 (a) 1. A variable is restricted away using Schroder s elimination principle [57]: #[1 x] #[0 x] Note that existential quantification is both monotonic and extensive on Bfun . The other Boolean connectives and quantifiers are handled similarly. For notational convenience, when X VI , we inductively define if X = #; if x X . Bfun consists ....

E. Schroder. Der Operationskreis des Logikkalkuls. B. G. Teubner, Leibzig, 1877.

....# such that, for each a # Bval, #(a) 1 if and only if a(x) 1. For # 1 , # 2 # Bfun, we write # 1 # # 2 to denote the function # such that, for each a # Bval, #(a) 1 if and only if both # 1 (a) 1 and # 2 (a) 1. A variable is restricted away using Schroder s elimination principle [21]: #x . # def = #[1 x] # #[0 x] Note that existential quantification is both monotonic and extensive on Bfun. The other Boolean connectives and quantifiers are handled similarly. Pos # Bfun consists precisely of those functions assuming the true value under the everything is true ....

E. Schroder. Der Operationskreis des Logikkalkuls. B. G. Teubner, Leibzig, 1877.

....# such that, for each a # Bval, #(a) 1 if and only if a(x) 1. For # 1 , # 2 # Bfun, we write # 1 # # 2 to denote the function # such that, for each a # Bval, #(a) 1 if and only if both # 1 (a) 1 and # 2 (a) 1. A variable is restricted away using Schroder s elimination principle [22]: #x . # def = #[1 x]##[0 x] Note that existential quantification is both monotonic and extensive on Bfun. The other Boolean connectives and quantifiers are handled similarly. Pos # Bfun consists precisely of those functions assuming the true value under the everything is true assignment, ....

E. Schroder. Der Operationskreis des Logikkalkuls. B. G. Teubner, Leibzig, 1877.

....# such that, for each a # Bval, #(a) 1 if and only if a(x) 1. For #1 , #2 # Bfun, we write #1 # #2 to denote the function # such that, for each a # Bval, #(a) 1 if and only if both #1 (a) 1 and #2 (a) 1. A variable is restricted away using Schroder s elimination principle [37]: #x . # def = #[1 x] # #[0 x] Existential quantification is both monotonic and extensive on Bfun. The other Boolean connectives and quantifiers are handled similarly. Pos # Bfun consists precisely of those functions assuming the true value under the everything is true assignment, ....

E. Schroder. Der Operationskreis des Logikkalkuls. B. G. Teubner, Leibzig, 1877.

....as shown in figure 1. We will not present the implementation of disj separately, as it is simply the dual of conj. The function implies, which computes node1 node2, is also very similar and will not be shown. 4 Restriction A variable is restricted away using Schroder s elimination principle [6]: conj(node1; node2) def if node1 = True then node2 elseif node2 = True then node1 elseif node1 = False then False elseif node2 = False then False elseif node1 var node2 var then mknd(node1 var ; conj(node1 then ; node2) conj(node1 else ; node2) elseif node1 var node2 var then ....

E. Schroder. Der Operationskreis des Logikkalkuls. B. G. Teubner, Leibzig, 1877.

.... zs) false d 1 (xs ; ys ; zs) xs (ys zs) d 2 (xs ; ys ; zs) xs (ys zs) ys :zs) xs :ys zs) ys (xs zs) This turns out to be the required fixpoint and it immediately leads to the solution for q : 1 The first explicit statement of the principle appears to be by Schroder ([30] page 22) who derived it from Boole s principle of development : F = F [x 7 false] x ) F [x 7 true] x ) sometimes referred to as Boole s Expansion Theorem, or Shannon expansion ) Boole considered disjunction to be exclusive, so the elimination principle would have made little sense to ....

E. Schroder. Der Operationskreis des Logikkalkuls. B. G. Teubner, Leibzig, 1877.

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E. Schroder, Der Operationskreis des Logikkalkuls, B. G. Teubner, Leibzig, 1877.

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E. Schroder, Der Operationskreis des Logikkalkuls, B. G. Teubner, Leibzig, 1877.

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E. Schroder, Der Operationskreis des Logikkalkuls, B. G. Teubner, Leibzig, 1877.

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E. Schroder, Der Operationskreis des Logikkalkuls, B. G. Teubner, Leibzig, 1877.

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