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Can I say,

(a) The boy used an axe to chop wood.

(b) The boy used an axe to chop down the wood.

(c) The boy used an axe to chop the tree down / chop down the tree.
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(a) The boy used an axe to chop wood.
(b) The boy used an axe to chop down the tree.

(c) The boy used an axe to chop the tree down / chop down the tree.
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Yes to all 4 versions, However B is strange.

The boy used an axe to chop down the wood - he chopped down all the trees in the woodland area.
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I got confused.

We can say :

to chop wood, but cannot not say to chop the tree. Why?

==

How about this?

He used the anx to chop woods.
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to chop wood, but cannot not say to chop the tree. Why?-- We can chop a tree down and then chop it up, but we cannot simply chop a tree and get any result except the satisfaction of vandalism: chopping a tree does not bring the tree down to the ground.

(X) He used the axe to chop woods-- 'Wood' (= firewood or board material) is uncountable, so no 's'.
He used the axe to chop down the woods-- 'Woods' (= small forest) = He chopped down all the trees in the woodland area: not a very likely sentencce