Revision as of 22:48, 14 October 2012

First Order Logic

Chenkai Wang

0. Introduction

Mathematics is all about logic. Every insightful mathematical theory (among non-mathematical ones) is built up by certain kinds of rules (axiomatically). The rules which enable us to produce a theory system are introduced by formal theory. Don't mix formal theory with the actual theory you are playing with! We sometimes call formal theory a metatheory. The study of formal theory falls into the realm of mathematical logic, which is the novel frontier of the study of logic from the late 18th century until nowadays. The philosophical inception such as the relating ontology was investigated by true logicians which will not be covered here. Among all different formal systems, first order theories are the most useful one. OK, we now onto the main subject: first order logic.

1. The Language

We call our first order logic a language because it simulates our natural language. The main difference between a natural language and a formal language is our formal languages can never have "mistakes", that is, the ambiguity. You can never have one thing representing two different other things based on your context. (However, in our informal discussion, we usually omit this rule for convenience, but it is still better than our natural languages.) A first order theory MUST consists all of the followings: 1. some constant symbols 2. some object variable symbols 3. some well-formed formula (wff) variable symbols 1-3 are called alphabet, you can define new type of alphabet later on, but that's not primitive. 1-3 are not inter-definable. 4. some rules (can be recursive) which acts on constants and object variables for constructing well-formed formula 5. some axioms, must be well-formed formula 6. some inference rules which acts on well-formed formulas for constructing new valid sentences from axioms. 4-6 are called syntax of our first order language.

2. Some Useful Metatheorems

3. miscellanea

"SUCH THAT" is confusing!