# SET THEORY AND FURTHER LOGIC

Last Updated 28/04/05

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1. The Paper
2. Textbooks

# 1. The Paper

This paper covers basic axiomatic set theory plus a subject within the ambit of formal logic but beyond the introductory logic of a standard first year course. In recent years that further subject has been modal logic (the logic of necessity and possibility). Another candidate for the ‘further logic' component is intuitionistic logic, another would be probability theory. Students would of course be advised of any change in the ‘further logic' component , and the paper in any changeover year would include questions on set theory and on both ‘further logic' subjects. This paper has a sister paper, Mathematical Logic. The common parent was a paper, now defunct at the B.A. level, called Symbolic Logic. Once modal logic became part of the staple diet it was felt that there was too much ground for an undergraduate to cover in any depth in one paper; hence the split. The courses are taught in alternate years: if Mathematical Logic is taught one year, Set Theory and Further Logic is taught the next. But B.A. exam papers for both are set every year.

What can set theory do for a philosophy student? Here is a four-part answer. (1) Set theory provides a bagful of conceptual tools useful in many areas of rigorous thought and teaches one to handle them with precision, e.g. partial ordering, equivalence relation, isomorphism, recursive definition and proof by (mathematical) induction. (2) It sharpens reflection on matters central to philosophy of thought and language, such as the nature of truth, satisfaction, predicate extensions and associated antinomies such as Russell's Paradox. (3) It is very important for philosophy of mathematics, for questions of ontology and the number systems. (4) It provides a proper understanding of infinity (and one of the deepest problems of mathematics, the continuum problem), and enables one to see through mystical twaddle wrapped in talk of the infinite.

The relevance of modal logic is more obvious. Questions of necessity and possibility are at the heart of metaphysics—no worthwhile discussion of identity, existence or causation can avoid them—and the questions become much sharper when formulated with the aid of modal logic. Though reasoning with modal terms (e.g. ‘must' and ‘can') is ubiquitous, it is not obvious what the correct principles are for modal reasoning; the study of formal modal logic is illuminating here because it clarifies the options. Other areas of semantic investigation also benefit greatly from modal logic, e.g. proper names, counterfactual conditionals and tense. Formal systems of modal logic are apt not only for the logic of necessity and possibility, but also for the logic of knowledge and the logic of obligation. So modal logic can be useful in epistemology, ethics and political philosophy, as well as metaphysics.

# 2. Textbooks

## Set Theory

• Enderton, H. 1977. Elements of Set Theory. New York: Academic Press.

• Hrabcek, K. and M. T. Jech. 1984. Introduction to Set Theory. 2nd ed. New York: M. Dekker.

In recent years the lecture course has used Enderton's book in conjunction with the lecturer's handouts. Both these books cover more than the course content; more elementary introductions leave out some of what is philosophically important. There are many texts on set theory, and one can rarely tell the level of the material from their titles alone. Lévy's masterful Basic Set Theory requires a fairly high level of sophistication, and Introduction to Axiomatic Set Theory by Takeuti and Zaring is onto graduate level material before half way!

It is usually best to stick to one text book as different books adopt different conventions and definitions; the net result is the same, but confusion may result from using two books. However, to get a feel for the area you might look at:

• Devlin, K. 1992. Sets, Functions & Logic: an Introduction to Abstract Mathematics. 2nd ed. London: Chapman & Hall.

An advanced book I recommend for subsequent study if you are interested in the philosophical questions raised by set theory is:

• Potter, M. 2004. Set Theory and its Philosophy . Oxford University Press.

## Modal Logic

Introductory modal logic textbooks are a mixed bag, almost all of them at present falling short of what is required. The lecture course was for a number of years given by Genoveva Marti, who with good results did not use a text book but distributed her own notes instead.

Here is a list of textbooks with my comments. No doubt there are some texts on modal logic that have escaped my attention; others are not on the list either because they are too bad or too advanced.

My first choice:

• Fitting , M. & Mendelsohn, R. First-Order Modal Logic . Dordrecht: Kluwer.

There is more in the Fitting & Mendelsohn than can be dealt with comfortably in one term; but that is fine, as one can be selective. It has a good balance of formal logic and philosophical discussion. The book is a paperback costing £24 online at Waterstones at the time of writing (April 2005).

• Hughes, G. & Cresswell, M. A New Introduction to Modal Logic.

The Hughes & Cresswell is an update of their classic Companion to Modal Logic . It is my second choice. Its chief virtue is that it is comprehensive. Its chief drawback is that the cramped layout makes it rather forbidding for first-time students.

• Beall, J. & Van Fraassen. B. Possibilities and Paradox . Oxford: Oxford University Press.

A smorgasbord of topics beyond introductory classical logic. While it could serve as a text for an interesting but superficial course, there is not enough modal logic in it to give a decent grounding in that subject. It touches lightly on a lot of different things, e.g. many-valued logic, intuitionistic logic, relevance logic, paradoxes of material implication & logic of conditionals, the Liar and the Sorites.

• Konyndyk, K. Introductory Modal Logic. Notre Dame, Indiana: University of Notre Dame Press.

This might be helpful as supplementary reading. But it is too light on the logic to support a proper term-long course on modal logic.

• Girle, R. Modal Logics and Philosophy.
This is only on the list because it was used the year before last. I don't like it all, chiefly because it muddies the syntax-semantics distinction. But it does have some virtues. It is good on the relations between the various formal systems of modal logic, and it shows the value of modal logic by describing briefly several quite differenct applications.