One of the novel features of MIE is Brandom’s philosophy of logic. He calls this the expressive theory of logic. On this view, the primary purpose of logic is to express certain things. It privileges the conditional and negation. The conditional expresses the acceptance of an inference from premises which form the antecedent to the conclusion which forms consequent. The conditional lets you say that a certain inference is acceptable. Of course, conditionals in different logical systems express different sorts of acceptance. Classical conditionals express a weak form of acceptance, intuitionistic conditionals a stronger acceptance in the form of saying there is a general method of transforming justification for the premises into justification for the consequence, and so on. Negation expresses incompatibilities, generally in the presence of a conditional so as to allow one to say that certain inferences are not kosher. Incompatibility can be used to create an entailment relation defined as inclusion on sets of incompatibles. Brandom suggests taking conjunction and disjunction as set operations on those sets of incompatibilities. This would work for languages with a conditional and negation. If one does not have negation, then, I suppose, come up with the sets of incompatibilities although one couldn’t say that the incompatibilities were such. Barring defining conjunction and disjunction in terms of incompatibilities, I’m not sure exactly what they would be accepting. Conjunction might express the acceptance of both conjuncts. Intuitionistic disjunction might express the acceptance or ability to demonstrate one of the disjuncts and you know which one. I’m not sure what classical disjunction would express exactly; possibly that one accepts one of the disjuncts although no further information is given about which.

There is nothing in Ch. 2, where logical expressivism is introduced, about quantifiers. At this point in the book, nothing has been said about objects, so I’m not sure what the quantifiers express since it is likely to be tied in with theoretical claims about objects. Modal operators aren’t addressed in MIE, although they are tackled in the Locke Lectures. I don’t remember that terribly well, although I think for the most part Brandom sticks to alethic modalities in S4 and S5. I could be wrong on this. I am almost positive he doesn’t get to non-normal modal operators (to use Restall’s terminology) such as the Kleene star. I currently have no idea what the Kleene star would express. I’m similarly unsure about intensional connectives as in relevant logic’s fusion. It might be an interpretation similar to Restall’s of application of data in the form of propositions to other data, although this is really speculative. There might be something in this.

Something that I’m a little more immediately curious about is the status of translations of formulas. There are certain systems of logic that can be translated into others, e.g. S4 into intuitionistic logic. The conditional in S4 is classical, but the translation (in Goedel’s translation at least) of the intuitionistic A->B looks like [](A’->B’) where A’ and B’ are the translations of A and B. The classical conditional then will express a weaker acceptance of an inference but it will be modified in some way by the necessity operator in front. If the view of logic is right, one would expect the translation to preserve what is expressed in some form. I will have to track down the relevant part of the Locke Lectures in order to further test this idea.