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The claim that every true proposition has a truthmaker leads to philosophical difficulties, such as the question of what the truthmaker for an ethical, modal or mathematical proposition could be. Of course someone who is deeply enough committed to truthmakers and who simultaneously doubts that a truthmaker could be found for a certain kind of proposition will simply deny that that proposition could be true. Those who find the Parmenidean insight sufficiently compelling often take it to be a particularly enlightening metaphysical pursuit to search for truthmakers of these kinds of propositions.
Another difficulty for the claim that every true proposition has a truthmaker is with negative existential propositions (or, equivalently, universal propositions). What entity makes it true that unicorns do not exist? Proposals include the totality of all things, or some worldly state of affairs such as x1's not being a unicorn, x2's not being a unicorn, ..., and everything's being x1, or x2, or ... (the latter suggestion is due to Richard Gale). David Lewis has proposed a more moderate version of the truthmaker theory on which truthmakers are only required for positive propositions. What makes a negative proposition p true is the lack of a falsemaker for it, i.e., the lack of a truthmaker for the negation of p. This may be what Aristotle was getting at when he said that to speak truly is to say of what is that it is and of what is not that it is not.
Truthmaker theorists differ as to what entities are the truthmakers of various propositions. Some say that the truthmaker of the proposition that Socrates is sitting (assuming he is!) is Socrates' being seated (whatever exactly that might turn out to be on the correct ontology) and in general the truthmaker of the proposition expressed by a sentence s can be denoted by the participial nominalization of s. Others will say that the truthmaker of the proposition that Socrates is sitting is just Socrates himself. In any case, the truthmaker is supposed to be something concrete, and on the first view is that whose existence is reported by the proposition and on the second view is that which the proposition is about.