Suppose we are in the year 1995. Suppose Mary believes that Pluto (at the time still considered a planet) is the farthest planet from the sun. Because of Plutoâs irregular orbit, the orbit of Pluto crossed the orbit of Neptune, so that in 1995, the farthest planet from the sun is Neptune. Suppose Mary does not know this fact.
If x = y and y = z, then substituting z for y, x = z.
(1) Mary believes that Pluto = the farthest planet from the sun.
(2) Neptune = the farthest planet from the sun.
Therefore, substituting âNeptuneâ for âthe farthest planet from the sunâ in (1), we get
(3) Mary believes that Pluto = Neptune.
However, Mary does not believe that Pluto is Neptune, a paradox.
The FregeâChurch ontology resolves this by saying the belief introduces an "intensional context" whereby the terms following the words "believes that" are in a context whereby they refer not to the denotation of the words, but to the
concept associated with the words for the believer. Each word has a name, a denotation, and a concept associated with it.
Terminology
Propositions, properties, and relationships
An object has properties. A banana has the property of being yellow.
A
proposition is a sentence that is either true or false. A proposition can be considered to be a
function, with
objects in it considered as
variables, and the value of the function being either
truth or
falsity, a
truth function. For example, write âx is yellowâ as Y(x), so that Y(x) = Truth, if and only if âx is yellowâ is true, and Y(x) = Falsity if and only if âx is Yellowâ is false. For example, Y(banana) = Truth, since a banana is yellow. However, Y(apple) = Truth also, since some apples are yellow.
Similarly a sentence expressing a relationship between two objects can be considered a truth function of two variables, that is, a relationship between two objects can be considered to be a truth function of two variables. For example, let S(x, y) = âx is smaller than yâ. So S(mouse, elephant) = Truth, since a mouse is smaller than an elephant, but S(mouse, ant) = Falsity, since a mouse is not smaller than an ant.
Object, name, concept
An object (referent, denotation) has a name, the name of the object. The object has a concept (sense), the concept of the object, associated with the name of the object. A name or concept are themselves objects, and have names, the name of the name of the object, and the name of the concept of the object. Similarly they have concepts as any other object. A name is said to
denote the object for which it is the name.
Resolution of the paradox of the name relationship using the FregeâChurch ontology
Ambiguities in ordinary language lead to confusion
The English
ordinary language has ambiguities that need to be clarified as we sometimes refer to an object with a word, e.g., a cat. We refer to the name by using scare quotes, the name of the cat, e.g., the word âcatâ. There is ambiguity in the language as regards referring to the cat as a concept, and cat as an object.
Intensional context
An expression such as âbelieves thatâ is said to introduce an
intensional context. In an intensional context, the names that occur denote the concepts of the objects for the believer. They do not denote the objects themselves.
Resolution
âThe farthest planet from the sunâ, as it appears in proposition (1) is Maryâs concept of âthe farthest planet from the sunâ, not about the actual farthest planet from the sun as it appears in (2), so the substitution cannot be done. A more rigorous and formal treatment of this is given by Church.[1]
References
^
abChurch, Alonzo. "A Formulation of the Logic of Sense and Denotation." In Structure, Method and Meaning: Essays in Honor of Henry M. Sheffer, edited by P. Henle, H. Kallen and S. Langer, 3â24. New York: Liberal Arts Press, 1951.
^Gottlob Frege. "Ăber Sinn und Bedeutung" in Zeitschrift fĂŒr Philosophie und philosophische Kritik 100: 25â50. Translation: "On Sense and Reference" in Geach and Black (1980).
^
Gottlob Frege. "Ăber Begriff und Gegenstand" in Vierteljahresschrift fĂŒr wissenschaftliche Philosophie 16: 192â205. Translation: "Concept and Object" in Geach and Black (1980).
Suppose we are in the year 1995. Suppose Mary believes that Pluto (at the time still considered a planet) is the farthest planet from the sun. Because of Plutoâs irregular orbit, the orbit of Pluto crossed the orbit of Neptune, so that in 1995, the farthest planet from the sun is Neptune. Suppose Mary does not know this fact.
If x = y and y = z, then substituting z for y, x = z.
(1) Mary believes that Pluto = the farthest planet from the sun.
(2) Neptune = the farthest planet from the sun.
Therefore, substituting âNeptuneâ for âthe farthest planet from the sunâ in (1), we get
(3) Mary believes that Pluto = Neptune.
However, Mary does not believe that Pluto is Neptune, a paradox.
The FregeâChurch ontology resolves this by saying the belief introduces an "intensional context" whereby the terms following the words "believes that" are in a context whereby they refer not to the denotation of the words, but to the
concept associated with the words for the believer. Each word has a name, a denotation, and a concept associated with it.
Terminology
Propositions, properties, and relationships
An object has properties. A banana has the property of being yellow.
A
proposition is a sentence that is either true or false. A proposition can be considered to be a
function, with
objects in it considered as
variables, and the value of the function being either
truth or
falsity, a
truth function. For example, write âx is yellowâ as Y(x), so that Y(x) = Truth, if and only if âx is yellowâ is true, and Y(x) = Falsity if and only if âx is Yellowâ is false. For example, Y(banana) = Truth, since a banana is yellow. However, Y(apple) = Truth also, since some apples are yellow.
Similarly a sentence expressing a relationship between two objects can be considered a truth function of two variables, that is, a relationship between two objects can be considered to be a truth function of two variables. For example, let S(x, y) = âx is smaller than yâ. So S(mouse, elephant) = Truth, since a mouse is smaller than an elephant, but S(mouse, ant) = Falsity, since a mouse is not smaller than an ant.
Object, name, concept
An object (referent, denotation) has a name, the name of the object. The object has a concept (sense), the concept of the object, associated with the name of the object. A name or concept are themselves objects, and have names, the name of the name of the object, and the name of the concept of the object. Similarly they have concepts as any other object. A name is said to
denote the object for which it is the name.
Resolution of the paradox of the name relationship using the FregeâChurch ontology
Ambiguities in ordinary language lead to confusion
The English
ordinary language has ambiguities that need to be clarified as we sometimes refer to an object with a word, e.g., a cat. We refer to the name by using scare quotes, the name of the cat, e.g., the word âcatâ. There is ambiguity in the language as regards referring to the cat as a concept, and cat as an object.
Intensional context
An expression such as âbelieves thatâ is said to introduce an
intensional context. In an intensional context, the names that occur denote the concepts of the objects for the believer. They do not denote the objects themselves.
Resolution
âThe farthest planet from the sunâ, as it appears in proposition (1) is Maryâs concept of âthe farthest planet from the sunâ, not about the actual farthest planet from the sun as it appears in (2), so the substitution cannot be done. A more rigorous and formal treatment of this is given by Church.[1]
References
^
abChurch, Alonzo. "A Formulation of the Logic of Sense and Denotation." In Structure, Method and Meaning: Essays in Honor of Henry M. Sheffer, edited by P. Henle, H. Kallen and S. Langer, 3â24. New York: Liberal Arts Press, 1951.
^Gottlob Frege. "Ăber Sinn und Bedeutung" in Zeitschrift fĂŒr Philosophie und philosophische Kritik 100: 25â50. Translation: "On Sense and Reference" in Geach and Black (1980).
^
Gottlob Frege. "Ăber Begriff und Gegenstand" in Vierteljahresschrift fĂŒr wissenschaftliche Philosophie 16: 192â205. Translation: "Concept and Object" in Geach and Black (1980).