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Major types of logic

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Syllogistic logic

Organon is Aristotle’s principal work of logic, including the Prior Analytics; it constitutes the first explicit work of formal logic, with in particular the introduction of syllogistics.

The works of Aristotle are considered in Europe and the Middle East in the classical, medieval period as the image of a fully developed system. However, Aristotle was not the only one, nor the first: the Stoics proposed a system of propositional logic that was studied by medieval logicians. Also, the problem of multiple generality was recognized in medieval times.

Propositional logic

Propositional calculus is a formal system in which formulas represent propositions that can be formed by combining atomic propositions and using logical connectors, and in which a system of formal proof rules establishes certain “theorems”.

Calculus of predicates

∀x.F(x) este adevărat(Gottlob Frege’s Begriffschrift introduced the notion of quantifiers into logical notation, which here represents the formula ∀x. F(x), which is true.)

A predicate calculus is a formal system, which can be either first-order logic, second-order logic, higher-order logic, or infinitary logic. It expresses through quantification a large sample of natural language propositions. For example, Bertrand Russell’s barber paradox, “there is a man who shaves all men who do not shave themselves” can be formalized by the formula: (∃x)(man(x) ∧ (∀y)(man(y) → (shave(x,y) ↔ ¬ shave(y,y)))) using the predicate man(x) to indicate that x is a man, the binary relation shave(x,y) to indicate that {y is shaved by x and other symbols to express quantification, conjunction, implication, negation, and equivalence.

Modal logic

In natural language, a modality is an inflection or addition to modify the semantics of a proposition.

For example, the statement “We are going to the games” can be changed to “We should go to the games”, or “We can go to the games” or “We will go to the games” or “We have to go to the games”.

More abstractly, modality affects the frame in which an assertion is satisfied.

In formal logic, a modal logic is a logic extended by the addition of operators, which are applied to propositions to modify their meaning.

Philosophical logic

Philosophical logic deals with formal descriptions of natural language. These philosophers consider that the essence of everyday reasoning can be transcribed into logic, if one or more methods succeed in translating ordinary language into this logic. Philosophical logic is essentially an extension of traditional logic that predates mathematical logic and is concerned with the connection between natural language and logic.

Therefore, philosophical logicians have contributed greatly to the development of non-standard logics (e.g. free logics, temporal logics) as well as various extensions of logic (e.g. modal logics) and the semantics of these logics (e.g., Kripke’s supervaluationism in the semantics of logic).

Basic notions of formal logic

A logical language is defined by a syntax, that is to say a system of symbols and rules for combining them in the form of formulas. Moreover, a semantics is associated with the language. It makes it possible to interpret it, that is to say to attach a meaning to these formulas as well as to the symbols. A deduction system makes it possible to reason by constructing demonstrations.

Logic typically includes:

  • propositional logic (also called propositional calculus),
  • predicate logic, which contains notations for entities with quantifications over those entities

Plus:

  • combinatorial logic based on the notions of function and application, in connection with lambda-calculus and intuitionistic logic.

Syntaxes

The syntax of propositional logic is based on propositional variables also called atoms that we denote with lowercase letters (p, q, r, s, etc.). These symbols represent propositions on which we do not pass judgment on their truth: they can be either true or false, but we can also mean nothing about their status. These variables are combined by means of logical connectors which are, for example:

  1. The disjunctive binary connector (or), with symbol: ∨;
  2. The connective binary connective (and), with symbol: ∧;
  3. The binary connector of implication, with symbol: →;
  4. The unary or monadic connector of the negation (no), with symbol: ¬.

These variables then form complex formulas.

The syntax of second-order logic, unlike that of first-order, considers:

  • the terms: representing the studied objects,
  • formulas: properties of those studied objects.

In the following we will note V the set of variables (x, y, z…), F the set of function symbols (f, g…) and P the set of predicate symbols (P, Q…). There is also a so-called m-arity map. The meaning of the formulas is the subject of semantics and differs according to the language considered.

