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Documentation on the Trivium Logic.

from Wikipedia

Logic is the branch of philosophy concerned with the use and study of valid reasoning.
The study of logic also features prominently in mathematics and computer science.
Logic was studied in several ancient civilizations, including Greece, India, and China.
In the West, logic was established as a formal discipline by Aristotle, who gave it a fundamental place in
philosophy.

The study of logic was part of the classical trivium, which also included grammar and rhetoric.
Logic was further extended by Al-Farabi who categorized it into two separate groups (idea and proof).
Later, Avicenna revived the study of logic and developed relationship between temporalis and the implication.

In the East, logic was developed by Hindus, Buddhists and Jains.

Logic is often divided into three parts: inductive reasoning, abductive reasoning, and deductive reasoning.

from Wikipedia

Abductive reasoning (also called abduction, abductive inference or retroduction) is a form of logical
inference which goes from an observation to a theory which accounts for the observation, ideally seeking to find the
simplest and most likely explanation.

In abductive reasoning, unlike in deductive reasoning, the premises do not guarantee the conclusion.
One can understand abductive reasoning as "inference to the best explanation".

The fields of law, computer science, and artificial intelligence research renewed interest in the subject of
abduction.
Diagnostic expert systems frequently employ abduction.

from Wikipedia
philpapers.org

Deductive reasoning, also deductive logic, logical deduction or, informally, "top-down" logic, is the process
of reasoning from one or more statements (premises) to reach a logically certain conclusion.
It differs from inductive reasoning or abductive reasoning.

Deductive reasoning links premises with conclusions.
If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion
reached is necessarily true.

Deductive reasoning (top-down logic) contrasts with inductive reasoning (bottom-up logic) in the following
way: In deductive reasoning, a conclusion is reached reductively by applying general rules that hold over the
entirety of a closed domain of discourse, narrowing the range under consideration until only the conclusion(s) is
left.

In inductive reasoning, the conclusion is reached by generalizing or extrapolating from, i.e., there is
epistemic uncertainty.
However, the inductive reasoning mentioned here is not the same as induction used in mathematical proofs –
mathematical induction is actually a form of deductive reasoning.

from Wikipedia
Properties of inductive reasoning

Inductive reasoning (as opposed to deductive reasoning or abductive reasoning) is reasoning in which the
premises are viewed as supplying strong evidence for the truth of the conclusion.
While the conclusion of a deductive argument is certain, the truth of the conclusion of an inductive argument is
probable, based upon the evidence given.

Many dictionaries define inductive reasoning as reasoning that derives general principles from specific
observations, though some sources disagree with this usage.

The philosophical definition of inductive reasoning is more nuanced than simple progression from
particular/individual instances to broader generalizations.
Rather, the premises of an inductive logical argument indicate some degree of support (inductive probability) for
the conclusion but do not entail it; that is, they suggest truth but do not ensure it.
In this manner, there is the possibility of moving from general statements to individual instances (for example,
statistical syllogisms).

Trivium Education

LOGIC — (Answers the Why of a subject.)

Developing the faculty of reason in establishing valid [i.e., non-contradictory] relationships among facts
yields basic, systematic Understanding– it is a guide for thinking correctly; thinking without contradiction.
More concisely, it is the art of non-contradictory identification.

The work of logic is proof.
Proof consists of establishing the truth and validity of a concept or proposition in correspondence with objective,
factual reality by following a self-consistent chain of higher-level thought back down to foundational, primary
concepts or axioms (i.e., Existence, Consciousness, and Causality).
It is a means of keeping us in touch and grounded to objective reality in our search for valid knowledge and
understanding.

Logic brings the rhythm of the subjective thoughts of the mind, and the subsequent actions of the body, into
harmony with the rhythm of the objective universe.
The intention is to amicably synchronize individual mental processes, and their attendant actions, with the
processes of our surrounding natural, factual existence over the period of a lifetime.

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