What is the definition of intension?

Question

I have read in philosophy books about intension. But no one has defined it rigorously. What does it actually mean? I asked in a previous question about identity of intension. If this question is answered, then we can also define identity of intension. Has anyone ever defined intension rigorously?

Answer 1

Informally,

The intension of an expression is something like its conceptual content, while its extension comprises all that exemplifies that conceptual content. Take the expression digit, for instance. The intension of the word (at least in the sense which it has in arithmetic) is the concept 'single symbol referring to a whole number', and its extension is the set of symbols {0, I, 2, 3, 4, 5, 6, 7, 8, 9}.

(L.T.F. Gamut: Logic, Language and Meaning, vol. 2: Intensional Logic and Logical Grammar (1991), which also discusses in more detail what I summarize below.)

As another side note, the intension/extension distinction is roughly the same as that of sense and reference.

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With a possible-world semantics, the intension of an expression can be defined as a function from possible worlds to the extension in that world:

[[α]]:  W → τ:  w ↦ [[α]]w

where α is an expression whose denotation is in τ (e.g. truth value, set of individuals), W is the set of possible worlds, [[α]] is the intension of α, and [[α]]w the extension of α at world w.

You can think of the set W of possible worlds w as the logical space: The set of all imaginable scenarios of what the world could look like. A possible world is a very large situation that is fully specified w.r.t. the statements that are true in it, and changing the truth value of a statement leads to a minimally different world, such that the universe of all these worlds covers the full combinatorial space of what is logically possible. Each world has its own set of people, things, properties of and relations between individuals, and thus its own distribution of truth values over statements. The extension of an expression is its denotation at a particular world; the intension is a generalization over these possible worlds collecting what people, pairs of lovers, US presidents etc. could look like throughout the logical space.

For instance, the intension of a one-place predicate like "dog" is a function from all the possible worlds to the set of individuals that are dogs in that world, the intension of a definite description like "the president of the US" is a function from possible worlds to whoever is the president of the US in that world, and the intension of a statement is a function from possible worlds to the truth value of the statement in that world.

The intension of a statement (the intension of a statement is also called "proposition") may alternatively be identified with the set of possible worlds in which it is true:

[[φ]] = {w ∈ W:   [[φ]]w = 1}

The intension of a tautological statement will be the entire logical space (= the set W), the intension of a contradictory one the empty set, and a contingent statement will have as its intension precisely the set of worlds (possible scenarios) in which it is true.

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It may now not be obvious to see how this relates to the informal notion of intension cited above. It helps to think of intesion and extension as algorithm and value: The intension function is a recipe that tells you how to compute the reference of an expression under each circumstance: If the world looks like this, "the president of the US" is this person; if circumstances were like that, "the president of the US" would be that person. Being able to identify in any given situation which entities are referred to by the word "dog" entails to have understood the concept of a dog. Understanding the intension (the propositional content) of a statement amounts to knowing exactly the combination of circumstances (=> set of possible worlds) in which it is true. This idea of intension comes closer to what we understand by "meaning" than the extension in a concrete situation (which is e.g. a single truth value) does.

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Identity of intension then comes down to identity of the above function (or in the case of statements, identity of the above set), both of which have a rigorous mathematical definition: Two functions are identical iff their set of argument-value pairs (here: <world, extension> pairs) is identical; two sets are identical iff every member of the first is also a member of the second and vice versa. Simply put,

two expressions are intensionally equivalent iff they have the same extension distribution across all possible worlds.

Note that ironically, under a classical set-theoretic treatment, this definition of intension is itself purely extensional: We are just collecting elements of the form <world,extension> into a set, and identify two intensions by membership of elements in the respective function sets -- rather than some kind of equality of conceptual content.

Answer 2

As with most terms in philosophy, different authors use the word in different ways. The older (and more precise) definition is that the intension of a category, C is the collection of categories that C implies. For example, the intension of "mammal" includes categories like chordate, "warm blooded", "has mammary glands", etc. At one time the extension was the opposite--the extension of C was the collection of categories that imply C, so the extension of "mammal" would include things like carnivore, primate, etc. This is very similar to the genus/species distinction in older philosophy.

