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Zeroth-order logic is a term in popular use among practitioners for the subject matter otherwise known as boolean functions, propositional calculus, or sentential calculus. One of the advantages of this terminology is that it institutes a higher level of abstraction in which the more inessential differences between these various subjects can be subsumed under the pertinent isomorphisms. By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type X × Y → B and abstract type B × B → B in a number of different languages for zeroth order logic. These six languages for the sixteen boolean functions are conveniently described in the following order: B2 → B by means of the sequence of four boolean values (f(1,1), f(1,0), f(0,1), f(0,0)). Such a sequence, perhaps in another order, and perhaps with the logical values F and T instead of the boolean values 0 and 1, respectively, would normally be displayed as a column in a truth table. ( ) & = & 0 & = & mbox \ (x) & = & ilde & = & x' \ (x, y) & = & ildey lor x ilde & = & x'y lor xy' \ (x, y, z) & = & ildeyz lor x ildez lor xy ilde & = & x'yz lor xy'z lor xyz' end It may also be noted that is the same function as and , and that the inclusive disjunctions indicated for and for may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function is not the same thing as the function .
Logical operators Related topics | ||||||||
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