Syllogistic Reasoning – Advanced Study Notes
(Designed for a university‑level course, e.g., MIT CS 181 / Philosophy 201)
1. Introduction
A syllogism is a deductive argument consisting of exactly two premises and a single conclusion, all of which are categorical propositions (statements that assert something about a class of objects).
Historically, the study of syllogistic logic dates back to Aristotle (3rd c. BCE) and was formalised by the catenae (catenas) of medieval logicians. In modern terms, syllogistic reasoning is a small, yet foundational fragment of first‑order logic (FOL).
Objective – We aim to derive a first‑principles account of syllogisms, understand the algebraic structure that governs their validity, and explore subtle edge‑cases that arise in practice.
2. Categorical Propositions – The Building Blocks
| Symbol | English | Formal Notation | Example |
|---|---|---|---|
| A | Universal affirmative | ∀x (P(x) → Q(x)) | “All dogs are mammals.” |
| E | Universal negative | ∀x (P(x) → ¬Q(x)) | “No dogs are reptiles.” |
| I | Particular affirmative | ∃x (P(x) ∧ Q(x)) | “Some dogs are friendly.” |
| O | Particular negative | ∃x (P(x) ∧ ¬Q(x)) | “Some dogs are not friendly.” |
- Quantifier: ∀ (all), ∃ (some).
- Subject: the first term (P).
- Predicate: the second term (Q).
2.1 Existential Import
- Universal propositions do not automatically assert the existence of their subjects (classical view).
- Particular propositions do assume existence.
- Modern (Fregean) view: A and E do not carry existential import; only I and O do.
Note: The choice of view affects the validity of certain syllogistic figures (see § 4.3).
3. Syllogistic Figures and Moods
3.1 Figures (Arrangement of Middle Term)
| Figure | First Premise | Second Premise | Middle Term |
|---|---|---|---|
| I | M – P | S – M | M appears in middle position |
| II | P – M | S – M | M appears in middle position |
| III | M – P | M – S | M appears in both premises |
| IV | P – M | M – S | M appears in both premises |
Notation: “X – Y” means X is the subject, Y the predicate.
3.2 Moods (Combination of Proposition Types)
A mood is a triple of letters (e.g., AEI). The first two letters denote the premises, the third the conclusion.
There are 256 possible moods (4⁴); only 24 are traditionally valid (see below).
| Figure | Valid Moods | Example |
|---|---|---|
| I | AEE, AIE, EAE, EIO, IIO, OII | All M are P; All S are M; therefore All S are P. |
| II | AEE, AIE, EAE, EIO, IIO, OII | All P are M; All S are M; therefore All S are P. |
| III | AEI, EIE, IIO, OII | All M are P; All M are S; therefore All S are P. |
| IV | AAI, EAI, IAI, OAI | All P are M; All M are S; therefore All P are S. |
Theorem (Aristotle) – A syllogism is valid iff it belongs to one of the 24 moods above.
4. Rigorous Derivation of Validity
4.1 Formal Translation
Consider a syllogism in Figure I:
- Premise 1: All M are P → ∀x (M(x) → P(x))
- Premise 2: All S are M → ∀x (S(x) → M(x))
- Conclusion: All S are P → ∀x (S(x) → P(x))
We prove validity by direct deduction:
- Assume ∀x (S(x) → M(x)) (Premise 2).
- Assume ∀x (M(x) → P(x)) (Premise 1).
- Let a be arbitrary with S(a).
- By Premise 2, M(a).
- By Premise 1, P(a).
- Thus, ∀x (S(x) → P(x)).
Since a was arbitrary, the conclusion holds for all a.
This derivation uses only universal instantiation and modus ponens.
4.2 Algebraic Approach
Introduce the cumulative operation ∩ (intersection) for classes.
Let S, M, *