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Although the language's propositional logic and relational logic are different,
many of the key concepts are the same. In particular, the concepts of validity,
contingency, unsatisfiability, and so forth have the same definitions in
relational logic as they do in propositional logic.
The sentence is valid, if and only, if it is satisfied by every truth assignment.
A sentence is unsatisfiable, if and only, if it is not satisfied by any truth
assignment. A sentence is contingent, if and only, if
there is some truth assignment that satisfies it and some truth assignment
that falsifies it. A sentence is satisfiable if and only if
it is valid or contingent, I.e., there is at least one truth
assignment that satisfies it. A sentence is falsifiable, if and only, if
it is either contingent or unsatisfiable. That is, if and only, if there is least
one truth assignment that falsifies it. Not only are the definitions of these
concepts the same, some of the results are the same as well.
If we substitute ground relational sentences or equations for propositions,
we get similar results for the two logics. A ground sentence in relational logic is
valid, contingent, unsatisfiable, if and only, if the corresponding sense in
propositional logic is valid, contingent or unsatisfiable.
Here, for example, are relational logic versions of some common propositional
logic, validities. The law of the excluded middle p of a or
not p of a. Double negation.
P of a, if and only, if not p of a. And deMorgan's Laws that allows us to
distribute negation over conjunction and disjunction.
Of course, not all sentences in relational logic are ground.
Here are some valid non-ground sentences of relational logic for which there are no
corresponding sentences and propositional logic.
The common quantifier reversal axioms tell us that reversing quantifiers of the same
type has no effect on truth assignment. Existential distribution allows us to move
an existential quantifier inside of a universal quantifier.
And note however, that the reverse is not true as we'll see a little later.
Finally, negation distribution allows us to distribute negation of quantification
provider type by flipping the quantifier and negating the scope of the quantified
sentence. This exercise test your understanding of
these concepts by asking you to say whether various sentences are valid,
contingent, or unsatisfiable.