In accepting the proposition that every truth has a truthmaker, one is accepting that there is something out in the world that makes a proposition true. There is no true statement in the world that is left without some truthmaker to support it. In accepting the entailment principle, one is accepting that truthmakers have a type of transitivity. If a statement <Q> is entailed by the statement <P>, then <Q> must be made true by the same truthmaker that makes <P> true. Both these claims seem straightforward and easy to accept on their faces, however, they also seem to lead to a contradiction when we accept both as true and consider the existence of necessary truths.
When we say that <P> entails <Q>, what we are really saying is that it is impossible for <P> to be the case while, at the same time, <Q> is not. If we assign a proposition with a necessary truthmaker to <Q>, <Q> will always be the case and there is no way for it to be otherwise. This means that necessary statements will always be true regardless of what <P> is, leading to the conclusion that any true proposition <P>, like <the grass is green>, entails the necessary proposition <Q>. Using the entailment principle, it becomes the case that any truthmakers for contingent truths also become the truthmaker for the necessary proposition <Q>. On the other hand, if we accept that all true propositions have truthmakers, it cannot be the case that any true proposition will entail <Q> because that leads to <Q> not having its own truthmaker. If someone asked me to prove a necessary statement true, I could respond with <The grass is green> and, because of the entailment principle, that statement would suffice to prove <Q>.
This contradiction, if it stands, forces the truthmaker theorist to accept one of three things. They either have to reject the claim that all truths have truthmakers and therefore some are just free truths, reject the claim that <Q> is made true by <P>’s truthmaker when <P> entails <Q>, or accept that some truths are made true by anything (a slightly different position from accepting there are free truths, but still not preferable).
One way for the truthmaker theorist to respond is to take another look at what he means when he talks about the entailment principle. When we talk about the entailment principle as it is presented, truthmaker T is said to make <Q> true through <P>, but the principle isn’t specific as to whether the truthmaker makes <Q> true because <Q>’s truthmaker is the same as T or because there is something about T that aligns with <Q>’s truthmaker. Due to the lack of specificity in the entailment principle, there is cause to investigate what it really means when we say that <P> entails <Q>.
One common case of entailment uses a disjunctive statement, where <P> is an “or” statement whose truth could lead to <Q>’s truth. For example, have <P> be the statement <Jim is a student or Jim is an actor>, where the truthmaker of <P> is that “Jim is a student.” This combination leads to our <Q> that <Jim is a student>. There would be no issue in stating that <P> and <Q> share the exact same T in this case. However, the previous case could have been different as the truthmaker for <P> may have been “Jim is an actor.” If it were the case that “Jim is an actor”, then <Q> wouldn’t have been entailed by <P>. This means that it is possible for <P&~Q> to be true while <P> still entails <Q>. Due to this, we must say that our original definition for entailment isn’t correct.
Due to this error of definition, we must find a new way to define entailment. If we look at another case of entailment, we find a potential guiding light to our new definition. This other case of entailment has the truth of <P> always requiring the truth of <Q>. This is the constitutive entailment, because <P> that is constituted, at least in part, of <Q>. In order to see an example of this, have
be the statement <Jim is a student and Jim is writing> and <Q> be the statement that <Jim is a student>. In this scenario, it is clear that if <P> is true, then <Q> must be.
To further understand how this works, let’s continue assuming that P is <Jim is a student and Jim is writing> and Q is <Jim is a student>. What do we mean when we say <Jim is a student and Jim is writing> is true? It seems that we are assenting to the fact that there are at least three truthmakers: “There is a ‘Jim’”, “He has the property of being a student”, and “He is writing”. If all three of these truthmakers exist, then it would seem we necessarily have P. In a similar manner, when we say <Jim is a student>, we are assenting to a similar group of truthmakers. The only difference (in the listed truthmakers) is the relation of Jim to writing. From here, we can make the claim that P’s truthmaker could be considered a type of conjunction of truthmakers (TP) and Q’s truthmaker is another conjunction of truthmakers (TQ). In order for TP to exist, there must be several truthmakers that exist. If TP is a superset of TQ, then it is safe to reach the conclusion that Q is made true whenever TP is true. This is not because Q needs TP, it is only because TP happens to have everything that TQ has. In symbolic logic terms: P entails Q ↔ TP⊃TQ. From this, it follows that a more accurate definition of entailment is that <P> has a truthmaker that includes <Q>’s truthmaker in some respect.
