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Concept
Intensional logic
Summary
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 as their value (intensions). The distinction between intensional and extensional entities is parallel to the distinction between sense and reference.
Logic is the study of proof and deduction as manifested in language (abstracting from any underlying psychological or biological processes). Logic is not a closed, completed science, and presumably, it will never stop developing: the logical analysis can penetrate into varying depths of the language (sentences regarded as atomic, or splitting them to predicates applied to individual terms, or even revealing such fine logical structures like modal, temporal, dynamic, epistemic ones).
In order to achieve its special goal, logic was forced to develop its own formal tools, most notably its own grammar, detached from simply making direct use of the underlying natural language. Functors (also known as function words) belong to the most important categories in logical grammar (along with basic categories like sentence and individual name): a functor can be regarded as an "incomplete" expression with argument places to fill in. If we fill them in with appropriate subexpressions, then the resulting entirely completed expression can be regarded as a result, an output. Thus, a functor acts like a function sign, taking on input expressions, resulting in a new, output expression.
Semantics links expressions of language to the outside world. Also logical semantics has developed its own structure. Semantic values can be attributed to expressions in basic categories: the reference of an individual name (the "designated" object named by that) is called its extension; and as for sentences, their truth value is their extension.
As for functors, some of them are simpler than others: extension can be attributed to them in a simple way.
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Two-dimensionalism
Two-dimensionalism is an approach to semantics in analytic philosophy. It is a theory of how to determine the sense and reference of a word and the truth-value of a sentence. It is intended to resolve the puzzle: How is it possible to discover empirically that a necessary truth is true? Two-dimensionalism provides an analysis of the semantics of words and sentences that makes sense of this possibility. The theory was first developed by Robert Stalnaker, but it has been advocated by numerous philosophers since, including David Chalmers.
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 another symbol. In the case of a word, the word's definition often implies an intension. For instance, the intensions of the word plant include properties such as "being composed of cellulose (not always true)", "alive", and "organism", among others. A comprehension is the collection of all such intensions.
Temperature paradox
The Temperature Paradox or Partee's Paradox is a classic puzzle in formal semantics and philosophical logic. Formulated by Barbara Partee in the 1970s, it consists of the following argument, which speakers of English judge as wildly invalid. 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 systems of logic.