Some questions about the basic concepts in semantics

Question

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'?

Answer 1

The terminology is somewhat vague and, to make it worse, sometimes used quite differently across authors and episodes. The SEP article gives a historical overview of the usage of some of these terms, but even today it's a bit of a mess and there is no agreement on the exact meaning of these terms.

A good place to start is the distinction between extension and extension, which 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}.

As a very rough first approximation,

• meaning ≈ sense ≈ connotation ≈ algorithm ≈ intension ≈ the conceptual content/dictionary definition of an expression
• reference ≈ denotation ≈ value ≈ extension ≈ the things in the real world that the expression refers to

But a number of more fine-grained distinctions or modifications can be made:

• meaning = (sense vs. denotation), and denotation = (intension vs. extension), where sense = the aspects of meaning independent of concrete things in the real world and revolving around such issues as dictionary meaning (paraphrases or translations), word sense relations (like synonymy) and features (like [+/- animate]), and denotation = the link between the expression and the objects in the real world, encompassing both intension and extension
• meaning = (denotation vs. connotation), where denotation = literal/core meaning and connotation = the social/communicative, emotional, stylistic, ... associations attached to the expression
• sense/denotation/reference vs. intension/extension, where sense/denotation/reference is predominantly used for expressions that denote individuals (e.g. proper names, definite descriptions), and intension/extension for the rest (e.g. predicates, sentences)
• reference vs. denotation, where reference = what the speaker intends to talk about and denotation = what the expressions used by the speaker actually denote, in case the speaker errs in the expressions they use

For a beginning in semantics, I would say it is good enough to go with the two-fold distinction between intension and extension in the sense as explained in Gamut.

Intension and extension are then somewhat complementary concepts, but in a way the intension includes the extension and thus the extension of an intension is determined by the intension, namely in that the intension 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.

Heavily simplified, intension + context = extension.

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.