First Order Logic

Logic is the study of reasoning, a fundamental part of human cognition. Logic is used almost everywhere in mathematics. Being sure about reasoning is an indispensable part of logic.

One of the most important questions of philosophy is "What is the source of truth?". This question may answered differently by different people.

Source of Truth

TODO: There are different logical systems.

Formal Systems

A formal system is consists of;

• A formal language which is describes the well-formed formulas of the system. It is also called the syntax of the system.
• A set of rules that are used to manipulate the formulas of the language. These rules are used to derive new formulas from the given formulas. These rules are also called the semantics of the system.

A formal language is constructed by specifying an alphabet and a set of rules that are used to build formulas. Here are the definitions:

• An alphabet $A$ is a set of symbols.
• A string or formula of an alphabet $A$ is an ordered sequence of symbols from the alphabet and is denoted by $A^{\ast }$.
• A well formed formula (wff) is a formula that is formed according to the (syntax) rules of the language.

TODO: Truth semantics and proof semantics.