Formal logic

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Logic is a method by which valid conclusions are drawn from given premises. It can be applied to natural-language arguments or to formal, symbolic systems. Today logic provides the foundation upon which all areas of modern mathematics and science are ultimately based.

Types of logic

Informal logic is the study of natural-language arguments, especially the identification of logical fallacies.

Formal logic, on the other hand, involves the systematic study of logical reasoning usually within the framework of a symbolic system of terms, operations, quantifiers, axioms, theorems, etc.

Formal logic can be further subdivided into several related areas, including:

  1. Aristotelian logic (or syllogistic logic) — simple, deductive logic based on syllogisms.
  2. Propositional logic (or sentential calculus, etc.) — simple, deductive logic based on statements called propositions and the logical operators/connectives "not", "and", "or", "implies", etc.
  3. Predicate logic (or first-order logic, predicate calculus, etc.) — extends propositional logic by introducing the logical quantifiers "for all" and "for some" (or "there exists").
  4. Modal logic — extends predicate logic by introducing modal operators such as "possibly" or "formerly".
  5. Mathematical logic — reduces all of the above to symbolic manipulation based on well-defined rules.

The term classical logic may refer to the first type of logic listed above or the first two or three types; it generally does not include the forth or fifth type.

Deductive vs. inductive reasoning

The kinds of reasoning described by logic can be roughly classified as deductive or inductive.

  • Deductive reasoning — conclusions follow necessarily from premises; based on strict implications; sometimes placed in the context of reasoning "from the general to the specific".
    1. Example: "All men are mortal. Socrates is a man. Therefore, Socrates is mortal." (a syllogism)
    2. Example: "A person can't be both a woman and a man. Socrates is a man. Therefore, Socrates is not a woman."
Notice that each conclusion must be true if we assume the premises are true. The first premise of the second example, however, may not be true, depending on how you define "man" and "woman". If you don't accept the truth of either of the premises, then the argument is still logically valid, but is not sound.
  • Inductive reasoning — conclusions may often, but not always, follow from the premises; inferences based on incomplete or imperfect information; sometimes described in the context of reasoning "from the specific to the general".
    1. Example: "Every man I've known has had a beard. Socrates is a man. Therefore, Socrates must have a beard."
    2. Example: "If you're a woman, it is very unlikely you have a beard. Socrates has a beard. Therefore, Socrates is probably a woman."
The first argument looks very much like the first deductive example above. But notice that the first premise doesn't say "all" men have beards, only all men "I've known". Even if the first premise is true, that doesn't mean there's not a man out there you haven't met yet who doesn't have a beard; and Socrates may be such a man. Thus, the conclusion isn't necessarily true. As for the second example, it is a probabilistic argument of the kind that is often encountered in statistics (specifically, in hypothesis testing, also known as significance testing).
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