Truth-making facts?

I have been reviewing Bill's Facts category.  Bill is a friend of facts, seen as 'truth-makers'.

2. Now what is the nature of this external truth-maker? It can't be Al by himself, and it can't be fatness by itself. Nor can it be the pair of the two. For it could be that Al exists and fatness exists, but the first does not instantiate the second. What's needed, apparently, is the fact of Al's being fat. So it seems we must add the category of fact to our ontology, to our categorial inventory. Veritas sequitur esse is not enough. It is not enough that 'Al' and 'Fat' have worldly referents; the sentence as a whole needs a worldly referent. Truth-makers cannot be 'things' or collections of same, but must be entities of a different categorial sort. (Or at least this is so for the simple predications we are now considering.) [here]

3. The argument I have just sketched, the truth-maker argument for facts, is very powerful, but it gives rises to puzzles and protests. There is the Strawsonian protest that facts are merely hypostatized sentences, shadows genuine sentences cast upon the world. Butchvarov quotes Strawson's seminal 1950 discussion: "If you prise the sentences off the world, you prise the facts off it too. . . ." ("Facts," 73-74) Strawson again: "The only plausible candidate for what (in the world) makes a sentence true is the fact it states; but the fact it states is not something in the world." Why aren't facts in the world? [here]

Could Tom by himself be the truth-maker of 'Tom is tired'? No. For if he were, then he would also be the truth-maker of 'Tom is manic' -- which is absurd. [here]

Bill's reasoning here seems to be that 'Tom exists' entails neither 'Tom is tired' nor 'Tom is manic'. That's right, but it is not the question at issue. The insight of those who would say that Tom is the truth-maker of 'Tom is tired' is that we need look no further than Tom himself to verify that Tom is indeed tired. The statement is about Tom and nothing else. Sometimes Bill seems to require some object or entity that we can label 'Tom's tiredness' within Tom. We might identify this with the high level of lactic acid in Tom's muscles. But we can't see this directly. What we can see is Tom's sluggishness of movement and yawning which is caused by the lactic acid. So perhaps we can say that Tom's sluggishness is the truth-maker for 'Tom is tired'. Let's ignore the possibility that this is feigned. We can observe Tom's slowness of movement and if necessary measure it objectively. The question is then, How do we get from the sluggishness to the words 'Tom is tired' and the judgement that this statement is true? Perhaps that is what the truth-making puzzle is about. One answer might be that there is a causal chain running from Tom's slowness, through our observation of it, and our having learned that 'Tom is tired' is the kind of thing one thinks and says in these circumstances. But this may be thought too behaviouristic an understanding of truth-making. And the causal connections vastly too complicated to verify. Nevertheless, it can't be denied that on seeing Tom's movements the thought expressed as 'Ah, poor Tom is tired', is likely to occur to us involuntarily, though it may not be said.  This causal chain must surely be part of what makes 'Tom is tired' true when Tom is tired.  But Bill never considers this, as far as I can see.

And if there are no facts, then how do we explain the truth of contingently true sentences such as 'The cat is on the mat'? There is more to the truth of this sentence than the sentence that is true. The sentence is not just true; it is true because of something external to it. And what could that be? It can't be the cat by itself, or the mat by itself, or the pair of the two. For the pair would exist if the sentence were false. 'The cat is not on the mat' is about the cat and the mat and requires their existence just as much as 'The cat is on the mat.' The truth-maker, then, must have a proposition-like structure, and the natural candidate is the fact of the cat''s being on the mat. This is a powerful argument for the admission of facts into the categorial inventory.[here]

I don't understand why explaining the truth of 'the cat is on the mat' is any more problematic than explaining the truth of  'the cat exists'. After all, the latter involves seeing that the cat's head is on the cat's shoulders. To see disconnected cat parts is to not see the cat and hence to deny the cat's existence.  To put this the other way round, if seeing the cat on the mat is seeing a fact, then seeing the cat's head on its shoulders is also seeing a fact.  But seeing this fact has somehow been absorbed into seeing the cat.  So maybe there are no objects and it's 'facts all the way down'.  With my apologies for the grizzliness of the example.

Perhaps properties and relations are what's left over once we have 'factored out' objects from the world. This is a mathematical idea and one that is hard to explain in an everyday context.