In traditional logic (also called classical logic or “excluded middle” logic), a formula is either true or false. More formally, the set of truth values ​​is a set B of two booleans: true and false. The meaning of connectors is defined using Boolean to Boolean functions. These functions can be represented in the form of a truth table.

The meaning of a formula therefore depends on the truth value of its variables. We are talking about interpretation or affectation. However, it is difficult, in the sense of algorithmic complexity, to use semantics to decide whether a formula is satisfactory (or not) or even valid (or not). It would be necessary for that to be able to enumerate all the interpretations which are exponential in number.

An alternative to semantics is to examine well-formed proofs and consider their conclusions. This is done in a deduction system. A deduction system is a couple (A, R), where A is a set of formulas called axioms and R a set of rules of inference, that is to say relations between sets of formulas (the premises) and formulas (the conclusion).

A derivation from a given set of assumptions is a non-empty sequence of formulas that are: either axioms or formulas deduced from previous formulas in the sequence. A proof of a formula ϕ from a set of formulas Γ is a derivation from Γ whose last formula is ϕ.

Quantification

Essentially two quantifiers are introduced into modern logic:

  • ∃: there is at least one, called an “existential quantifier”;
  • ∀: for everything, called “universal quantifier”.

Thanks to negation, existential and universal quantifiers play dual roles and therefore, in classical logic, one can base the computation of predicates on a single quantifier.

Equality

A binary predicate, called equality, states that two terms are equal when they represent the same object. It is managed by specific axioms or schemas of axioms. However among the binary predicates it is a very particular predicate, whose usual interpretation is not only constrained by its properties stated by the axioms: in particular there is usually only one possible equality predicate per model, the one that corresponds to the expected interpretation (the identity). Its addition to the theory preserves some good properties like the completeness theorem of classical predicate calculus. We therefore very often consider that equality is part of the basic logic and we then study the calculus of egalitarian predicates.

In a theory that contains equality, a quantifier, which can be defined from the previous quantifiers and equality, is often introduced:

  • ∃! (there is one and only one).

Other quantifiers can be introduced into the calculus of egalitarian predicates (there exists at most one object verifying such a property, there exist two objects…), but useful quantifiers in mathematics, such as “there is an infinity…” or “there is one finite number…” cannot be represented there and require other axioms (such as those of set theory).

Non-binary logic

It wasn’t until the early 20th century that the principle of bivalence was clearly challenged in several different ways:

  • The first way considers trivalent logics that add indeterminate value, they are due to Stephen Cole Kleene, Jan Łukasiewicz and Bochvar and generalize to general-purpose logics.
  • The second way insists on the demonstrable. So there is what is demonstrable and the rest. In this “remainder” there can be refutable propositions, i.e., whose negation is demonstrable, and propositions of uncertain status, neither demonstrable nor refutable. This approach, due in particular to Gödel, is entirely compatible with classical bivalent logic, and one can even say that one of the contributions of 20th century logic is to have clearly analyzed the difference between demonstrability and validity, which is based on an interpretation in terms of truth values. But intuitionist logic is based on an interpretation of demonstrations, Heyting’s semantics — thus a proof of implication is interpreted by a function which associates a proof of the hypothesis with a proof of the conclusion, rather than on a interpretation of statements by truth values. We were however able afterwards to give semantics which interpret the statements, like that of Beth, or that of Kripke in which the basic concept is that of the possible world. Intuitionist logic is also used to analyze the constructive character of demonstrations in classical logic. Linear logic goes even further in the analysis of proofs.
  • The third way is due to Lotfi Zadeh who develops a fuzzy logic, in which a proposition is true according to a certain degree of probability (degree to which a degree of probability is assigned).
  • The fourth way is that of modal logic which, for example, attenuates (possible) or reinforces (necessary) propositions. If Aristotle was already interested in modalities, the 20th century, under the initial impetus of Clarence Irving Lewis, brings a more in-depth study of these, and Saul Aaron Kripke gives an interpretation of the statements of modal logics using possible worlds.

Translated and adapted from Wikipedia by Nicolae Sfetcu

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