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Could all systems of logic (not only classical logic 1, but also non-classical logics like intuitionistic logic 2, quantum logic 3, many-valued logic 4, modal logic 5, paraconsistent logic 6...etc) be "derived" from propositional logic?

In this lecture from a software engineering teacher 7, it says that although there are hundreds of different logics that have been studied throughout history, propositional logic can be considered as the most simple form of logic. Does it mean that it is the most basic one? Does it mean that it's the "bedrock" of every studied logic?

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  • "derived" no. It is "more basic" only in the sense that the "model" of language used is very very simplified: no analysis of statement into its components: subject, predicate, etc. but only sentential connectives. Nov 8, 2021 at 7:14
  • Usually, it is the first part to be studied because it is quite simple but it is still enough to introduce the basic concepts, like validity, interpretation, inference. Nov 8, 2021 at 7:15
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    No. There are alternative formalizations of logic that are differently structured and do not go through propositional logic (then possibly adding quantifiers and/or operators). For example, Aristotle's syllogistic was term logic, and there are modern extensions of it that go by the name term-functor logic or relational syllogistic, see Sommers-Englebretsen book. Another alternative is Peirce-Taski relation algebra.
    – Conifold
    Nov 8, 2021 at 10:35

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Propositional logic is basic in the sense that because it is zero-order and does not include quantification, it is a fragment of quantifier logic. But it is not a basic form of all logics in the sense you are asking. Intuitionistic logic, quantum logic, etc., all have their propositional fragments, which are distinct from each other. Classical logic, for example, includes ¬¬P ⊢ P (double negation elimination) while intuitionistic logic does not. The lecture notes in your link relate to classical propositional logic, since they use boolean truth tables.

The various non-classical logics have sufficiently different algebras that it is not feasible to identify a common core of rules that applies to all. There is even a relevance logic that does not have P ⊢ P as a theorem. In some cases, logics can be partially ordered, e.g. the theorems of minimal logic are a proper subset of those of intuitionistic logic, which in turn are a proper subset of those of classical logic. But there is little if anything that all logics have in common.

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Propositional logic offers obvious congenial features to us, being a calculus solely on a notion residing at the heart of logic. Its status can be likened to Euclidean geometry with respect to non-Euclidean ones, which is historically prior and pedagogically familiar, however, mathematically speaking, does not bear an inherent privilege. In general, it is better to assess logical systems by their metalogical properties.

The following metalogical properties are the ones I can cite at once:

  • Consistency
  • Soundness
  • Completeness
  • Decidability
  • Compactness
  • Löwenheim-Skolem property
  • Categoricity
  • Expressiveness (of the associated formal language)
  • Finite axiomatisability.

One may count in others, count out some of them, or ramify into types according to one's purposes.

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