The Temperature Paradox or Partee's Paradox is a classic puzzle in
formal semantics and
philosophical logic
Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical ...
. Formulated by
Barbara Partee
Barbara Hall Partee (born June 23, 1940) is a Distinguished University Professor Emerita of Linguistics and Philosophy at the University of Massachusetts Amherst (UMass).
Biography
Born in Englewood, New Jersey, Partee grew up in the Baltimore ...
in the 1970s, it consists of the following argument, which speakers of
English
English usually refers to:
* English language
* English people
English may also refer to:
Peoples, culture, and language
* ''English'', an adjective for something of, from, or related to England
** English national ide ...
judge as wildly
invalid
Invalid may refer to:
* Patient, a sick person
* one who is confined to home or bed because of illness, disability or injury (sometimes considered a politically incorrect term)
* .invalid, a top-level Internet domain not intended for real use
As t ...
.
# The temperature is ninety.
# The temperature is rising.
# Therefore, ninety is rising. (invalid conclusion)
Despite its obvious invalidity, this argument would be valid in most formalizations based on traditional
extensional In any of several fields of study that treat the use of signs — for example, in linguistics
Linguistics is the science, scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, obj ...
systems of logic. For instance, the following formalization in
first order predicate logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantif ...
would be valid via
Leibniz's law:
# t=90
# R(t)
# R(90) (valid conclusion in this formalization)
To correctly predict the invalidity of the argument without abandoning Leibniz's Law, a formalization must capture the fact that the first premise makes a claim about the temperature at a particular point in time, while the second makes an assertion about how it changes over time. One way of doing so, proposed by
Richard Montague
Richard Merritt Montague (September 20, 1930 – March 7, 1971) was an American mathematician and philosopher who made contributions to mathematical logic and the philosophy of language. He is known for proposing Montague grammar to formalize th ...
, is to adopt an
intensional logic
Intensional logic is an approach to predicate logic that extends first-order logic, which has quantifiers that range over the individuals of a universe (''extensions''), by additional quantifiers that range over terms that may have such individuals ...
for natural language, thus allowing "the temperature" to denote its
extension
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* E ...
in the first premise and its
intension
In any of several fields of study that treat the use of signs — for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language — an intension is any property or quality connoted by a word, phrase, or anoth ...
in the second.
# extension(t)=90
# R(intension(t))
# R(90) (invalid conclusion)
Thus, Montague took the paradox as evidence that nominals denote ''individual concepts'', defined as functions from a
world
In its most general sense, the term "world" refers to the totality of entities, to the whole of reality or to everything that is. The nature of the world has been conceptualized differently in different fields. Some conceptions see the worl ...
-time pair to an individual. Later analyses build on this general idea, but differ in the specifics of the formalization.
Notes
External links
*
Non-classical logic
Philosophical logic
Predicate logic
Formal semantics (natural language)
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