In
Aristotelian logic
In logic and formal semantics, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly b ...
, ''dictum de omni et nullo'' (
Latin
Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
: "the maxim of all and none") is the principle that whatever is affirmed or denied of a whole kind K may be affirmed or denied (respectively) of any subkind of K. This principle is fundamental to
syllogistic logic in the sense that all valid syllogistic argument forms are reducible to applications of the two constituent principles ''dictum de omni'' and ''dictum de nullo''.
''Dictum de omni''
''Dictum de omni'' (sometimes misinterpreted as
universal instantiation
In predicate logic, universal instantiation (UI; also called universal specification or universal elimination, and sometimes confused with '' dictum de omni'') is a valid rule of inference from a truth about each member of a class of individual ...
)
[See “Dictum de omni et nullo.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/dictum%20de%20omni%20et%20nullo. Accessed 18 Jul. 2023.] is the principle that whatever is universally affirmed of a kind is affirmable as well for any subkind of that kind.
Example:
(1) Dogs are mammals.
(2) Mammals have livers.
Therefore
(3) dogs have livers.
Premise (1) states that "dog" is a subkind of the kind "mammal".
Premise (2) is a (
universal affirmative) claim about the kind "mammal".
Statement (3) concludes that what is true of the kind "mammal" is true of the subkind "dog".
''Dictum de nullo''
''Dictum de nullo'' is the related principle that whatever is denied of a kind is likewise denied of any subkind of that kind.
Example:
(1) Dogs are mammals.
(4) Mammals do not have gills.
Therefore
(5) dogs do not have gills.
Premise (1) states that "dog" is a subkind of the kind "mammal".
Premise (4) is a (
universal negative) claim about the kind "mammal".
Statement (5) concludes that what is denied of the kind "mammal" is denied of the subkind "dog".
Discussion
In
Aristotelean syllogistic, these two principles correspond respectively to the two argument forms, Barbara and Celarent.
These principles correspond roughly to a valid argument form known as
universal hypothetical syllogism in
first-order predicate logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...
. Nevertheless, Aristotelean syllogistic does not employ the formal machinery of first-order quantification. This by itself accounts for why it is incorrect to identify ''Dictum de omni'' as universal instantiation. The latter mistake is more easily explained by the fact that universal instantiation is a single-premise form of deduction that is not even a syllogism.
See also
*
Aristotle
Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
*
Syllogism
A syllogism (, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true.
In its earliest form (defin ...
*
Term logic
In logic and formal semantics, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by ...
*
Class (philosophy)
*
Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Classes act as a way to have set-like ...
*
Natural kind
In the philosophy of science and some other branches of philosophy, a "natural kind" is an intellectual grouping, or categorizing of things, that is reflective of the actual world and not just human interests. Some treat it as a classification ide ...
*
Type (metaphysics)
Type may refer to:
Science and technology Computing
* Typing, producing text via a keyboard, typewriter, etc.
* Data type, collection of values used for computations.
* File type
* TYPE (DOS command), a command to display contents of a file.
* ...
*
Downward entailing In linguistics, linguistic semantics, a downward entailing (DE) propositional operator is one that constrains the meaning of an expression to a lower number or degree than would be possible without the expression. For example, "not," "nobody," "few ...
*
Monotonic function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of or ...
References
*
Aristotle
Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
, ''
Prior Analytics'', 24b, 28–30.
Notes
External links
*{{cite SEP , url-id=logical-form , title=Logical Form , last=Pietroski , first=Paul
Latin logical phrases
Arguments
Term logic
Ancient Greek logic
Inference