# Future Contingents

## The Problem of Future Contingents

This problem has a celebrated history and has motivated seminal developments in philosophy and in logic throughout time. The problem is stated, its ramifications are explored, and an attempt to solve it is undertaken and abandoned in what we have as the 9th chapter of Aristotle's text *On Interpretation*. The main concerns of the series of lectures collected in this text bear on the subject of modal logic (indeed, Aristotle's logic of categorical propositions or classes, which dominated the study of logic for thousands of years, extends to modalities.) Apparently, Aristotle took logic and philosophy of logic to be seamlessly interwoven with metaphysics and epistemology and other areas of philosophic inquiry. We can detect this in this case because the problem of future contingents is one that arises from within the study of a deductive argument; the problem itself, however, is metaphysical in the broad sense. It appears, then, that Aristotle takes the abstract study of deductive reasoning to possess heuristic fecundity: this means that the construction of arguments with abstract premises, which do not need empirical verification, is considered sufficient for yielding conclusions about empirical and in-principle verifiable matters. This approach is rejected root and branch by the school of thought known as Empiricism; one could even say that one way of defining Empiricism is by underscoring its rejection of what Aristotle does in this case. The Ontological argument for proving the existence of God is another instance in the history of philosophy in which deductive reasoning with no empirically dependent premises is offered to prove a matter that ought to depend on empirical verification. Aristotle considered nature, the structured and teleological or purposefully driven totality of things, to be the ultimate source of any subject of investigation, including reasoning. His first formulation of the logical law of non-contradiction is to the effect that no thing can both have and not have the same attribute predicated of it.

In the problem of future contingents, Aristotle deploys a deductive modal argument to prove, as he thinks he can, that future events are pre-determined. As we have hinted, reliance on logic is taken as sufficient for establishing conclusions with respect to empirical reality in this case. Because the conclusion entails further - Aristotle thinks - that human actions are impotent to affect a deterministic universe of not only past but also future events, the argument is a significant moment for the school of thought known as Logical Fatalism. The most common presentations of this school tend to caricature it - and the impression is easily created that the view is so hopelessly absurd that no one of sound mind could accept it. Of course, there are more and less sophisticated versions of the position and it is instructive that so central a philosophic figure as Aristotle would be associated with the history of Logical Fatalism. Logical Fatalism appears utterly confused about the relationship between logic and reality. Logical truths are trivial, depending simply on what logic is at work or, in the updated version, depending on how certain key particles of the language like "not" and "and" and "all" happen to be defined. In early modern philosophy, such truths would be distinguished as depending on relations of concepts or "ideas." The broader notion is that of analytic truths. A logical truth like "either it is raining or it is not raining - at a specified location and for a specified time" - is trivially true; any statement making a claim whose logical form is "either p or not-p" is true. Alternatively, the meanings of "not" and "either-or" in the sentence are such that the meaning expressed cannot possibly not be true. The broader sense of analytic statement contemplates examples like the classic textbook sentence "a bachelor is an unmarried man" which is true trivially, again, because knowing the definitions of the words in the sentence suffices for determining its truth value which is inevitably true. The inevitability stems from language and not from the way the world works. Analytic statements cannot be informative, except in the attenuated sense in which they provide information about how words are defined in a language. In contrast, we have the informative statements that make contingent propositions: such meanings can be possibly true and possibly false, though not both in the same context. It ought to be clear that a sentence like Aristotle's regarding whether there will be a sea-battle or not is contingent and as such it is of an entirely different type from the analytic truths (or analytic falsehoods). If, however, a proof that utilizes exclusively analytic premises is furnished to obtain a conclusion as to whether there will be a sea-battle or not, then the latter sentence - about the battle - ought to be also analytic. Yet, it cannot possibly be. Thus, the lapse seems to be utterly vicious. On the other hand, Aristotle seeks the ultimate origins of how things work in Nature and this applies also in the case of logic. A statement of the form "it is true of all x either that they F or that they don't F" is considered analytically true by Aristotle not because of the meanings of words like "all" but because Nature works in such a way that any statement of this form has to be true. (Aristotle understands that deductive reasoning is a matter of logical form and this indeed qualifies him for being the honorary "Father" of Logic.) Logical Fatalism seems excusable as a position of choice for someone who subscribes to this view. Of course, claims about nature or about events in historical time cannot be credited if they conflict with logic; it is the converse that seems to misunderstand the relationship between logic and reality (the converse consisting in the claim that "if claims about things or events are to be credited or not should depend on mere consistency with logical laws.")

Aristotle thinks that he can derive the conclusion that it is necessarily the case that p or necessarily the case that not-p (where p stands for the target proposition regarding the occurrence of the sea-battle on the morrow). He finds this alarming and sets out, as we will see, to detect the error in the proof itself.

A feature of the discussion of future contingents in Aristotle's text with seminal significance for the philosophy of logic is this: while the argument that is presumed to be establishing a deterministic conclusion is __modal__, the search for a solution (or identification of error) follows a different path. The attempted resolution is pursued __extensionally__: by returning to non-modal or truth-functional logic and considering addition of a __third truth-value__.

