Trivial Truth

In order to balance convenience for the programmer with precision, CORE has a notion of when a sentence is trivially true.

Trivially True

Suppose P is a sentence. To determine whether P is trivially true, search for the first applicable rule from the beginning in the following list:

  1. If P is of the form A -> B for sentences A and B, P is trivially true if A trivially implies B.

  2. If P is of the form A <-> B, P is trivially true if A trivially implies B and B trivially implies A

  3. If P is of the form A & B, P is trivially true if A is trivially true and B is trivially true.

  4. If P is of the form A | B, P is trivially true if A is trivially true or B is trivially true.

  5. If P is of the form *X(A(X)), P is trivially true if A is trivially true.

  6. If P is of the form ~A, P is trivially true if A is trivially false.

  7. If P is the sentence true, P is trivially true.

One should interpret the trivial truth of a sentence with or without unbound variables to mean that no matter which objects are substituted for the unbound variables, the sentence can easily be prove true. A case for the trivial truth of a sentence beginning with an existential quantifier does not exist because not all models may have objects.

Trivially False

Suppose P is a sentence. To determine whether P is trivially false, search for the first applicable rule from the beginning in the following list:

  1. If P is of the form A -> B for sentences A and B, P is trivially false if A is trivially true and B is trivially false.

  2. If P is of the form A <-> B, P is trivially false if either A -> B or B -> A is trivially false.

  3. If P is of the form A & B, P is trivially false if either A is trivially false or B is trivially false.

  4. If P is of the form A | B, P is trivially false if A is trivially false and B is trivially false.

  5. If P is of the form ^X(A(X)) or the form *X(A(X)), P is trivially false if A is trivially false.

  6. If P is of the form ~A, P is trivially false if A is trivially true.

  7. If P is the sentence false, P is trivially false.

One should interpret the trivial falsity of a sentence with or without unbound variables to mean that no matter which objects are substituted for the unbound variables, the sentence can easily be proven false.

Trivial Implication

Suppose P and Q are sentences. To determine whether P trivially implies Q, search for the first applicable rule from the beginning of the following list. At most one rule is applied, so if a rule is applied and the rule does not give trivial implication, then P does not trivially imply Q.

  1. If P and Q have mismatching numbers of unbound variables or predicates, then P does not trivially imply Q.

  2. If P is of the form A | B, P trivially implies Q if A trivially implies Q and B trivially implies Q.

  3. If Q is of the form A & B, P trivially implies Q if P trivially implies A and P trivially implies B.

  4. If P is trivially false, then P trivially implies Q.

  5. If Q is trivially true, then P trivially implies Q.

  6. If P is of the form ~A and Q is of the form ~B, P trivially implies Q if B trivially implies A.

  7. If Q is of the form ~~A, P trivially implies Q if P trivially implies A.

  8. If P is of the form A -> B and Q is of the form C -> D, then P trivially implies Q if C trivially implies A and B trivially implies D.

  9. If P is of the form *X(A(X)) and Q is of the form *X(B(X), then P trivially implies Q if A trivially implies B.

  10. If P is of the form ^X(A(X)) and Q is of the form ^X(B(X)), then P trivially implies Q if A trivially implies B.

  11. If P is of the form A <-> B and Q is of the form C <-> D, then apply any of the following rules:

    • If A trivially implies C, C trivially implies A, B trivially implies D, and D trivially implies B, then P trivially implies Q.

    • If A trivially implies D, D trivially implies A, B trivially implies C, and C trivially implies B, then P trivially implies Q.

    • If C trivially implies D and D trivially implies C, then P trivially implies Q.

  12. If P and Q are identical relation sentence terms, then P trivially implies Q.

  13. If P and Q are identical predicate sentence terms, then P trivially implies Q.

  14. Otherwise, apply any of the following rules:

    • If P is of the form A & B and either A trivially implies Q or B trivially implies Q, then P trivially implies Q.

    • If Q is of the form A | B and either P trivially implies A or P trivially implies B, then P trivially implies Q.

Regardless of the number of unbound variables of A and B, one should interpret “A trivially implies B” to mean that no matter which objects are substituted for the unbound variables, one can easily prove that A implies B.

Trivial Equivalence

Suppose P and Q are sentences. To determine whether P is trivially equivalent to Q, determine whether P trivially implies Q and Q trivially implies P. If so, then P is trivially equivalent to Q, otherwise P is not trivially equivalent to Q.