Zerothorder logic is a term in popular use among practitioners for the subject matter otherwise known as boolean functions, monadic predicate logic, 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. In mathematics, a boolean function is usually a function F(b1, b2, ..., bn) of a number n of boolean variables bi from the twoelement boolean algebra B = {0, 1}, such that F also takes values in B. A function on an arbitrary set X taking values in B is...
In mathematics, a monadic logic is one that employs only unary relations. ...
In mathematical logic the propositional calculus or sentential calculus is a formal deduction system whose atomic formulas are propositional variables. ...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...
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. Table 1. Propositional Forms on Two Variables L_{1}  L_{2}  L_{3}  L_{4}  L_{5}  L_{6}  .  x :  1 1 0 0  .  .  .  .  y :  1 0 1 0  .  .  .  f_{0}  f_{0000}  0 0 0 0  ( )  false  0  f_{1}  f_{0001}  0 0 0 1  (x)(y)  neither x nor y  ~x & ~y  f_{2}  f_{0010}  0 0 1 0  (x) y  y and not x  ~x & y  f_{3}  f_{0011}  0 0 1 1  (x)  not x  ~x  f_{4}  f_{0100}  0 1 0 0  x (y)  x and not y  x & ~y  f_{5}  f_{0101}  0 1 0 1  (y)  not y  ~y  f_{6}  f_{0110}  0 1 1 0  (x, y)  x not equal to y  x + y  f_{7}  f_{0111}  0 1 1 1  (x y)  not both x and y  ~x ∨ ~y  f_{8}  f_{1000}  1 0 0 0  x y  x and y  x & y  f_{9}  f_{1001}  1 0 0 1  ((x, y))  x equal to y  x = y  f_{10}  f_{1010}  1 0 1 0  y  y  y  f_{11}  f_{1011}  1 0 1 1  (x (y))  not x without y  x => y  f_{12}  f_{1100}  1 1 0 0  x  x  x  f_{13}  f_{1101}  1 1 0 1  ((x) y)  not y without x  x <= y  f_{14}  f_{1110}  1 1 1 0  ((x)(y))  x or y  x ∨ y  f_{15}  f_{1111}  1 1 1 1  (( ))  true  1  These six languages for the sixteen boolean functions are conveniently described in the following order:  Language L_{3} describes each boolean function f : B^{2} → 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.
 Language L_{2} lists the sixteen functions in the form f_{i}, where the index i is a bit string formed from the sequence of boolean values in L_{3}.
 Language L_{1} notates the boolean functions f_{i} with an index i that is the decimal equivalent of the binary numeral index in L_{2}.
 Language L_{4} expresses the sixteen functions in terms of logical conjunction, indicated by concatenating function names or proposition expressions in the manner of products, plus the family of minimal negation operators, the first few of which are given in the following variant notations:
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 . Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...
 Language L_{5} lists ordinary language expressions for the sixteen functions. Many other paraphrases are possible, but these afford a sample of the simplest equivalents.
 Language L_{6} expresses the sixteen functions in one of several notations that are commonly used in formal logic.
See also
