## McGee on Necessitation and Conditional Proof

### August 19, 2015

I’ve been taking great pleasure in reading Ole Hjortland and Colin Caret’s recent edited collection of papers from the long running Arche ‘Foundations of Logical Consequence’ project. So far I’ve greatly enjoyed every paper in the collection. What I want to point out here is a rather curious passage in Van McGee’s contribution “The Categoricity of Logic”. After claiming that the meanings of the sentential-connectives are given by their natural deduction rules McGee goes on to make the following, rather curious, comment.

“In proposing this, I don’t want to go to the extreme of saying that any inferential system whose methods of reasoning with the classical connectives don’t include the rules is deficient or defective. Normal systems of modal logic include the rule of necessitation (From $\varphi$, you may infer $\Box\varphi$). If they also included conditional proof, you could assume $\varphi$, derive $\Box\varphi$ by necessitation, then discharge the assumption to derive the invalid conclusion $(\varphi \supset \Box\varphi)$.”

McGee then goes on to argue that this is not a problem, because the rules of normal systems of modal logic are “intended to preserve modal validity”, while natural deduction rules like conditional proof preserve ‘being entitled to accept’ but not ‘modal validity’.

There are a few strange things going on in this passage. Firstly, we have a rather awkward comparison of logical frameworks. Typical presentations of normal modal logics are in the FMLA framework[1], where we convieve of logics as consisting of sets of valid formulas closed under various rules. In this setting necessitation is the closure condition which tells us that whenever $A$ is in the set (i.e. is a theorem of the logic) then so is $\Box A$. In settings like this we cannot even talk about rules in the SET-FMLA framework like Conditional Proof. To do this we would need to consider appropriate consequence relations for modal logics where we can properly ask about the status of rules like conditional proof. Lifted to such a setting the rule of necessitation becomes the rule which takes us from $\emptyset\vdash A$ to $\emptyset\vdash\Box A$ – i.e. if $A$ follows from the empty set of premises, then so does $\Box A$. This is because necessitation is (to use terminology popularised by Timothy Smiley) a rule of proof, telling us that if we have proved $A$ then we can conclude $\Box A$. By contrast conditional proof, the rule which takes us from $\Gamma, A \vdash B$ to $\Gamma\vdash A\supset B$ is a rule of inference, telling us that if we can deduce $B$ on the basis of some assumptions along with $A$ then we can conclude $A\supset B$ on the basis of those assumptions. What McGee is appealing to in the above quotation, though, is the rule which allows us to transition from $\Gamma\vdash A$ to $\Gamma\vdash\Box A$.

I think this is the most natural way of reading the above passage, and in this case our diagnosis of the above ‘problem’ is quite simple: there is none, any appearance to the contrary arising due to a confusion between rules of inference and rules of proof. This is a somewhat under-recognised distinction, in part due to the phenomena, pointed out in section 4 of Lloyd’s paper ‘Smiley’s Distinction between Rules of Inference and Rules of Proof[2], of logicians thinking in terms of undifferentiated axiomatisations (where an axiomatisation consists solely of a collection of axioms and rules) rather than dividing up our rules into rules of inference and rules of proof).

This is not the only way to read this passage, though. There are presentations of normal modal logic which are closed under the rule McGee seems to have in mind in the above quotation. Fix on a class of modal frames and understand a model as being a model on one of those frames and let us define the following two notions of validity:

• $\Gamma\vdash A$ is locally valid iff, for all models, $A$ is true at a world if all the formulas in $\Gamma$ are true there.

• $\Gamma\vdash A$ is globally valid iff, for all models, $A$ is true at all worlds in the model if all the formulas in $\Gamma$ are true at all worlds in that model.

So local consequence preserves truth at a point in a model, while global consequence preserves being true throughout (i.e. at all worlds in) a model. Now the rule mentioned above which takes us from $\Gamma\vdash A$ to $\Gamma\vdash \Box A$ is globally valid, but is not locally valid. By contrast, conditional proof preserves local validity but not global validity. If we think in terms of global validity then we can say a bit more about what McGee might have in mind by ‘modal validity’: preservation of modal validity is just preservation of truth in all possible worlds. Reading McGee in this way will make this passage come out true, but personally I’m dubious about whether this is the intended reading of this passage.

The moral of the story here? People ought to be more attentive to the rule of inference/rule of proof distinction. Also, comparing logics typically concieved of in different frameworks can quite quickly get you into trouble.

NOTES
[1] The notion of ‘logical framework’ which I have in mind, and the labels FMLA and SET-FMLA for particular logical frameworks is due to Lloyd Humberstone. More information on this (and much more besides) at http://plato.stanford.edu/entries/connectives-logic/.

[2] L. Humberstone, “Smiley’s Distinction between Rules of Inference and Rules of Proof”, In T. J. Smiley, Jonathan Lear & Alex Oliver (eds.), The Force of Argument: Essays in Honor of Timothy Smiley. Routledge 107–126 (2010)