# Introduction to Logic

* *Logic* studies *arguments* and argumentative *forms of reasoning*, and is concerned with the development of standards and criteria to evaluate arguments.
* An *argument* is a group of statements, called *premises*, assembled with intention of deriving the truth of the final statement, called *conclusion*.

Logic is concerned with *declarative sentences* from a natural language because only those can be assigned a *truth value* and thus translated into a *symbolic formal language*, where an argument maintains the same form but is expressed differently.

* The *logical form* of an argument can be translated from a natural into a symbolic language.

formally defined arguments are constructed in math, independently of any natural language.

* All *systems of logic* have a *formal language* used for writing *logical statements* or *propositions*.
* Propositions consist of logical *formulae* that comes in a form of *atomic formulae*, such as *logical constants* (true, false, ⟙, ⟘) and *logical variables* (p, q, r, etc.), and *compound formulae* that is build from atoms and *logical connectives* (¬, ∧, ∨, →, etc.).

Normally, all formulas have some intended interpretation. For example, a formula may assert a true property about the natural numbers, or some property that must be true in a data base. This implies that a formula has a well-defined meaning or semantics.

But how do we define this meaning precisely? In logic, we usually define the meaning of a formula as its truth value. A formula can be either true (or valid) or false.

Defining rigorously the notion of truth is actually not as obvious as it appears. We shall present a concept of truth due to Tarski.

Roughly speaking, a formula is true if it is satisfied in all possible interpretations.