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.