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What's the difference between, apposition, metonymy, and synecdoche? May apposition include metonymy or synecdoche?

These three words are very simple to understand. Addressing your sub-question: May apposition include metonymy or synecdoche? No, apposition does not include metonymy or synecdoche. In fact, apposition is a grammatical construct, whereas, metonymy and synecdoche are figure of speech.

World Wide Words has lucid explanation on metonymy and synecdoche.

Both¹ are figures of speech used in rhetoric. They’re not the same thing, though metonymy is often interpreted so widely that synecdoche can be regarded as a special case of it.

Explanation about Synecdoche:

Let’s take synecdoche first. You use this when you speak of a part of something but mean the whole thing. When Patrick O’Brian has Captain Jack Aubrey tell his first lieutenant to “let the hands go to dinner” he’s employing synecdoche, because he’s using a part (the hand) for the whole man. You can also reverse the whole and the part, so using a word for something when you only mean part of it. This often comes up in sport: a commentator might say that “The West Indies has lost to England” when he means that the West Indian team has lost to the English one. America is often used as synecdoche in this second sense, as the word refers to the whole continent but is frequently applied to a part of it, the USA.

Further example,

Example: "I have four mouths to feed at home."

It simply means, "I have four family members to feed at home."

Here,

Mouths (the part) is used to represent people (the whole thing).

Explanation about Metonymy:

Metonymy is similar, but uses something more generally or loosely associated with a concept to stand in for it. When Americans speak of the Oval Office, for example, they are really referring to the activity within it, the position or function of the President. It’s a linked term, and so a metonym. British writers refer similarly to the Crown, when they’re really discussing the powers, authority and responsibilities of the monarchy, which is symbolized by the crown. The difference between synecdoche and metonymy is that in metonymy the word you employ is linked to the concept you are really talking about, but isn't actually a part of it. Another example is the turf for horse racing. But the distinction isn't always obvious and often can’t be rigorously applied, and many people use metonymy to mean both.

Example: "The White House declared that we are at war with Mars."

Here,

The White House is a metonymy for the President of the United States.

Explanation about Apposition by Dartmouth Uni.

Placing side by side two coordinate elements, the second of which serves as an explanation or modification of the first

Vocabulary.com:
When you put two things right next to each other, you can call that apposition. The apposition of your dog and your cat makes an adorable photograph.

A grammatical relation between a word and a noun phrase that follows. Though you can use the noun apposition to talk about the positioning of objects — or words — close together, it's usually found in scientific or technical writing. In grammar, an apposition occurs when two words or phrases are placed beside each other in a sentence so that one describes or defines the other.

An example is the phrase "my dog Woofers," in which "my dog" is in apposition to the name "Woofers." "My dog" and "Woofers" are placed side by side and represent same pet.

So, what is the difference between Apposition and Metonymy/Synecdoche?

-> Apposition is used literally and Metonymy/Synecdoche are used figuratively. In apposition, you have two nouns or noun phrases side by side. In which, second noun/noun-phrase describes the first noun/noun-phrase.

-> Whereas, in metonymy/synecdoche there is only one noun in sentence presenting other noun figuratively. We need to understood other noun figuratively, it is not present in the given sentence.

I hope these explanation help!

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