Talk:Zeroth order logic

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Zeroth order logic is an informal term that is sometimes used to indicate the common principles underlying the algebra of sets, boolean algebra, boolean functions, logical connectives, monadic predicate calculus, propositional calculus, and sentential logic.  The term serves to mark a level of abstraction in which the more inessential differences among these subjects can be subsumed under the appropriate isomorphisms.

Propositional forms on two variables

By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type \(X \times Y \to \mathbb{B}\!\) and abstract type \(\mathbb{B} \times \mathbb{B} \to \mathbb{B}\!\) in a number of different languages for zeroth order logic.

\(\text{Table 1.} ~~ \text{Propositional Forms on Two Variables}\!\)
\(\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\!\) \(\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\!\) \(\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\!\) \(\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\!\) \(\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\!\) \(\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\!\)
  \(x\colon\!\) \(1~1~0~0\!\)      
  \(y\colon\!\) \(1~0~1~0\!\)      

\(\begin{matrix} f_{0} \'"`UNIQ-MathJax1-QINU`"' It may also be noted that \((x, y)\!\) is the same function as \(x + y\!\) and \(x \ne y\), and that the inclusive disjunctions indicated for \((x, y)\!\) and for \((x, y, z)\!\) may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function \((x, y, z)\!\) is not the same thing as the function \(x + y + z\!\).

  • Language L5 lists ordinary language expressions for the sixteen functions. Many other paraphrases are possible, but these afford a sample of the simplest equivalents.
  • Language L6 expresses the sixteen functions in one of several notations that are commonly used in formal logic.



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Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.