Foundations of Logic: Deduction, Signs, and Knowledge
Linguistic Signs and Their Nature
Signs are the basic elements of a code.
The designated refers to the set of persons, animals, or things of any kind that are represented by the signs.
The denotated refers to the set of people who are the target audience for the signs, and who can interpret the transmitted information. (e.g., ‘I love you’)
Considering the relationship between a sign and what it designates, there are two main types of signs:
Natural Signs (Indices)
Natural signs, also known as indices, have a causal link between the sign and what it designates. The sign acts as the effect, and the designated as the cause.
Conventional Signs
Conventional signs are established voluntarily and freely, without any inherent causal relationship between the sign and what is designated.
Natural Deduction and Fundamental Logical Laws
Understanding Natural Deduction in Logic
Natural deduction is a method for studying the formal validity of arguments. It involves inferring a conclusion from a set of given premises by applying the laws and rules of propositional calculus. This includes the use of logical constants, connectives, and propositional variables.
Key Logical Laws for Deduction
Logical laws are inference schemas that are always valid (tautologies). While endless, some are given special names due to their common use:
Law of Identity
Any proposition is identical to itself:
p → porp ↔ p.Law of Contradiction
No proposition can be simultaneously affirmed and denied under the same circumstances:
¬ (p ∧ ¬ p).Law of Excluded Middle
Between affirming or denying a proposition, there is no third possibility:
p ∨ ¬ p.Modus Ponens
Given a conditional statement and the affirmation of its antecedent, the consequent can be affirmed:
[(p → q) ∧ p] → q.Modus Tollens
Given a conditional statement and the negation of its consequent, the negation of the antecedent follows:
[(p → q) ∧ ¬ q] → ¬ p.Law of Double Negation
The negation of a negation is an affirmation:
¬ (¬ p) ↔ p.
The Natural Deduction Process
The natural deduction process begins with a set of premises from which a conclusion is inferred by applying determined rules.
The Scope and Limits of Human Knowledge
Immanuel Kant defined the Enlightenment as humanity’s emergence from its self-imposed immaturity.
Ancient Philosophical Perspectives
Socrates, Plato, and Aristotle believed that human reason is capable of revealing the intimate causes of what truly exists. Plato asserted that Ideas are the proper object of rational knowledge, while things that are born, die, and are transformed through the senses are not known rationally. From particulars, one can only have opinion, not science.
Aristotle, contrary to Plato, argued that the universal and necessary are the essences of things (which constitute the proper object of rational knowledge), not Platonic Ideas.
Modern Philosophical Perspectives
René Descartes, a key representative of rationalism, denied the cognitive value of the senses and celebrated the supreme value of reason.
John Locke and David Hume, representing empiricism, argued that knowledge is not hidden from us but is revealed through experience.
Enlightenment Philosophy
Kant concluded that objective knowledge cannot be obtained either solely from sense experience or solely from pure reason.
19th and 20th Century Philosophy
Georg Wilhelm Friedrich Hegel represented idealism and rationalism, while Karl Popper offered a critical perspective.