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Zeroth order logic

This page belongs to resource collections on Logic and Inquiry.

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 variablesEdit

By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type   and abstract type   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:

  • Language   describes each boolean function   by means of the sequence of four boolean values,          Such a sequence, perhaps in another order, and perhaps with the logical values   and   instead of the boolean values   and   respectively, would normally be displayed as a column in a truth table.
  • Language   lists the sixteen functions in the form   where the index   is a bit string formed from the sequence of boolean values in  
  • Language   notates the boolean functions   with an index   that is the decimal equivalent of the binary numeral index in  
  • Language   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 be noted that   is the same function as   and   The inclusive disjunctions indicated for   and for   may be replaced with exclusive disjunctions without affecting the meaning, since the terms disjoined are already disjoint. However, the function   is not the same thing as the function  

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

TranslationsEdit

SyllabusEdit

Document historyEdit

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.