While this might look like a game of semantics, the disjunctive entailment shows that it is possible for <P> to be true while <Q> is not, disproving the original definition. Meanwhile, this new definition immediately follows from the constitutive case. It also doesn’t cause the same issue our original definition caused, as <Jim is a student> can only be entailed by <Jim is a student or Jim is an actor> if “Jim is a student” is the truthmaker of the latter. Therefore, this change in definition is a substantial one.
If we accept this analysis of how entailment should be defined, then we can accept that <P> entails necessary truths, while necessary truths still have their own truthmakers. This is because whenever we make a statement <P>, there are many facts that we are implicitly stating. Some of those implicit facts, for any P, will be the necessary statements. For example, when someone says <Jim is a student>, they are relying on the fact that “Jim is Jim” is true and “Jim is Socrates” is not true. In other words, the truthmaker of proposition <P> entails <N> because TP contains the truthmaker TN. In this solution, every necessary truth still has an independent truthmaker, it is just that the truthmaker is included in TP, for all P. The entailment principle is still a valid principle, but the way it was originally defined didn’t correctly consider how the same truthmaker should be applied with respect to two different statements. There are no truths that have to be made true by anything, just an acknowledgement that some truths are everywhere in a logically consistent system.
One could make the argument that this analysis is really just a roundabout way of agreeing with the statement that some things are made true by everything. This objection misses the point. It doesn’t rely on TP as a whole to entail <N>. It relies on the fact that any TP must include the necessary truthmakers to exist in a logically consistent system. If statement P doesn’t contain the necessary truthmakers in a system implicitly, then it can’t be a logical statement. It is more accurate to say that this understanding of the issue acknowledges that if a statement really is necessary, one must be able to find that every proposition is based on that statement at least implicitly.
There are two objections that could be made to this analysis that boil down to a very similar issue. The first is objecting to the idea that any proposition requires a conjunction of truthmakers in order to be true. The second is objecting to the idea that any proposition could actually have more than one truthmaker. In this analysis of entailment, it is impossible to have multiple truthmakers lead to the truth of a proposition without using a conjunction. Due to this, once it is shown that multiple truthmakers can be used to make one proposition true, then it must be the case that the proposition requires a conjunction. Because of this fact, responding to the second objection will suffice in responding to the first.
The first, and simpler, response is a matter of the truthmaker theorist preferring less truthmakers in the world. If every proposition that is true has a truthmaker directly associated with it, then there would be a world of limitless truthmakers. This is because I could simply conjunct any two (or more) truthmakers to make a whole new truthmaker. Instead of the proposition
Of course, assuming my reader would prefer a lightweight world may be asking too much, so here is argument based on the logic of the matter. If we reject the idea of any proposition having multiple truthmakers, then we have a scenario where R is the proposition <P&Q>, where <P> and <Q> have their own independent truthmakers. Now where does R’s truthmaker come from without somehow relying on the existence of those truthmakers? It seems contradictory to say that R’s truthmaker exists completely independent of the truthmakers of P and Q. Because it seems that R’s truthmaker is dependent on the truthmakers of P and Q, a person making the claim that R has its own truthmaker also has to accept that truthmakers have truthmakers. In that case, what would be the truthmaker of a truthmaker? Could there be a second layer of “truthmakers” that support the original truthmaker? If that is the case, what is making those truthmakers true that didn’t make the original truthmakers true? In making the claim that proposition R has only one truthmaker, the objector accepts a world where truthmakers are supported by an infinite regress of “truthmakers” that has no supporting foundation.
Someone may point out that this new way of looking at entailment could lead to an issue regarding necessary truths. In order for this vision to work, every necessary truth must contain every other necessary truth. This, in my view, is a nonissue. When one looks at the ideas that are necessary, it seems accurate to say that they do rely on other necessary ideas. For example, the law of noncontradiction is at least somewhat reliant on the idea that something cannot not be itself, otherwise known as the law of identity. If the law of identity did not exist, the law of noncontradiction would not exist. In the same way, the law of identity would be nonexistent without the ideas of the law of noncontradiction.
From the arguments above, the entailment problem was merely a matter of definitions that were too focused on the necessity of the statements made and not the foundations that they are made on. When we discuss true propositions, we must not forget that any proposition is based on some truth of the world, so any definitions that fail to mention those truths must be studied intently and used cautiously for fear of creating paradoxes with no real basis.
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