Relations between statements in logic

Main Thing

Logical relations between statements describe the connections between simple propositions that can be true or false. The main types are:

1. Based on comparability of subjects and predicates:

   • Comparable - Statements with the same subjects and predicates but differing in quantifiers and connectives. Example: "All students study math" and "Some students do not study math."

   • Incomparable - Statements with different subjects and predicates. Example: "All students study math" and "Some athletes are Olympic champions."

2. Based on the ability to be true together:

   • Compatible - Statements that can be true together. Example: "Some people are athletes" and "Some people are not athletes."

   • Incompatible - Statements that cannot be true together. Example: "All students study math" and "Some students do not study math."

3. Based on identity of components:

   • Equivalence - Equivalent statements with identical subjects, predicates, quantifiers, and connectives. Example: "Moscow is an ancient city" and "The capital of Russia is an ancient city."

   • Subordination - Statements with the same predicates and connectives but subjects in a genus-species relation. Example: "All plants are living organisms" and "All flowers (some plants) are living organisms."

   • Partial Overlap - Statements with the same subjects and predicates but different connectives. Example: "Some mushrooms are edible" and "Some mushrooms are not edible."

4. Based on the truth values:

   • Opposition - Statements with the same subjects and predicates but opposite connectives that cannot be true together but can be false together. Example: "All people are truthful" and "No people are truthful."

   • Contradiction - Statements with the same predicates, opposite connectives, and subjects in a genus-species relation where one must be true and the other false. Example: "All people are truthful" and "Some people are not truthful."

The logical square is a diagram representing the relations between these four types of categorical statements: A (universal affirmative), E (universal negative), I (particular affirmative), and O (particular negative).

Terms

• Statement - A sentence that can be judged as true or false, such as "The sky is blue."

• Subject - The term in a statement about which something is being claimed or asserted. In linguistics, the subject corresponds to the noun or noun phrase that is the focus of the statement.

• Predicate - The part of a statement that says something about the subject. It expresses the action, state or property being attributed to the subject. In linguistics, the predicate corresponds to the verb or verb phrase.

• Universal Affirmative (A) - A statement of the form "All S are P", affirming that the entire subject is included in the predicate.

• Particular Affirmative (I) - A statement of the form "Some S are P", affirming that at least part of the subject is included in the predicate.

• Universal Negative (E) - A statement of the form "No S are P", denying that any part of the subject is included in the predicate.

• Particular Negative (O) - A statement of the form "Some S are not P", denying that the entire subject is included in the predicate.

Analogy

The relations between statements can be visualized using Euler circles or Venn diagrams, where the subjects and predicates are represented as overlapping or non-overlapping sets. Compatible statements are overlapping sets, while incompatible statements are non-overlapping sets.

A Main Misconception

Many people confuse the relations of opposition and contradiction between statements. Opposite statements cannot be true together but can be false together, while contradictory statements cannot be simultaneously true or false.

Example: "All birds can fly" and "No birds can fly" are opposite but can both be false if some birds can fly and some cannot. However, "All birds can fly" and "Some birds cannot fly" are contradictory - if one is true, the other must be false.

The History

1. In the Middle Ages (around 1200 CE), logicians introduced the concept of the logical square to represent relations between categorical statements.

2. In the 19th century, logicians developed the theory of relations between complex statements (conjunctions, disjunctions, etc.).

3. In 1879, Gottlob Frege introduced symbolic propositional calculus to formalize logical relations between statements.

"Logic is the anatomy of thought" (John Locke, English philosopher and founder of empiricism, 1632-1704).

Three Cases of Application

1. When analyzing arguments, one must determine the relations between the constituent statements to assess their logical coherence and consistency. For example, if two premises contradict each other, the argument is invalid.

2. In computer programming, logical operations (AND, OR, NOT) are based on relations between propositions and are used to construct complex conditions. For example: "If age >= 18 AND isStudent == true, then grantDiscount()."

3. In law, it is crucial to identify contradictions between laws and regulations to resolve conflicts. For instance, if one law states "X is legal" and another states "X is illegal," there is a contradiction that needs resolution.

Interesting Facts

• There are 6 distinct relations between statements in the logical square.

• The maximum number of simple statements in a complex one is 2^n, where n is the number of variables.

• In 1847, George Boole introduced Boolean algebra, representing statements with 0 and 1.

• The law of excluded middle does not hold for infinite sets, e.g., "All natural numbers are even or odd."

• Quantum logic uses fuzzy statements with truth values ranging from 0 to 1.