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According to the Semantics (Kate Kreans, 2011), there are two kinds of denotation for predicates. For example, the word 'dog', has extension (the set of all dogs in the actual world), and intension (the set of all dogs in all possible worlds). But I'm just wondering, does this mean denotation includes extension and intension? Shouldn't intension be equal to connotation? And can we say the 'possible worlds' here means 'different contexts'?

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The terminology is somewhat vague and ambiguous.
In 20th century philosophy, the denotation -- connotation concept pair is indeed roughly the same as that of extension -- intension.
In modern linguistics, while "denotation" would still usually be understood as synonymous to "extension", one could also use it as a cover term for both extension and intension to contrast it with the contemporary linguistic concept of connotation: While the denotation of an expression is its literal meaning, the connotation are the stylistic, emotional or cultural associations attached to it. For instance, the words "psychotherapist" and "shrink" have the same denotation -- the same definition and the same referents -- but "shrink" has a more colloquial and perhaps negative connotation.

The difference between extension and intension is summarized nicely in L.T.F. Gamut: Logic, Language and Meaning, vol. 2: Intensional Logic and Logical Grammar (1991):

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}.

The intension/extension distinction is roughly the same as that of sense and reference, though the latter are more typically used for expressions that denote individuals, which are then called the referent of the expression, whereas the intension/extension concept also applies to e.g. sentences, whose extensions are truth values.

Intension and extension are used somewhat complementary, but in a way one could say that the intension of an expression includes the extension, in that it is an abstraction over the extensions in all possible worlds.

A possible world may best be thought of as an alternate universe: a very large actual or hypothetical situation that is fully specified w.r.t. the statements that are true in it. 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, such that each change in the truth value a sentence leads to a minimally different world. There is one designated acutal world, the one we live in, and infinitely many other possible worlds. For instance, there may be one possible world which is just like ours except that Hillary Clinton instead of Donald Trump is the president of the US; there may be a world in which humanity developed different dog breeds and the set of dogs looks slightly different; there is a possible world which is five minutes ago where my coffee cup was still half full; there is a world in which Hillary Clinton is president and my coffee cup is still half full, etc.

The extension of an expression is its denotation at a particular world; the intension is a generalization over these possible worlds collecting what dogs, 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 claim in the book that the intension of "dog" is "the set of all dogs in all worlds" is non-standard at best or plain wrong at worst (I've never seen it defined that way anywhere else); normally, one would want to define the intension not as the set union (I suppose that's what she means) of the extension across all possible worlds, but rather as a function from possible worlds into the respective extension. For an outline on how to define this formally, see here.

The intension comes closer to what we would consider the meaning of an expression, whereas the extension is what it refers to in the concrete world which accidentally looks the way it does: We wouldn't want to take "true" as the meaning of the sentence "Dogs have four legs", or Donald Trump as the meaning of the description "the president of the US": Rather, these are what we refer to in the real world (the extension), while the meaning, or sense, is a recipe (= a function -- the intension) that tells us how to determine the truth of the sentence or the person the description points to in any given situation, depending on what the world currently looks like.

Also note that while extensions are defined against possible worlds, natural language expressions in usage context are typically evaluated relative to particular utterance situations, that is, relevant fractions of the actual world: When Jane Doe says that "I took the dogs out", she doesn't mean to say that she went on a walk with all the dogs in the world, but the set of dogs relevant in the given situation, e.g. her neighbor's dogs that she is currently taking care of.

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  • Thank you for your kind reply! But may I ask one more question? In the last sentences, as is mentioned, Jane Doe said "I took the dogs out" in a specific situation. In this situation, "dog" is not an extension but an intension, right? Can you give me another example when something is in "possible world"? I'm still confused about what the possible world is. Thank you in advance! @lemontree♦ – ronghe Aug 29 at 12:45
  • "Dog" is just a word. Every word (and every complex expression) has both an extension and an intension. The extension in that particular situation is the set of Jane's neighbor's dogs; the intension is the concept of a dog abstracting over what dogs could look like in any hypothetical situation. – lemontree Aug 30 at 9:38
  • The idea of possible worlds is needed to account for the fact that extensions are subject to chance, whereas the concept (the intension) of a word should abstract away from such accidentals. E.g. the extension of "dog" in the real world depends on which dogs happen to be alive at this moment, and which breeds happened to evolve throughout history, but one could imagine that there are slightly different looking dogs around that one would still subsume under the concept "dog" even if they may not exist in the real world. That's what the possible worlds are for. – lemontree Aug 30 at 9:46
  • A possible world is an abstract concept. One doesn't need to believe in the actual physical existence of parallel universes or the like, but that's what comes closest to the idea: Any logically possible circumstance, real or hypothetical. There is exactly one such possible world which is the actual world we live in, and indefinitely many other alternative hypothetical scenarios that are slightly or very different from the real world. In each possible world, the set of dogs may look different; the intension then collects what "dog" could refer to in all hypothetical varieties of the world. – lemontree Aug 30 at 9:52
  • Thank you for your kind reply! Now I understood it :) @lemontree♦ – ronghe Aug 31 at 23:34

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