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 Failed to parse (Missing texvc executable; please see math/README to configure.):

and abstract type Failed to parse (Missing texvc executable; please see math/README to configure.): 
in a number of different languages for zeroth order logic.


Failed to parse (Missing texvc executable; please see math/README to configure.):
Failed to parse (Missing texvc executable; please see math/README to configure.): Failed to parse (Missing texvc executable; please see math/README to configure.): Failed to parse (Missing texvc executable; please see math/README to configure.): Failed to parse (Missing texvc executable; please see math/README to configure.): Failed to parse (Missing texvc executable; please see math/README to configure.): Failed to parse (Missing texvc executable; please see math/README to configure.):
  Failed to parse (Missing texvc executable; please see math/README to configure.): Failed to parse (Missing texvc executable; please see math/README to configure.):      
  Failed to parse (Missing texvc executable; please see math/README to configure.): Failed to parse (Missing texvc executable; please see math/README to configure.):      

Failed to parse (Missing texvc executable; please see math/README to configure.):

Failed to parse (Missing texvc executable; please see math/README to configure.):

Failed to parse (Missing texvc executable; please see math/README to configure.):

Failed to parse (Missing texvc executable; please see math/README to configure.):

Failed to parse (Missing texvc executable; please see math/README to configure.):

Failed to parse (Missing texvc executable; please see math/README to configure.):

Failed to parse (Missing texvc executable; please see math/README to configure.):

Failed to parse (Missing texvc executable; please see math/README to configure.):

Failed to parse (Missing texvc executable; please see math/README to configure.):

Failed to parse (Missing texvc executable; please see math/README to configure.):

Failed to parse (Missing texvc executable; please see math/README to configure.):

Failed to parse (Missing texvc executable; please see math/README to configure.):


These six languages for the sixteen boolean functions are conveniently described in the following order:

  • Language Failed to parse (Missing texvc executable; please see math/README to configure.):
describes each boolean function Failed to parse (Missing texvc executable; please see math/README to configure.): 
by means of the sequence of four boolean values, Failed to parse (Missing texvc executable; please see math/README to configure.): 
Failed to parse (Missing texvc executable; please see math/README to configure.): 
Failed to parse (Missing texvc executable; please see math/README to configure.): 
Failed to parse (Missing texvc executable; please see math/README to configure.): 

  Such a sequence, perhaps in another order, and perhaps with the logical values Failed to parse (Missing texvc executable; please see math/README to configure.):

and Failed to parse (Missing texvc executable; please see math/README to configure.): 
instead of the boolean values  and  respectively, would normally be displayed as a column in a truth table.
  • Language Failed to parse (Missing texvc executable; please see math/README to configure.):
lists the sixteen functions in the form Failed to parse (Missing texvc executable; please see math/README to configure.): 
where the index  is a bit string formed from the sequence of boolean values in Failed to parse (Missing texvc executable; please see math/README to configure.): 


  • Language Failed to parse (Missing texvc executable; please see math/README to configure.):
notates the boolean functions Failed to parse (Missing texvc executable; please see math/README to configure.): 
with an index  that is the decimal equivalent of the binary numeral index in Failed to parse (Missing texvc executable; please see math/README to configure.): 


  • Language Failed to parse (Missing texvc executable; please see math/README to configure.):
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:

Failed to parse (Missing texvc executable; please see math/README to configure.):

It may be noted that Failed to parse (Missing texvc executable; please see math/README to configure.):

is the same function as  and  The inclusive disjunctions indicated for Failed to parse (Missing texvc executable; please see math/README to configure.): 
and for Failed to parse (Missing texvc executable; please see math/README to configure.): 
may be replaced with exclusive disjunctions without affecting the meaning, since the terms disjoined are already disjoint.  However, the function Failed to parse (Missing texvc executable; please see math/README to configure.): 
is not the same thing as the function 
  • Language Failed to parse (Missing texvc executable; please see math/README to configure.):
lists ordinary language expressions for the sixteen functions.  Many other paraphrases are possible, but these afford a sample of the simplest equivalents.
  • Language Failed to parse (Missing texvc executable; please see math/README to configure.):
expresses the sixteen functions in one of several notations that are commonly used in formal logic.

Translations

Syllabus

Focal nodes

Peer nodes

Logical operators

Related topics

Relational concepts

Information, Inquiry

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Document history

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.