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I am clear on everything that precedes the grey box. I just can't wrap my mind on how we can possibly get more valid inferences from a proper subset of interpretations than we could from the original set of interpretations. Can anyone explain this to me?

  • This is in modern categorical logic due to the presence of adjunctions. – Mozibur Ullah Apr 23 '15 at 7:53

In nuce: if there are many models/interpretations, there are many occasions for falsifying a formula, i.e. the set of tautologies is small. If there are fewer models/interpretations, more formulae stand unrefuted, so the set of tautologies is bigger.

For sake of illustration, consider, like Mauro Alleganza recommends, only formulae (instead of the more complicated inferences), and only the most extreme cases:

  1. There are all kinds of (classical) interpretations: only classical tautologies are valid.
  2. There is only one interpretation: everything true according to the interpretation is valid, too. Why? The definition of "valid" says a formula is valid iff it is true in every (= the only one) interpretation.

PS: Perhaps this is still a more extreme case:

2a. There is no interpretation whatsoever: everything is valid, since something p is valid iff for every x: if x is an interpretation, then p is true according to x.

PPS: There are more extreme (non-classical) cases one could consider instead of 1, i.e. interpretations that do not obey classical constraints like bivalence or consistency... In logics based on theses interpretations even fewer formulae are considered valid, and likewise for inferences. (In the three-valued logic K3 there are no valid formulae, for example.)

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Forget for a moment about "inferences" and consider only formulae.

Tautologies (or valid formulae) are formulae that are true in all interpretations, like A → A or ∀x(x=x).

If we consider one of Peano's axiom for arithmetic, like ∀x(x+0=x), it is not a "logical truth" (i.e. a valid formula) : it is true only in the "arithmetical" interpretation.

But the "logical laws", like ∀x(x=x), are still true in the "arithmetical" interpretation (they are ture in all interpreations !)

Of course, in the arithmetical interpretation also the theorem deriving from the arithmetical axioms will be true (they are logical consequences of the axioms).

Conclusion : adding axioms will reduce the number of possible interpretations, and vice versa.

The same for modal logics; we "specify" a normal modal systems adding some axiom to the "basic system K.

This amounts to adding "restrictive" requirements to the class of interpretations, and thus "reducing" the number of interpretations ...

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My answer here is basically this answer: https://philosophy.stackexchange.com/a/23278/9166 in a more general application.

Adding acceptable conditions on an alternate world is just adding a generalized axiom. Consider what happens when you add an axiom to a system.

All of the consequences of this new fact are now inferences, so the number of inferences has gone up. At the same time interpretations that were valid before, are now no longer valid, because every interpretation of the new system is an interpretation of the original system, but there are interpretations of the original that are now ruled out by the new facts.

If you back of from whole logics to simpler axiomatic systems, maybe it is clearer. From an algebraic point of view a 'world' or 'interpretation' is to a logic (modal or otherwise) what a concrete instance of a given group is to the axioms of groups, or what a given (concrete instance of a) field is to the axioms of fields.

There are abelian groups of any finite size, for example the cyclic group on that number of elements. And you can extend the axioms of an abelian group to get the axioms of a field. But we can only construct fields out of these groups when the size of the group has only one prime (perhaps repeatedly) in its factorization. More rules, less instances.

At the same time we can say a more useful things about the interactions of the elements of a field than of the group that underlies it. More rules, more inferences.

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