This suggests that "Subtract everything false and what remains must be Truth" is probably false. Which further implies that there is probably a category of statements which are neither true nor false. And, given some other premises I won't bother enumerating here because they are contingent on history, that that third category of statements is actually in many cases more influential and important than the truth, for good or ill-- and it really can be either good or ill, not just ill.
Reasoning from premises which are neither true nor false is necessary to come up with moral conclusions if argumentative reasoning is to be used at all, as purely factual premises can make no moral claim nor can they by themselves justify a moral conclusion (Hume, 1751 ruthlessly paraphrased). Moral conclusions are in the category of statements which are neither true nor false even though when they are used as premises in deductive or inductive reasoning they typically function as if they were true statements. Statements about the value of an object (X has such and such a value) are neither true nor false if their scope goes beyond what will a particular entity (or group) exchange or how a particular entity regards the object; what a particular entity or collection of entities is willing to exchange for a thing can be a properly factual claim, the subjectivity of which is implied in the statement because particular subjects (those who are willing to make the exchange) are the object of inquiry (what the statement is ultimately about).
Value conceived of as something inherent to an object, event, or whatever else, lacks that anchor to any particular subject or group of subjects and thereby becomes neither true nor false; it can become true or false by appending "I say..." or "They say..." to its front, at which point it is no longer a general statement about the object or event but instead one of an uncountable number of possible statements about how a particular entity or collection of entities regards some object or event.
All of this is to say that there are a large number of very important statements which can (and often should) cause human action (or inaction, as the case may be) which logic and a dispassionate study of reality by themselves are insufficient to justify. The values of things are in that collection.