Logic

Logic is the systematic study of valid inference — what follows from what. It originated with Aristotle's Organon (4th c. BCE), was extended by Stoics and medievals, and was revolutionised by Gottlob Frege in the late 19th century with the development of modern symbolic logic.

Argument: structure and evaluation

An argument consists of:

• Premises — propositions offered as evidence.
• Conclusion — proposition supposedly supported by premises.
• Inferential link — the claim that premises support the conclusion.

Two principal types:

Type	Aim	Strength
Deductive	Conclusion necessarily follows	Valid / invalid
Inductive	Conclusion probably follows	Strong / weak

A valid deductive argument has the property: if premises are true, conclusion must be true. A sound argument is valid + true premises.

Aristotelian / syllogistic logic

Aristotle's logic — codified in the Organon (Categories, De Interpretatione, Prior Analytics, Posterior Analytics, Topics, Sophistical Refutations) — dominated until the late 19th century.

Four categorical propositions (A, E, I, O)

Letter	Form	Example
A	Universal affirmative: All S are P	All humans are mortal
E	Universal negative: No S are P	No humans are immortal
I	Particular affirmative: Some S are P	Some humans are wise
O	Particular negative: Some S are not P	Some humans are not wise

Syllogism

A syllogism is an argument with two premises and a conclusion, with three terms (major, minor, middle):

All men are mortal. (major premise) Socrates is a man. (minor premise) Therefore, Socrates is mortal. (conclusion)

There are 256 possible syllogistic forms in four figures; only 15 (or 24 if existential import included) are valid. Medieval mnemonic: Barbara, Celarent, Darii, Ferio…

Square of opposition

A and O are contradictories; E and I are contradictories; A and E are contraries; I and O are subcontraries; A implies I; E implies O.

Propositional (sentential) logic

Modern logic begins with George Boole (The Laws of Thought, 1854) and is systematised by Frege, Russell, Whitehead (Principia Mathematica 1910–13).

Connectives

Symbol	Name	Reading
¬	Negation	not p
∧	Conjunction	p and q
∨	Disjunction	p or q
→	Conditional	if p then q
↔	Biconditional	p if and only if q

Key inference rules

Rule	Form
Modus Ponens	p → q; p; ∴ q
Modus Tollens	p → q; ¬q; ∴ ¬p
Hypothetical Syllogism	p → q; q → r; ∴ p → r
Disjunctive Syllogism	p ∨ q; ¬p; ∴ q
Constructive Dilemma	(p → q) ∧ (r → s); p ∨ r; ∴ q ∨ s

Truth tables test validity of propositional forms; an argument is valid iff there is no row where premises are all true and conclusion false.

Predicate (first-order) logic

Frege's Begriffsschrift (1879) introduced quantifiers — ∀ (universal) and ∃ (existential) — and predicates with variables. This allows expression of propositions like:

• All humans are mortal: ∀x (H(x) → M(x))
• Some humans are wise: ∃x (H(x) ∧ W(x))
• There exists exactly one God: ∃x (G(x) ∧ ∀y (G(y) → x=y))

First-order logic is sound and complete (Gödel's Completeness Theorem 1929). Higher-order logics quantify over predicates or sets.

Induction

Inductive reasoning moves from particular cases to general conclusions or to specific predictions. Types:

• Enumerative: from instances to general — "every observed swan is white, so all swans are white".
• Analogical: from similarity in known cases.
• Causal: identifying cause-effect relationships.
• Statistical: from sample to population.

Hume's problem of induction (1748): induction itself cannot be justified deductively, and to justify it inductively is circular. Modern responses include Popper's falsificationism (Logic of Scientific Discovery 1934 / 1959), Bayesianism (probabilistic inference), and pragmatism.

Mill's methods of induction

John Stuart Mill (A System of Logic, 1843) identified five canons for identifying causes:

1. Agreement — common factor in all positive instances.
2. Difference — different factor between positive and negative.
3. Joint (Agreement and Difference combined).
4. Concomitant Variation — variation of factor with effect.
5. Residues — subtracting known causes leaves residue caused by remaining factor.

Informal fallacies

Fallacies are common errors of reasoning that may persuade despite their invalidity:

Fallacy	Pattern
Ad hominem	Attacking the person, not argument
Straw man	Misrepresenting opponent's position
Appeal to ignorance	Treating absence of evidence as evidence of absence
Appeal to authority	Citing inappropriate authority
Begging the question	Assuming conclusion in premise
False dilemma	Presenting only two options when more exist
Slippery slope	Asserting chain of consequences without basis
Post hoc ergo propter hoc	Confusing sequence with causation
Hasty generalisation	From inadequate sample
Equivocation	Shifting meaning of a term
Tu quoque	"You also" — dismissing argument by pointing to similar fault in arguer

Modal and non-classical logics

20th-century logic has proliferated:

• Modal logic — necessity (□) and possibility (◇); C.I. Lewis (1932), Saul Kripke (1959 — possible-worlds semantics).
• Deontic logic — obligation, permission, prohibition (von Wright 1951).
• Epistemic logic — knowledge and belief (Hintikka 1962).
• Temporal logic — past, present, future (Prior 1957).
• Many-valued logics — Łukasiewicz, fuzzy logic.
• Intuitionistic logic — Brouwer, Heyting; rejects law of excluded middle.
• Paraconsistent logic — handles contradiction without explosion (Priest).

Logic in Islamic philosophy

Islamic philosophers built on Aristotelian logic:

• Al-Farabi — extensive commentary on the Organon.
• Ibn Sina — al-Najat and al-Shifa logical sections; modal innovations.
• Ibn Rushd — Aristotle commentaries, central to Latin Averroism.
• Ibn Sina–Razi controversies on modal logic, contingency, necessity.
• The post-classical madrasa logic (Tahdhib, Sullam al-Ulum) persists in many Pakistani religious schools today.