On a Misguided Application of Excluded Middle

By | August 10, 2009

Many discussions of logical fatalism and of the compatibility of divine foreknowledge and future contingency turn on the question of whether propositions about future contingents are true in advance. More exactly, they raise questions about whether any ‘will’ or ‘does’ propositions about events which have an intermediate chance of occurring (i.e., a current single-case objective probability greater than zero and less than one) are true. One common argument contends that the ‘open future’ position, which denies that any propositions about future contingents are now true, leads to a denial of the law of excluded middle (LEM). For example, in a recent collection, David Hunt writes (p. 276):

Either I will call my mother tomorrow, or I won’t call my mother tomorrow. One or the other of these statements about the future must be true. The principle that either a given statement or its denial is true is called the “Law of Excluded Middle.”

According to Hunt, LEM necessitates that some propositions about future contingents are true. But he’s simply mistaken if he thinks his example gives us a clear instance of LEM. It doesn’t, and it’s easy to show this.

LEM states that, for all propositions P, either P or its denial, Not-P, is the case. This can be given either a truth-functional or a supervaluationist reading.

  • Truth-functional LEM: For all P, either P is true or Not-P is true.
  • Supervaluationist LEM: For all P, ‘either P or Not-P’ is true.

In general, though, the point behind LEM (on either reading) is that P and Not-P are mutually exclusive and jointly exhaustive. Between them, they completely exhaust logical space. Every possible scenario is one in which one of those two holds, and no possible scenario is one in which both hold.

But Hunt’s example doesn’t correspond to either reading of LEM. To show this we need only describe a logically possible scenario in which neither (1) ‘Hunt calls his mother tomorrow’ nor (2) ‘Hunt does not call his mother tomorrow’ obtains. Here’s one: Hunt doesn’t exist. In that case, Hunt isn’t around either to call his mother or to refrain from calling his mother. So neither (1) nor (2) is true. (Compare with ‘The present king of France is bald’ and ‘The present king of France is not bald’. Neither of those is true if there is no present king of France.)

Hunt might respond by suggesting that we should read (2) as (2*) ‘It is not the case that Hunt calls his mother tomorrow’. On that reading we do indeed have an instance of LEM with (1) and (2*). But a new problem arises: (2*) isn’t about a future contingent. It is true right now simply in virtue of the fact that tomorrow hasn’t happen yet. Hence its current truth doesn’t depend on anything future. What’s more, its truth doesn’t depend on Hunt’s existence, the existence of his mother, or even the existence of any created thing whatsoever. Of course, if tomorrow Hunt should call his mother, (2*) will then have become false. But that in no way licenses the inference that it is now false.

In sum, either we have a choice between propositions about future contingents, but LEM fails to apply, or LEM applies, but we are no longer forced to choose between two propositions about future contingents. Either way, Hunt’s argument has zero force against the open future position.

6 thoughts on “On a Misguided Application of Excluded Middle

  1. Rafal Urbaniak

    Hi Alan,

    Interesting. Here's a thing that I'm not really convinced of.

    You consider:

    (2*) 'It is not the case that Hunt calls his mother tomorrow'.

    Now, you say a few things about it:

    1. “(2*) isn't about a future contingent. It is true right now simply in virtue of the fact that tomorrow hasn't happen yet.”

    2. “its current truth doesn't depend on anything future.”

    3. “its truth doesn't depend on Hunt's existence, the existence of his mother, or even the existence of any created thing whatsoever.”

    These need some extra support: prima facie it seems like you have a very unusual reading of negation, or a very unusual view of truth-making.

    For the sake of simplicity, let's say we have two one-place predicates “Huntizes”, H for short and “Motherizes” (M) that are truly predicable only of David Hunt and his mother, if they exist (so I went Quinean about names here).

    Also, we have a three-place predicate “Calls” (C), “x calls y at time t”, the T relation (Ttv = t is earlier than v) for our time ordering and a constant n for the present time.