Today, the extension of a category, C is almost always the set of individuals that fall under C, and the word extensional has been appropriated to mean that a thing is defined entirely by its members--or something analogous to that. For example, a mathematical set is extensional because two sets, A and B are equal if and only if every A is a B and every B is an A. Analogously functions are extensional if in your theory of functions, f=g iff ∀x.fx=gx.

The word "intensional" is used to mean "not extensional". For example if in your theory of functions, λx.2x is a different function from λx.x+x, then your functions are intensional. In this case, the intension of a function must be whatever makes two functions different even though they have the same extension. I don't know what that could be other than the function itself, so I get the impression that when some modern authors refer to "the intension of X" they just mean X except that they want to focus on x not being extensional.

Answer 3

While I make no attempt to define it rigorously, as so aptly done by lemontree: Intension is an old term always contrasted with extension. As I recall, it was J.S. Mill that replaced that distinction, by that of the connotation [intension] and denotation [extension].

Roughly, traditionally. the extension [or later denotation] of a general term, predicate or concept is made up of all those entities to which the term, predicate or concept correctly applies, or which fall under the concept/class.

The definition of intension [or later connotation], always more complex, has become an increasingly contentious issue in a post-positivist/realist world. A world where the "correspondence theory" has, to a great extent rightly, waned in significance. A world where a [simple, complex, general] terms meaning is reducible to its use [see Wittgensten's later work, and also pragmatism/instrumentalism], and the semantic anchor known as the the analytic/synthetic distinction is no longer available [compliments of Quine's Two Dogmas of Empiricism].

But in simpler times, the term intension/connotation simply meant the term’s meaning, or significance, (often, in addition to and as distinct from how it is defined). For instance, "having interior angles equal to the sum of two right angles," is part of the intension of the term “triangle,” though not necessarily part of its definition [a three side geometric object].

Aside: A simple Google search yields: Intension and extension, in logic, correlative words that indicate the reference of a term or concept: “intension” indicates the internal content of a term or concept that constitutes its formal definition; and “extension” indicates its range of applicability by naming the particular objects that it denotes.

Answer 4

In Semantics the intention of a term has a particular context --as well as the term you may already be familiar with extension. Intention & extension are typically taught together as the relationship is close. Extension refers to all the objects that correctly refer to a term. A term is defined by its properties to make it distinct from other terms in existence. The extension of a term refers to ALL the objects that have all of those properties at any period of time (past, present & future). Intention refers to the specific & distinct instances of the extension. So if LAKES were a topic we would say the extension of LAKES is all the lakes that correctly refer to the term LAKES. Lake Superior would clearly be in the extension of LAKES, but lake Superior would also be in the INTENTION of the term LAKES. So would lake Michigan, lake Tahoe, lake George,etc. The intention of a term qualifies itself by having the necessary properties it must have to be in THAT particular extension, in this example LAKES.

So the extension refers more to the idea of quantity. The quantity of HOW MANY items have these properties of a term --ALL of x whatever x happens to be. The intention of x identifies specific instances of x where what you are NAMING as part of the intention must have and share the identical properties of the extension of x. I want to emphasize the properties a term refers to MUST HAVE to be that thing. So a member of a set of properties must have identical properties. Let's say set X has exactly 10 properties that make X a distinct set in the entire Universe that we live in. ALL members of set X MUST HAVE those 10 identical properties in order to be placed in that set. The price of admission is those 10 properties set X is known to have. Along comes a K that has 8 out of the 10 properties, but K is missing 2 REQUIRED properties and admission to set X is denied. This is like saying, in a different example, a shirt cost $25 dollars and you don't have the $25 dollars plus the tax to purchase the shirt. If you are $2 dollars short you can't afford to purchase the shirt right? You must have the $25 dollars plus the tax to purchase the item period or come back with the right amount of money -- no shirt for you.

