Introduction to Logic: Arguments, Reasoning, and Syllogisms
1. The General Concept of Logic
Logic is the science that studies arguments and determines which are correct and incorrect. It is the science of correct deductions. To distinguish correctness, it is necessary to discover the rules of correct reasoning—this is the work of logic.
2. Validity and Truth of an Argument
It is crucial to distinguish between the correctness and truth of an argument. An argument is valid if it respects the laws of deduction and invalid if it violates them. The rules refer to the ways of linking the components of reasoning, not their content (logic studies how to reason, not the subject matter itself).
- Modus Ponens: A correct way of reasoning; the conclusion may be false.
- Fallacy of Affirming the Consequent: An incorrect form of reasoning; the conclusion may be true (a corrupt form of modus ponens).
3. Notion of Reasoning
Reasoning is a sequence of two or more premises (propositions) that lead to another proposition called the conclusion.
4. Aristotle’s Contribution to Logic
Aristotle is considered the creator of logic as a science, although Plato intuited the need for a science capable of establishing the laws of correct reasoning. Aristotle significantly developed the categorical syllogism.
5. Logical Term: Extension and Intension
Every term has a meaning, which is the set of attributes or qualities conveyed by the term (intension). Extension refers to the group of individuals to which a term applies.
6. Logical Judgment: Quantity and Quality
The medieval trial defined judgment as that logical form by which a concept (predicate) is affirmed or denied of a subject (grammatical sentences have truth or false values). Propositions are affirmative or negative based on their quality, and universal or particular based on their quantity.
7. Classification of Propositions by Quality and Quantity
- A: Universal Affirmative
- E: Universal Negative
- I: Particular Affirmative
- O: Particular Negative
8. Proposition Conversion
Propositions can be converted into others while maintaining their logical value. This must be done according to strict rules.
9. Proposition Conversion Modes
There are three modes of conversion:
- Simple Conversion (E, I): Swapping the subject and predicate.
- Conversion by Accident (E, A): The universal subject becomes particular.
- Conversion by Opposition (A, O): Swapping the subject and predicate, and negating the proposition while keeping the original quality.
10. Opposition of Propositions
Two propositions are opposed when they have the same subject and predicate but different quantity and quality.
11. Notion of Categorical Syllogism
A categorical proposition is one that makes an absolute assertion, without any conditions.
12. Structure of the Categorical Syllogism: Moods and Figures
A categorical syllogism is composed of a subject, predicate, and middle term. Placing the middle term in different positions creates different figures of the syllogism. The possible combinations of propositions within each figure are called moods. There are 19 valid moods.
13. Notion of Conditional and Hypothetical Syllogism
In a conditional syllogism, the conclusion is derived from two premises, the first of which is conditional. Two valid inference forms are:
- Modus Ponens: If A, then B. A is true. Therefore, B is true.
- Modus Tollens: If A, then B. B is not true. Therefore, A is not true.
14. Boolean Logic and Mathematicism
George Boole, with his work The Mathematical Analysis of Logic (1847), further developed Aristotelian syllogistic logic. He discussed terms and syllogisms as sharing properties. This initiated the program of mathematicism, which considers logic as a chapter of mathematics.
15. Frege’s Contribution to Logic
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