    Then, (1) is something like:

    (1') (Ex)(Ey)(H(x)&M(y)& (Et)(Tnt & C(x,y,t)))

    It's negation, as you point out, doesn't depend on the existence of Hunt or his mother (well, in the sense, it doesn't require their existence), because it says:

    (2') (x)(x)(~H(x)v~M(y)v~(Et)(Tnt & C(x,y,t)))

    My intuition is, (2') expresses your(2*). But then, I really don't see how its truth does not depend on the future. If Hunt and his mother exist, then (2') is true iff
    there is no future moment where he calls his mother. And this does seem to depend on the future.

    I mean, sure, you can say that you require the present existence of a truth-maker and that future doesn't exist currently, and that undetermined sentences about future, if there are such sentences, in general have no currently existing truth-makers.

    But then, it seems, you're postulating something that few competent users of a language would agree with: the more common intuition is that the truth of (2') depends (at least partially) on the future existence (or non-existence) of a certain state of affairs, which neither has to exist presently, nor has to be causally determined presently to make this sentence true.

    One reason why some people thought that the present existence of a
    truth-maker is needed is because they thought “truth-making” is “making” – that is, that it is similar to a causal relation. And how can something cause anything if it doesn't exist? But another, at least equally legitimate point of view is that truth-makers don't make sentences true through a causal link and that a truth-maker doesn't have to exist now to make a sentence true any more than it has to exist here rather than somewhere else to do that.

  2. Alan Rhoda

    Hi Rafal,

    Interesting. There may be some confusion about the sense in which (2*) may be said to “depend” on future events. There is a relevant distinction between truth-conditions and truthmakers. Truth-conditional dependence is a relation that holds among propositions. Truthmaker dependence is a relation that holds between a proposition and what exists.

    A truth-condition of (2*) is that Hunt not call his mother tomorrow. If tomorrow he should call his mother, then (2*) is false. So in the truth-conditional sense of “depends” (2*)'s truth depends on what happens in the future.

    But that's consistent with my claim that (2*)'s truth doesn't depend on any future events in the truthmaking sense of “depends”. As your (2') makes clear, there don't have to be exist any future events, ever, for (2*) to be true. It's truthmaker could simply be, say, God's deciding not to create.

    My main point against Hunt is that *Not-(E will occur)* is not the same as *E will not occur* because what's required to make the latter true is not the same as what's required to make the former true.

  3. Rafal Urbaniak


    Well, you said:

    “(2*) isn't about a future contingent. It is true right now simply in virtue of the fact that tomorrow hasn't happen yet.”

    This strongly suggests the reading on which the fact that no truth-makers of 1 exist right now is the sole reason why 2* is true.

    This, however, doesn't seem too intuitive to me: 2* is true (if it is) because of something happening or not happening in the future, not because of the current state of affairs.

    Anyway, your main point seems, in a sense, right: there is an interpretation on which “I will not call my mother tomorrow” suggests that I will exist tomorrow. But I don't think it's the part of the meaning. At best, it seems like a Gricean implicature. This is because there are clear cases where an existential assumption of this sort can be denied without contradicting the sentence. Here's an example:

    – I will not call my mother in year 2132.
    – Well, of course you won't – you won't be alive!

    Compare it with a fairly awkward response:

    – I will not call my mother in year 2132.
    – That's false! You won't be alive!

  4. Alan Rhoda

    Hi Matthew,

    Suppose states A and B are superposed, that A entails ~B, and that B entails ~A. We don't have a violation of LEM unless we further suppose that the A/B superposition is a case in which both A and B obtain. But we could just as easily construe A/B as a state in which neither A nor B obtain. Provided that A and B are only contraries and not contradictories, that's perfectly consistent with LEM.

  5. Alan Rhoda


    I'm basically arguing that “will not” sentences are ambiguous because the scope of “not” may be either wide or narrow. Hence, “E will not occur” can mean either “It will be that case that E fails to occur” or “It is not the case that E will occur”, and these are not equivalent.

    I take it that we normally construe the “not” in “will not” as having narrow scope (i.e., as ranging over only the predicate).

    On either reading, however, “I will not call my mother in 2132” seems to be true. On the wide-scope reading it's true simply because 2132 hasn't happened yet. On the narrow-scope reading it's true because, human longevity being what it is, the event described is (arguably) physically impossible.


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