zeroth order logic
calculus \PMlinkescapephraseCalculus \PMlinkescapephrasecolumn \PMlinkescapephraseColumn \PMlinkescapephrasecolumns \PMlinkescapephraseColumns \PMlinkescapephrasecompact \PMlinkescapephraseCompact \PMlinkescapephrasedegree \PMlinkescapephraseDegree \PMlinkescapephraseformal logic \PMlinkescapephraseFormal logic \PMlinkescapephraselevel \PMlinkescapephraseLevel \PMlinkescapephraseorder \PMlinkescapephraseOrder \PMlinkescapephrasereference \PMlinkescapephraseReference \PMlinkescapephrasetranslation \PMlinkescapephraseTranslation
Note. This entry overlaps to some degree with other entries on boolean functions (http://planetmath.org/BooleanValuedFunction) and propositional logic (http://planetmath.org/PropositionalCalculus), but serves as a compact reference and a translation manual for several different styles of notation.
Zeroth order logic is a term in popular use among practitioners for 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 inessential differences among these subjects can be subsumed under the appropriate isomorphisms.
1 Propositional forms
Table 1. Propositional Forms on One Variable 1 0 0 0 false 0 1 not 1 0 1 1 true
Table 2 lists equivalent expressions for the sixteen functions of concrete type and abstract type in the same set of languages.
Table 2. Propositional Forms on Two Variables 1 1 0 0 1 0 1 0 0 0 0 0 false 0 0 0 1 neither nor 0 0 1 0 and not 0 0 1 1 not 0 1 0 0 and not 0 1 0 1 not 0 1 1 0 not equal to 0 1 1 1 not both and 1 0 0 0 and 1 0 0 1 equal to 1 0 1 0 1 0 1 1 not without 1 1 0 0 1 1 0 1 not without 1 1 1 0 or 1 1 1 1 true
The columns of Tables 1 and 2 are conveniently described in the following order:
In Table 1, describes each boolean function by means of the sequence of two boolean values .
In Table 2, describes each boolean function by means of the sequence of four boolean values .
Sequences of these forms, perhaps in another order and perhaps with the logical values F and T instead of the boolean values 0 and 1, would normally be displayed vertically in a truth table under the column head for .
Language lists the functions in the form , where the index is a bit string formed from the sequence of boolean values in .
Language notates the functions with an index that is the decimal equivalent of the binary numeral index in .
Notice that the sense of the binary and decimal codings is highly dependent on context. One needs to know the number of variables in the function and the sequence of points over which it is evaluated in order to decode the indices properly.
Language expresses the boolean functions in terms of two families of logical operations:
Language lists ordinary language expressions for the propositional forms. Many other paraphrases are possible, but these afford a sample of the simplest equivalents.
Language expresses the propositional forms in one of the several notations that are commonly used in formal logic.
|Title||zeroth order logic|
|Date of creation||2013-03-22 17:55:47|
|Last modified on||2013-03-22 17:55:47|
|Owner||Jon Awbrey (15246)|
|Last modified by||Jon Awbrey (15246)|
|Author||Jon Awbrey (15246)|