#makinson

9 DEC 1883. Austin Daily Statesman.

The Makinson Trial.

A jury was secured without a great deal of trouble, and the introduction of evidence began at once. Our readers are so familiar with all the facts in the case that we deem it uninteresting to again rehearse them by repetition of the evidence. 

Dr. Smoot related the facts of his robbery and identified the defendant by his voice and general appearance. Dr. Swearingen related his little experience with the young man and identified him in positive terms as the prisoner. The facts of the prisoner’s discovery and arrest, the finding of the stolen property, and the identification of the robber were all brought out in detail together with his contradictory statements, and of course a positive identification of the stolen property found upon him when arrested. The evidence was short and sweet. The promised alibi didn’t make connections and the evidence on the part of the defendant was confined to proving the juvenile character of the defendant, to the effect that he was but nineteen years of age and innocent and confiding. Both his father and mother so testified. Here the case rested. 

The district attorney, Mr. Sheeks, made a wonderfully clever and pointed argument. He made no attempt at oratory, but every word he uttered was said just right, and his analysis of the evidence was as systematic and logical as it could have been made. He was not lengthy, but when he closed there was nothing unsaid. On this trial Mr. Sheeks has proved himself a lawyer of excellent metal, for no case has been tried in the Travis court house more closely than he tried this one, and a Statesman representative heard compliments for him from many sources.

We say that one formula tautologically implies another iff there is no assignment of truth values to propositional letters upon which the first formula comes out true and the second comes out false.

So writes Makinson in Topics in Modern Logic, p. 2. This definition seems to coincide with our normal idea of model-theoretic consequence: q is a model-theoretic consequence of p iff in all models where p is true, q is true as well. The only difference is that Makinson is dealing with propositional logic at this stage, so he talks about assignments of truth-values instead of models. In other words, his tautological entailment coincides with what we would symbolize as \(\models\).

So far so good. But then he starts talking about axiomatizing this relation with axiom schemata such as \[\alpha \wedge\ \beta \rightarrow \alpha\] \[\neg \neg \alpha \rightarrow \alpha\] where \(\rightarrow\) is not the conditional, but tautological implication. Which just seems to be wildly confused. You can't have axiom schemata of the form \[\neg \neg\ \alpha\ \models \alpha\] because axioms are meant to be purely proof-theoretic, and \(\models\) is a model-theoretic notion.

So there are two questions here: How did this confusion arise in the first place, and how are we meant to interpret these axioms sensibly?

How did this confusion arise in the first place?

This seems very similar to another confusion in Kleene: http://math.stackexchange.com/questions/577072/virtues-of-presentation-of-fo-logic-in-kleenes-mathematical-logic/577175#577175. In this case, Kleene mixes up

a Hilbert-style axiomatic proof system with an overlay of derived rules which look rather natural-deduction-like...the way he presents FOL perhaps reflects a transitional stage in our understanding of the relations between different types of logical theory.

So I think something similar is going on here, except with proof-theoretic and model-theoretic ideas: where Kleene mixes up Hilbert and Gentzen, Makinson mixes up proof theory and model theory.

How are we to interpret this sensibly?

Makinson also gives us a few derivation rules with his axiom system, for example

\[\textrm{From } \alpha \rightarrow \beta \textrm{ and } \beta \rightarrow \gamma \textrm{ to } \alpha \rightarrow \gamma\]

So it seems like we can charitably take Makinson to be presenting a Hilbert-Gentzen mashup like Kleene. So I suggest we take all of the axiom schemas to be of the form \(\Gamma \supset \beta\) whenever he writes \(\Gamma \rightarrow \beta\). And we can take his derivation rules to be of the form \(\frac {\alpha \quad \beta}{\gamma}\), such that the above rule turns out to be \[\frac {\alpha\rightarrow\beta\quad\beta\rightarrow\gamma}{\alpha\rightarrow\gamma}\]

My reading in logic is rather inconsistent, as is my knowledge - I have a tendency to start books, work halfway through them, and not finish them. (The same is true of my non-logic reading, but anyways.) So my plan is to go through a few short intermediate books thoroughly. Intermediate, because that's the level I'm at, and short, so that I actually work through them. I plan to go through

Philosophical Devices: Proofs, Probabilities, Possibilities and Sets, by David Papineau (Oxford UP, 2012), and

Topics in Modern Logic, by David Makinson (Methuen, 1973).

The Papineau book is rather basic and introductory, and I'm probably already familiar with most of the material, but I think it's a good idea for me to go through it. Topics in Modern Logic is no longer quite so modern, but I've heard good things about it and I think it should give me a firm grounding for further study. Then, there's

Philosophical Logic, by John Burgess (Princeton UP, 2009).

I plan to read that for non-classical logic; printing errors are listed on the website. For introductory computability theory, there's

Gödel's Incompleteness Theorems, by Raymond M. Smullyan (Oxford UP, 1992).

Incompleteness in the Land of Sets, by Melvin Fitting (College Publications, 2007).

There are two other which don't really fit in anywhere, at the moment:

How to Play Dialogues: An Introduction to Dialogical Logic, by Juan Redmond and Matthieu Fontaine (College Publications, 2011).

Logic: The Basics, by JC Beall (Routledge, 2010).

There's another rather more specific reason why I'm going through Philosophical Devices and Topics in Modern Logic first: apart from them being at the right level, they also have extremely convenient chapter lengths. Philosophical Devices has twelve chapters; Topics in Modern Logic has fifteen sections. I'm hoping that the routine of doing a chapter a week for Papineau and a section a week for Makinson will help me be steady with my studying.

There are omissions on the list, most notably proof theory - but I'll remedy that when I feel like I've understood enough of the basics: all things excellent are as difficult as they are rare.

Why am I noting this here? As http://www.logicmatters.net/2014/05/30/parsons-1-predicativity/ puts it: ". . .promising to comment here is a good way of making myself read through the book reasonably carefully. Whether this is actually going to be a rewarding exercise — for me as writer and/or you as reader — is as yet an open question: here’s hoping!"