Introduction

Author: Protyasha Roy

Created: 2025-01-07

Last Modified: 2025-01-07

Book: A Mathematical Introduction to Logic

Introduction

Symbolic logic(the use of symbols to denote propositions, terms, and relations in order to assist reasoning that the model is based on) is a mathematical model of deductive thought/reasoning. So, basically symbolic logic is deductive reasoning but with symbols denoting the propositions/terms. But over time, like other branches of mathematics it has grown beyond of its scope of birth.

How are models constructed?

Models are constructed based on properties of an object or some abstract idea where some properties are called essential and others are considered to be irrelevant in building the model. For example consider an airplane. If you want to build an airplane the essential features/properties would be its shape and design, whereas the irrelevant property will be its size. Whether or not the resulting model meets its intended purpose(what you want from this model) will largely depend on the selection of properties of the original thought/object.

Logic is one abstract concept. More than the airplanes. The real-life objects are certain “logically correct” deductions. Such as-

All men are mortal

Socrates is a man

Socrates is mortal

Here the validity of inferring the third sentence(the conclusion) from the first two(the assumptions) doesn’t depend on the idiosyncrasies of Socrates. Here the conclusion depends on the form of sentences rather than by empirical facts about mortality. It doesn’t really matter what “Mortal” means, what “All” means do matter.

See another example-

Borogoves are mimsy whenever it is brillig.

It is now brillig, and this thing is a borogove.

Hence this thing is mimsy

Here the third sentence is justified by the previous two assumptions following the form of it deductively without having a slightest idea of what mimsy borogoves might look like.

Logically correct deductions are of more interest than the contents of an example. Axiomatic mathematics consists of many such deductions. The deductions made by a working mathematician constitute real-life originals whole features are to be mirrored in our model.

The logical correctness depends on the form of these deductions rather than the contents. This criterion is vague, but it is just this sort of vagueness that prompts us to turn to mathematical models. To give a precise version of this criterion within a model there are few questions we initially be most concerned with. These are-

1. What does it mean for one sentence to “follow logically” from certain others?

2. If a sentence does follow logically from certain others, what methods of proof might be necessary to establish this fact?

3. Is there a gap between what we can prove in an axiomatic system(say for natural numbers) and what is true about the natural numbers?

4. What is the connection between logic and computability?

This book presents two models. The first - sentential logic - will be very simple and will be woefully inadequate for interesting deductions. It is because it preserves only some crude properties of real-life deductions. The second model - first-order logic - is suited to deductions encountered in mathematics.

This book does not include - valued logic, modal logic, intuitionistic logic - cause these represent different selections of properties of real-life deductions.