In tackling metaphysics, Aristotle starts with Plato's views, and more specifically of Plato's science of Forms and the Good. Thus Aristotle holds metaphysics to be the science of the first and most universal causes, rather than the attributes of things that have a contingent existence. Metaphysics is the study of "being qua being", or the study of attributes that belong to things merely insofar as they exist, e.g. existence, unity, sameness and difference.
The fundamental question posed by metaphysics is: what is true of existents insofar as they exist?
But by inheriting this definition of metaphysics, Aristotle is confronted with a problem, since he rejected Plato's theory of Forms and the good, and instead suggested that there are many senses of 'good', depending on what it is that you are evaluating. So too, he said, are there many senses in which things can be said to exist. Thus, it seems, there can be no science of existence and of universal causes, and so there can be no metaphysics.
Aristotle's solution is to demonstrate that there is a single, 'fundamental' sense of 'exist' from which the other senses derive, and that that sense of 'exist' is the subject of metaphysics. To do this, he first sets up some distinctions to distinguish between the various senses of 'exist':
Homonymous things are those which are both described by the same predicate but with different definitions, e.g. "I am Polish" and "I polish the floor"
Synonymous things are those which are both described by the same predicate and each with the same definition of that predicate, e.g. "I polish the floor" and "I buff the floor"
In some cases, one thing can be hononymous with another because it is dependent upon it, in what is called non-coincedental homonymy. Where this is the case, there will be one word that has focal meaning amongst the homonyms, in that it is the one from which all other meanings are derived.
Thus, for Aristotle, the central question of metaphysics becomes: what is true of substance insofar as it exists?
Aristotle held 'existence' to be a collection of homonyms, with a focal meaning that applies to substances (e.g. the meaning it has in 'Socrates exists'). The two most distinct meanings ot 'exist' that Aristotle drew out in relation to substances were:
1) Primary: a substance that exists in its own right, independent of others
2) Secondary: a substance that exists by virtue of its relation to a substance, dependent upon primary substances, e.g. colour (that flower is red), jogging (I am jogging)
Thus Aristotle draws two distinct areas of study for metaphysics.
Aristotle also tackled the question of mathematics; Plato held "1+1=2" to be a statement of substance, but Aristotle disagreed, thinking that numbers can only exist in relation to other primary substances, and so only a statement like "1 dog + 1 dog = 2 dogs" could be a statement about substance (in this case in particular it is a statement about dogs qua unity)
This marked a fundamental departure from Plato, since Aristotle suggested that we cannot understand truth without reference to physical things, in contrast to Plato's emphasis on the immaterial Forms.