Concepts of classical logic

Categories: Home -> Philosophy -> Logic

The idea of this page is to show very briefly what the main concepts of classical logic are that we need to know. As explained before, these concepts are only useful when we are doing serious analysis, not when lying prevails (for that one only responds with pure data).

Fundamental Principles (Laws of Thought)

These are the foundations on which all classical logical reasoning is built:

Principle of Identity:
- A thing is identical to itself. (A is A).
- Any object of knowledge (including a proposition) is exactly what it is. If a proposition is true, it is true.
Principle of Non-Contradiction:
- Nothing can be and not be at the same time and under the same aspect. It is not possible for A to be true and the negation of A (not A) to be true at the same time.
- It is not possible for a proposition to be true and false simultaneously.
Principle of the Excluded Middle:
- Every proposition is true or false; there is no third option. A is true or not-A is true. There is no third option.
- Given a proposition, it must have one of two truth values: True or False.
Principle of Sufficient Reason (Added by Leibniz):
- Everything that exists has a reason for being, a sufficient explanation of why it is as it is and not otherwise.
- It is the principle that requires that, to consider something true, there must exist a reason or foundation that supports it necessarily or sufficiently.
- Every fact or truth has a sufficient reason or cause for being so and not otherwise. Nothing occurs without a sufficient reason.
- This principle asks that every fact, claim, or existence have a foundation, justification, or explanation.

Note: The first three are accepted by classical logic. The principle of sufficient reason is more philosophical and is not universally accepted in modern logic (it is metaphysical, not formal-logical).

Categorical Logic (Syllogisms)

A Categorical Syllogism is a deductive argument composed of exactly three categorical propositions (two premises and a conclusion) that contain exactly three terms, each of which appears in two of the propositions.

Term	Name	Where it appears first	Where it appears in the conclusion
P	Major term	In the major premise (it is the predicate of the conclusion)	Predicate of the conclusion
S	Minor term	In the minor premise (it is the subject of the conclusion)	Subject of the conclusion
M	Middle term	Appears in both premises, but never in the conclusion	Does not appear

Example:

Major premise: All mammals (M) are animals (P). M = mammals
Minor premise: All dogs (S) are mammals (M). S = dogs
→ Conclusion: All dogs (S) are animals (P). P = animals

Categorical Propositions (A, E, I, O)

A categorical proposition is an affirmation or negation about the relation between two categories or classes: a Subject (S) and a Predicate (P).

The four standard forms are identified with vowels (taken from the Latin words AffIrmo and nEgO):

Type	Letter	Formula	Name
Universal Affirmative	A	All S are P.	Affirms that the entire class S is included in the class P.
Universal Negative	E	No S are P.	Affirms that the class S and the class P are completely separate.
Particular Affirmative	I	Some S are P.	Affirms that at least one member of S is also a member of P.
Particular Negative	O	Some S are not P.	Affirms that at least one member of S is excluded from the class P.

Quality and Quantity

Every categorical proposition has two distinctive characteristics:

Quantity

Indicates how many members of the subject class are affected:

Universal (A, E): Refers to all members of the subject class.
Particular (I, O): Refers to at least one (some) member of the subject class.

Quality

Indicates whether the predicate is affirmed or denied of the subject:

Affirmative (A, I): Affirms the membership of S in P (the subject is included, totally or partially, in the predicate).
Negative (E, O): Denies the membership of S in P (the subject is excluded, totally or partially, from the predicate).

Type	Quantity	Quality
A	Universal	Affirmative
E	Universal	Negative
I	Particular	Affirmative
O	Particular	Negative

Distribution of Terms

A term (Subject or Predicate) is distributed if the proposition refers to all members of the class designated by that term. If the proposition only refers to some members, the term is undistributed.

Type	Subject (S)	Predicate (P)	Mnemonic Rule
A	Distributed (All S)	Undistributed	All Subjects Distributed
E	Distributed (No S)	Distributed (No P)	Entire Subject and Predicate are Distributed
I	Undistributed (Some S)	Undistributed	Identically Indistributed
O	Undistributed (Some S)	Distributed (Not any P)	Only the Predicate is Distributed

Proposition	Classification
“All dogs are mammals.”	A
“No square is a circle.”	E
“Some student is Argentine.”	I
“Some politician is not honest.”	O

Key Rules:

Universal propositions (A, E) distribute the Subject.
Negative propositions (E, O) distribute the Predicate.

Practical meaning:

In A: I speak of all S, but not of all P.
In E: I speak of all S and all P as separate.
In I: I do not speak of all of either.
In O: I do not speak of all S, but I do exclude all P from what I deny.

The 8 Rules of Syllogism Validity

A syllogism is valid if and only if it satisfies all of these rules, guaranteeing that the structure of the argument is correct:

Rules of Structure (Quantity and Quality of the Propositions)

It must have exactly three terms: Major ($P$), Minor ($S$), and Middle ($M$). (Failure: Four Terms).
The Middle Term ($M$) must be distributed at least once. (Failure: Undistributed Middle).
Any term distributed in the conclusion must also be distributed in its premise. (Failure: Illicit Major or Illicit Minor).
There cannot be two negative premises. (If there are, the conclusion cannot be established).
If one of the premises is negative, the conclusion must be negative, and vice versa. (If the conclusion is negative, one premise must be so).
From two particular premises no conclusion follows. (At least one premise must be universal).
If both premises are universal, the conclusion cannot be particular. (This is the modern rule and applies when the "existential fallacy" is rejected).
If one premise is particular, the conclusion must be particular. (Derived from the previous rules).

Typical Syllogism Failures

Failure to comply with any of the previous rules results in a formal fallacy that invalidates the syllogism.

Name of the Failure	Rule Violated	Description
Fallacy of Four Terms	Rule 1	Four terms are used instead of three. This occurs through ambiguity or the metaphorical use of a term.
Fallacy of the Undistributed Middle	Rule 2	The Middle Term ($M$) is not distributed in either premise. There is no valid connection between $S$ and $P$.
Fallacy of the Illicit Major	Rule 3	The Major Term ($P$) is distributed in the conclusion, but not in its (major) premise.
Fallacy of the Illicit Minor	Rule 3	The Minor Term ($S$) is distributed in the conclusion, but not in its (minor) premise.
Fallacy of Negative Premises	Rule 4	Both premises are negative (E, O).
Existential Fallacy	Rule 7 (Modern)	The premises are universal and the conclusion is particular, forcing an implication of existence that is not guaranteed by the premises.

There are several more topics related to the syllogism and classical logic, but I think up to here is what is necessary to have an idea about classical logic. There are many sites and places where the topic is studied completely, and here I only want to give an idea that serves the purpose of the wiki.

To comment on or discuss this article

Click the link to access the discussions on the different networks

Concepts of Informal Logic

Categories: Home -> Philosophy

Last updated: 2025-12-04

| | | |

This site was written based on publicly available free articles from the internet.
The content of this site is available under the: Creative Commons Attribution 4.0 International (CC BY 4.0) license.
You can freely copy, redistribute and modify the material, as long as the source is mentioned: liberwiki.com