The various possible three-valued propositional logics of implication validate different formulae. For example, depending on the truth-table for their implicational connective, some validate the axiom of contraction while others do not1.
Thus for any pairing of a set of implicational formulae with a single implicational formula one may ask, "does every three-valued logic of formulae that validates all the formulae in that set also validate that single formula?". For example, does every possible three-valued logic that validates both the axiom of identity and the axiom of prefixing also validate the axiom of permutation2?
If we restrict ourselves to nonempty three-valued logics of formulae whose truth-tables for the implicational connectives preserve the classical evaluations for true and false, we can view this as a relation that holds between a set of formulae and a single formula whenever every such logic that validates each of the formulae in the set also validates that single formula. It turns out that, according to established criteria for what makes a logic, this relation qualifies as a logic itself3. For any set of formulae together with any single formula, this logic countenances the consequence relation from the set to the single formulae if and only if every logic that validates all of the formulae in the set also validates that single formula. It gives us a logic of the relationships between logics.
Footnotes:
- The classical tautology, '(p -> (p -> q)) -> (p -> q)' (an instance of the W schema).
- The classical tautologies,
- 'p -> p' (identity) (instance of the I schema)
- '(q -> r) -> ((p -> q) -> (p -> r))' (prefixing) (instance of the B schema)
- '(p -> (q -> r)) -> (q -> (p -> r))' (permutation) (instance of the C schema)
- It qualifies as a consequence relation characterized by the family of nonempty three-valued logics of formulae whose truth-tables for the implicational connective preserve the classical evaluations for true and false. For criteria for logical consequence relations, see D. Shoesmith and T. Smiley Multiple-conclusion Logic (Cambridge: Cambridge, 1978).
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I presume the consequence relation so generated is atheorematic, given that there is no need to have all the three valued implicational logics agree on a given formula simpliciter?
ReplyDeleteActually, no.
ReplyDeleteIt's got, for instance:
|- (p -> p) -> (p -> p)
It's also fun to look for sets of two or more formulae such that (a) every logic (of the type considered) contains _some_ of those formulae, and yet that (b) no single formula in the set belongs to all of those logics.
ReplyDeleteFor example, { '(p -> (p -> p)) -> (p -> p)', '((p -> (p -> p)) -> p) -> (p -> p)' } is such a set. Every logic (of the type considered) contains one or the other of those fomulae, but neither formula belongs to all of them.
Since such sets exist, one can prove that the sort of logics under consideration are _not_ closed under intersection.
And also, equally as interestingly, that the generalised consequence relation we get in the standard way (i.e. whenever all the formulas on the left are given the designated value, at least one formula on the right is given it also) will have consequences C1,...,Cn which follow from A1,...,Am without us having that A1,...,Am |- Ci for any 1=< i <= n.
ReplyDelete(I appear to have not absorbed the `preserves classical evaluations for true and false' bit before. That said, isn't (p -> p) -> (p -> p) going to end up not being gauranteed to get the designated value if v(p) is the intermediate value? i.e. couldn't we consider
the table for -> according to which the (writing 2 for the intermediate semantic value) v(2->2) = 2? In which case when v(p) = 2 then v((p->p)->(p->p)) = 2. That was my reason for thinking the logic was going to be atheorematic.)