Fundamentals of Logic: Principles and Applications

Principles of Logic

Principle of Identity

An object is the same as itself: A is A → A = A.

Principle of Contradiction

Nothing can both be and not be in the same sense at the same time. Contradictory statements cannot both be true: Nothing can be A and not A → ¬(A ∧ ¬A).

Principle of Excluded Middle

Everything must either be or not be. Every statement must be either true or false: Everything is A or not A → A ∨ ¬A.

Logical Paradoxes, Fallacies, and Invalid Arguments

Consider the statement: "This statement is false." This proposition creates a paradox. If we assume it's true, then its content declares it false. Conversely, if we assume it's false, then its content implies it's true. This self-contradictory statement challenges basic logical principles.

Formal Logic

Propositional Logic

Interprets statements as a whole, without decomposition.

Predicate Logic

Analyzes the internal structure of statements, distinguishing between the subject and its predicates.

Logic of Classes

Treats individuals as belonging to sets sharing specific properties.

Logic of Relationships

Deals with relationships between elements within a statement.

Symbolic Representation

Letters (p, q, r, s...)

Represent statements or propositions in reasoning. Example: "If it rains, the street gets wet" becomes: If p, then q.

Signs (→, ∨, ∧, ⊢...)

Represent relationships between propositions. Example: "If it rains, the street gets wet" becomes: p → q.

Propositional Logic: Statements

Simple (Atomic) Statements

Cannot be decomposed into other statements. Examples: "I study Philosophy," "The sun is shining."

Complex (Molecular) Statements

Can be decomposed into simpler statements. Example: "He's Antonio and I'm Antonia."

Truth Tables and Argument Validity

Applying truth tables to assess argument validity yields three possible outcomes:

• Contradiction: The resulting formula is always false, regardless of component statements' truth values.
• Contingency: The formula's truth value depends on the truth or falsity of its component statements.
• Tautology: The formula is always true, regardless of component statements' truth values. Only tautologies represent formally valid inferences.