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Logic (MATH0050) Notes

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Module
Logic (MATH0050)

Institution
University College London (UCL)

Explore Logic in the second year at UCL. Unravel the intricacies of this fundamental field, delving into the principles and frameworks that govern logical reasoning. Please note that these materials are intended for personal use only and should be used in accordance with academic integrity guidelin...

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Uploaded on December 1, 2023
Number of pages 114
Written in 2022/2023
Type Lecture notes
Professor(s) Dr isidoros strouthos
Contains All classes

logic

sujansathiendran Member since 3 year 5 documents sold

MATH 0050

Logic

Department of Mathematics
UCL

Lecture Notes
2022-2023

Isidoros Strouthos

7 January 2023

1

,Contents

1 Language 3

1.1 First order predicate language . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Conventional functional first order predicate language . . . . . . . . . . . . . . . . 13

2 Propositional Logic 19

2.1 Semantic aspects of propositional logic . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Syntactic aspects of propositional logic . . . . . . . . . . . . . . . . . . . . . . . 41

2.3 Completeness theorem for propositional logic . . . . . . . . . . . . . . . . . . . . 47

3 First Order Predicate Logic 58

3.1 Semantic aspects of first order predicate logic . . . . . . . . . . . . . . . . . . . . 58

3.2 Syntactic aspects of first order predicate logic . . . . . . . . . . . . . . . . . . . . 72

3.3 Completeness theorem for first order predicate logic . . . . . . . . . . . . . . . . . 77

4 Computability 87

4.1 Computable (partial) functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.2 Recursive (partial) functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.3 Encoding and the halting problem . . . . . . . . . . . . . . . . . . . . . . . . . . 104

2

,Chapter 1

Language

In order to be able to study the structure of mathematical objects and ideas, we will first provide a
framework in which mathematical objects can be defined and analysed.

1.1 First order predicate language

The symbols which we will use to construct our language consist of:

1) A countably infinite set of variable symbols: {x1 , x2 , x3 · · · } or {w, x, y, z, w0 , x0 , y 0 , z 0 , · · · }.

2) For each non-negative integer n, a countably infinite set of predicate symbols {P1 , P2 , P3 · · · }
or {P, Q, R, P 0 , Q0 , R0 , · · · }, each of which has arity n. If P is a predicate symbol of arity n,
then we call P an n-ary predicate symbol.

3) The symbols ¬, ), 8.

The set of all strings of symbols from the above list is denoted by Lstring .

For example, suppose that x, y, z, x1 , x2 , x3 are variable symbols, P is a 1-ary predicate symbol, and
Q is a 2-ary predicate symbol. Then x, P x, P xy, Qx, Qxy, Qxx, xQy, Qx1 x2 x3 , ¬P x, ) P xQyz,
8x8yQxy, 8x, 8x ) ¬P xQxy, 8x ) P ¬xQxy are all in Lstring .

Let us now consider the subset of Lstring that will be especially useful:

3

,

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