MATH20302 - 2010/2011

 

Unit specification

Aims

The programme unit aims to introduce the student to the idea of formalising arguments, both semantically and syntactically, and to the fundamental connection between these approaches.

Brief description

Logic is the study of arguments, what they are,and what it means to say they are sound. As such it is central to Mathematics, Philosophy, and, to an increasing extent in recent years, Computer Science.

Most of this course unit will deal with the most basic sort of logical argument (i.e., in everyday parlance, what we mean by 'A follows from B'), namely those logical arguments which depend for their soundness simply on the commonly agreed interpretation of the logical connectives 'not', 'and', 'or' and 'implies'. This subject is known as propositional logic.

We shall characterise the above notion of 'follows' in two fundamentally different ways, firstly in terms of preservation of truth (semantically), and secondly in terms of the formal rules it obeys (proof theoretically, or syntactically). The highlight of the course unit will be the Completeness Theorem, which tells us that these two quite different characterisations are equivalent. This is a fundamental result for Mathematics; its essence is that if something isn't formally provable then there must exist a counterexample.

In the final few lectures the student is introduced to some basic ideas about the predicate calculus, a language which allows the use of variables together with the quantifiers "for all..." and "there exists...". This language is of far greater expressive power than that of the propositional calculus, and using it one can in principle formalise any mathematical argument. This part of the course unit serves as a brief introduction to the level 3/4 units on the subject, which pursue it in geater depth.

Intended learning outcomes

On completion of this unit successful students will be able to:

• appreciate how arguments can be formalised semantically and syntactically and how these are connected (via the Completeness Theorem);
• in simple cases be able to show that 'A follows from B' both by giving a semantic argument and by constructing a formal proof;
• in simple cases be able to show that 'A does not follow from B' by using semantics.

Future topics requiring this course unit

The course unit forms a coherent subject on its own, and provides necessary background knowledge for the third and fourth level Logic course units.

Syllabus

1. Motivation, syntax, propositional variables, connectives, sentences. [2 lectures]
2. Valuations, logical consequence, logical equivalence, truth tables, satisfiability, Beth Trees. The Disjunctive Normal Form Theorem, expressibility, adequate sets of connectives. [6]
3. Rules of proof, formal proofs, the Correctness Theorem. [5]
4. Consistency, the Completeness and Compactness Theorems. [5]
5. The language of the Predicate Calculus. Structures and interpretations, with examples. Satisfaction, logical validity, and logical consequence for the predicate calculus. [4]

Textbooks

Course unit notes will be provided. It will not be necessary to buy any books, but there a number of good books around which the student might enjoy (although they all tend to use substantially different notation, so that they are definitely not alternatives to the course unit notes), for example:

• H.B. Enderton, A Mathematical Introduction to Logic, (second edition) Academic Press 2001, ISBN 0122384520.
• E. Mendelson, Introduction to Mathematical Logic, Wadsworth and Brooks 1997, ISBN 0534066240.
• E.J. Lemmon, Beginning Logic, Van Nostrand Reinhold (UK) 1971, ISBN 0442306768.

Learning and teaching processes

Two lectures and one examples class each week. In addition students should expect to do at least four hours private study each week for this course unit.

Assessment

Two take home tests; Weighting within unit 20%

2 hours end of semester examination; Weighting within unit 80%

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Arrangements