The intention of a dog is all of the properties and requirements of being a dog that makes dogs distinct from other animals or other life. The intention of a human being is all of the properties and requirements that make a human being distinct from other animals or other life. If anyone could answer that one about what is a human being one would resolve the whole abortion debate in a few moments. [We would then know all the properties something like a fetus MUST HAVE to be a member of the term human being.] Whatever the properties are that makes an individual distinct from all other things object X must have it to belong to the set you are talking about. If some properties are missing, then no admission to the intention of the term you are talking about.

The members of any intention will be less than the members of the same extension. The more specific details you bring to the table the smaller the set of members of that intention will be. You may and likely will have a set with just one member. The intention of the 43rd president of the United states has one member only: George W. Bush. You would not say the EXTENSION OF THE 43rd President . . . Extension of a term is never detailed or specific as the intention of a term. There is no such thing as the extension of the 43rd President of America. The President of the United States as a whole can have an extention. Intention is detailed and more specific whereas extension is desigend to be really broad [perhaps too broad like a generalization is]. All of the Presidents from the first to the current President will be in the extension of PRESIDENT of the United States. The INTENTION of a specific president would be recognized by name or chronological number. The INTENTION of a specific President means the set is INSIDE THE EXTENSION of human beings and on top of that, the human being MUST MEET REQUIREMENTS TO BE PRESIDENT. Being a human is not enough. The intention is all the properties x must have to be that thing in the extension. In this case of President the person must be a citizen, must be at least 35 years of age and so on. There are no intentions outside floating around with no extension. INTENTION is always a subset of some EXTENSION. You cannot HAVE AN EXTENSION INSIDE AN INTENTION. It makes no sense to say a parent set is inside of a child set. The parent set (a master set) is by definition larger than the child set (a subset of the master set). It is meaningless to say you can have an extention of a child set (a subset already in a master set). How can you have by definition a larger set fit inside of a smaller set? Can you fit a full sized skyscraper inside of a person or is it the other way around? If someone is pointing out a specific individuals in a class that is typically intention --not extention. If one is stating something UNSPECIFIC such as being broad, making a generalization that indicates EXTENTION --not intention.

What I am referring to above IS NOT IDENTICAL TO CONNOTATION AND DENOTATION. I really don't want to squabble over term names BUT the reader I hope is willing to understand and think about the CONCEPTS I have explained. You may call something x where I call that same item y but the CONCEPT or essence of the item we name is the same and has not changed. APPLE and APFEL refer to the same fruit, it is just named differently. Please consider the concepts defined if you do disagree about the specific terms used.

For those individual readers here who MUST have references or else what I say is irrelevant --and those same individuals think no one else has writtren about this topic-- look no further than Philosopher Westley C. Salmon. He has a book titled "LOGIC third edition". There is your reference for those people who think no one has written about what I just stated. [I would also check which people you CALL 'a philosopher' out--it doesn't go to every human] And No I did not plagiarize anything. This material and other concepts used to be taught in philosophy courses --not math.

Sorry to make a dig here for all of the math people on this site for logic BUT some so called logic terminology maybe spelled the same for you and me but the differences in context are not identical. [You do "Mathematical Logic" -- not logic]. Peter Smith who is a Philosopher writes a blog and in his writings he talks about how philosophers learn logic and how mathematicians learn logic. Google 'LOGIC MATTERS' and check out Peter Smith. Philosophers know it is not the SAME JOURNEY. It is about time everyone know they are two distinct fields. Philosopher Eric Steinheart wrote a book specifically for the people in Philosophy that are not so well versed in Mathematics: "More precisely: the math you need to do philosophy". If logic were math why would an author need to write such a book in the first place? Mathematical logic is MATH. If you don't agree than see the writings of actual philosophers --in this case I name two: Peter Smith and Eric Steinheart. They blatantly say so. I am sure those are not the only two philosophers that will agree. It also depends on what years people get there degree. Clearly there were such a thing of a philosophy PhD without much of a mathematics background; hence why certain books were written. How about in 1900 do you think Mathematical logic was common? Well, It wasn't. Now Mathematical logic is of frequent use, common place and a requirement. It wasn't ALWAYS like that.