Advanced Symbolic Logic

 

Boolean Schemata

 

 

Boolean Term Schemata

(i)  If a Boolean term schema is consistent, then in any universe containing a given object there is an interpretation of term letters that makes the schema come out true of that object.

 

(ii)  If a Boolean term schema is not valid, then in any universe containing a given object there is an interpretation of term letters that makes the schema come out false of that object.

 

(iii)  If one Boolean term schema fails to imply another, then in any universe containing a given object there is an interpretation of term letters that makes the one schema true of that object and the other false of it.

 

(iv)  If A1, . . . , An are Boolean term schemata and each of them separately is consistent, then in any universe containing distinct objects x1, . . . , xn there is an interpretation of term letters that simultaneously makes A1 true of x1, and A2 true of x2, and so on.

 

Boolean Existence Schemata

(v) A Boolean existence schema is valid if and only if its term schema is valid.

 

(vi)  A Boolean existence schema is consistent if and only if its term schema is consistent.

 

(vii)  One Boolean existence schema implies another if and only if the one term schema implies the other.

 

(viii)  A conjunction of Boolean existence schemata is consistent so long as each of them separately is consistent.

 

(ix)  A Boolean existence schema is implied by a conjunction of Boolean existence schemata only if it is implied by one of them in isolation.

 

Validity Tests for Boolean Existence Schemata

(i)   A Boolean existence schema is valid if and only if its term schema is valid.

 

(ii)  The negation of a Boolean existence schema is valid if and only if the term schema is inconsistent.

 

(iii)  An alternation of negations of Boolean existence schemata is valid if and only if one of those negations meets the above validity test.

 

(iv)  An existential conditional is valid if and only if the Boolean term schema in one of the existence schemata in the antecedent implies the Boolean term schema in the consequent.

 

(v)  A conjunction of Boolean statement schemata of any of the forms covered in (i)-(iv) is valid if and only if each of them comes out valid under the tests (i)-(iv).