kb/data/en.wikipedia.org/wiki/Logic-6.md

title	chunk	source	category	tags	date_saved	instance
Logic	7/11	https://en.wikipedia.org/wiki/Logic	reference	science, encyclopedia	2026-05-05T03:56:39.118518+00:00	kb-cron

In Aristotelian logic, the subject can be universal, particular, indefinite, or singular. For example, the term "all humans" is a universal subject in the proposition "all humans are mortal". A similar proposition could be formed by replacing it with the particular term "some humans", the indefinite term "a human", or the singular term "Socrates". Aristotelian logic only includes predicates for simple properties of entities. But it lacks predicates corresponding to relations between entities. The predicate can be linked to the subject in two ways: either by affirming it or by denying it. For example, the proposition "Socrates is not a cat" involves the denial of the predicate "cat" to the subject "Socrates". Using combinations of subjects and predicates, a great variety of propositions and syllogisms can be formed. Syllogisms are characterized by the fact that the premises are linked to each other and to the conclusion by sharing one term in each case. Thus, these three propositions contain three terms, referred to as major term, minor term, and middle term. The central aspect of Aristotelian logic involves classifying all possible syllogisms into valid and invalid arguments according to how the propositions are formed. For example, the syllogism "all men are mortal; Socrates is a man; therefore Socrates is mortal" is valid. The syllogism "all cats are mortal; Socrates is mortal; therefore Socrates is a cat", on the other hand, is invalid.

=== Classical ===

Classical logic is distinct from traditional or Aristotelian logic. It encompasses propositional logic and first-order logic. It is "classical" in the sense that it is based on basic logical intuitions shared by most logicians. These intuitions include the law of excluded middle, the double negation elimination, the principle of explosion, and the bivalence of truth. It was originally developed to analyze mathematical arguments and was only later applied to other fields as well. Because of this focus on mathematics, it does not include logical vocabulary relevant to many other topics of philosophical importance. Examples of concepts it overlooks are the contrast between necessity and possibility and the problem of ethical obligation and permission. Similarly, it does not address the relations between past, present, and future. Such issues are addressed by extended logics. They build on the basic intuitions of classical logic and expand it by introducing new logical vocabulary. This way, the exact logical approach is applied to fields like ethics or epistemology that lie beyond the scope of mathematics.

==== Propositional logic ====

Propositional logic comprises formal systems in which formulae are built from atomic propositions using logical connectives. For instance, propositional logic represents the conjunction of two atomic propositions

    P
  

{\displaystyle P}

and

    Q
  

{\displaystyle Q}

as the complex formula

    P
    ∧
    Q
  

{\displaystyle P\land Q}

. Unlike predicate logic where terms and predicates are the smallest units, propositional logic takes full propositions with truth values as its most basic component. Thus, propositional logics can only represent logical relationships that arise from the way complex propositions are built from simpler ones. But it cannot represent inferences that result from the inner structure of a proposition.

==== First-order logic ====

First-order logic includes the same propositional connectives as propositional logic but differs from it because it articulates the internal structure of propositions. This happens through devices such as singular terms, which refer to particular objects, predicates, which refer to properties and relations, and quantifiers, which treat notions like "some" and "all". For example, to express the proposition "this raven is black", one may use the predicate

    B
  

{\displaystyle B}

for the property "black" and the singular term

    r
  

{\displaystyle r}

referring to the raven to form the expression

    B
    (
    r
    )
  

{\displaystyle B(r)}

. To express that some objects are black, the existential quantifier

    ∃
  

{\displaystyle \exists }

is combined with the variable

    x
  

{\displaystyle x}

to form the proposition

    ∃
    x
    B
    (
    x
    )
  

{\displaystyle \exists xB(x)}

. First-order logic contains various rules of inference that determine how expressions articulated this way can form valid arguments, for example, that one may infer

    ∃
    x
    B
    (
    x
    )
  

{\displaystyle \exists xB(x)}

from

    B
    (
    r
    )
  

{\displaystyle B(r)}

.

=== Extended === Extended logics are logical systems that accept the basic principles of classical logic. They introduce additional symbols and principles to apply it to fields like metaphysics, ethics, and epistemology.

==== Modal logic ====