A fallacy is a defect in reasoning that renders an argument invalid. Unlike simple factual errors, fallacies are structural failures -- the logical scaffolding itself is compromised, regardless of whether the individual premises happen to be true. To study fallacies is to study the negative space of valid argument: by understanding how reasoning fails, we illuminate how it succeeds.
The formal fallacies violate the structural rules of deductive logic. Affirming the consequent -- reasoning that because Q is true and P implies Q, therefore P must be true -- mistakes a sufficient condition for a necessary one. Denying the antecedent commits the parallel error in the opposite direction.
P
Q
P → Q
T
T
T
T
F
F
F
T
T
F
F
T
✽
On the Nature of Proof
A proof is a finite sequence of propositions, each of which is either an axiom, a premise, or follows from previous propositions by a rule of inference. The final proposition in the sequence is the conclusion -- the theorem being proved. The beauty of formal proof lies in its mechanical verifiability: given the axioms and rules, any reader can check each step independently.
The distinction between syntactic proof (derivation within a formal system) and semantic proof (demonstration that a conclusion follows from premises in all interpretations) is central to mathematical logic. The completeness theorem of Godel establishes that, for first-order logic, these two notions coincide: everything semantically valid is syntactically provable.
P
¬P
P ∨ ¬P
T
F
T
F
T
T
✽
Syllogistic Reasoning
The syllogism, Aristotle's greatest contribution to logic, is a form of deductive reasoning consisting of two premises and a conclusion. Each proposition contains exactly two of three terms, and the conclusion draws a relationship between the terms not directly connected in the premises. The classic example: All men are mortal; Socrates is a man; therefore Socrates is mortal.
There are exactly 256 possible syllogistic forms, of which only 24 are valid. The medieval logicians catalogued these valid forms and gave them names -- Barbara, Celarent, Darii, Ferio -- mnemonic labels encoding the quality and quantity of each proposition. This taxonomy remains one of the most elegant achievements in the history of formal reasoning.