An extensional approach to logic, which has had many, philosophically motivated, famous adherents in our times, does not admit and cannot track any sensitivity of truth-values to variable contexts. The proposition expressed by a sentence like "it is possible that water freezes", symbolized, let's say, by W, has a truth-value, true or false (T or F), insofar as it is meaningful. In our actual physical universe, this proposition is true but extensionally no alternative states of affairs are available to be considered. The proposition W ought to be taken as meaning: "in such-and-such a universe, given the way natural kinds behave there, water freezes." Essentially, the possibilification has disappeared. "It is possible that water does not freeze" should be considered as a different proposition - and not as the negation of the former proposition. Let us symbolize this latter proposition by N. The proposition N is not the same as not-W. Let us take not-W. Both W and N can be true, although not in the same state of affairs. It is possible that water freezes and it is also possible that water does not freeze although this latter case is not the case of our universe as we know it with its natural kinds and their chemical compositions and properties. It is not inconceivable, however, that some alternative universe would have a compound with the same composition as water and the same properties except for the property of freezing. If we were to tell such a story as science-fiction, for instance, we would not be uttering absurdities. Only if we were to insist that this is the actual case, we would be absurd. In extensional logic, the context is considered packed into the proposition we consider: "it is possible that water freezes" has in it the indexical parameters of such-and-such a universe as ours. Because we banish context-sensitivity in this way, we have access to the simplest computational logic. This logic, as propositional logic to begin with, can be thought of replacing robust meaning (what meaningful sentences talk about) with two propositional types -- the True and the False (T/F). This proves sufficient for exploring and determining such properties as tautologousness, contradictoriness, contingency, consistency of theories, and validity of deductive arguments. The logic is characterized by the so-called Leibniz Law which is also, appropriately, called the Law of Extensionality: if A and B stand for any two propositions which have the same truth value (T or F), then they can be substituted for each other within any compound proposition in which they occur without changing the truth-value of the whole. This is not intuitive because it is simply not the case that, for instance, "Clinton was President" and "Bush II was President" can be inter-substituted within "Clinton was President before Bush II was President" without alteration of meaning: but, recall, logical meaning is a matter of T and F for propositional logic. The linguistic particle "before" in this example cannot be captured extensionally because it is not truth-functional: when "before" operates on two propositions, connecting them, we cannot compose the meaning of the whole based on the meanings of the components. We just saw that the components of this compound are both true (both Clinton and Bush II were Presidents) but - symbolizing true by "T" - we have: TbeforeT is T in one case (when the first sentence is about Clinton) and the same, TbeforeT, is not-T in the other case (when the first sentences is about Bush II.)

Modal logics extend propositional logic by adding context and relations among contexts. Now our current state of affairs (which means simply the standpoint from which we evaluate) can be designated as, for instance, @, and distinguished from other logically possible context which we can have from a non-empty set W = {w1, w2, ...} of indeces for different contexts. A relation R is imposed on W: this is the collection of pairs of contexts that access each other. There is an open number of modal propositional systems - while there is only one standard propositional logic: depending on what restrictions are placed on R, we get different modal logics and the motivation for picking one logic or another has to do with what kind of modality we are tracking: logical modality (necessity/possibility), epistemic modalities (knowing, believing), deontic modalities (moral obligatoriness, moral permissibility), physical modalities (physical necessity, physical possibility), temporal modalities (having-been-true always, having been true at least once; - similarly for the future)....

Aristotle thinks that his argument requires logical modalities. Aristotle has no distinction among different species of modalities as outlined above. He takes every modal proposition to be a matter of logical necessity or possibility. The system for this has as frame, as modal logicians say, <W, R> where the relation R has no restrictions (every context accesses or sees every context, including itself.) The frame <W, R> can be extended to a model if valuations for all propositions p are made available indexically (the valuations assigning truth values to the propositions for each world.)

On the other hand, if a third truth-value, besides T and F (true and false) is added to the standard propositional logic, we have a three-valued logic with logical connectives (not, either-or, and, if-then, if-and-only-if, etc.) defined now over three and not over two values. This means that the meanings of the connectives are considered different from those of the classical system. If we take the "not" in the proposition expressed by the sentence "it is not raining today" to have the standard meaning, then we cannot also consider it to have a non-standard meaning. A non-standard meaning of "not" can be argued for a proposition that makes the case for a future contingency -- and this is what tempts Aristotle. The example discussed in Aristotle's text is "there will not be a sea-battle tomorrow" - and, of course, also "there will be a sea-battle tomorrow." If we were to add a third value, semantically interpreting this third value as "indeterminate", then we would be constructing or defining a different connective: our "not" would not be the classical one. While the classical "not" reverses values (turning T to F and F to T), this non-classical "not" leaves at least one value, the indeterminate or I, unchanged. We are motivated to define it this way: the negation of an indeterminate proposition we may deem indeterminate. Interestingly, the non-classical "not" we are defining reverses T to F and F to T: for the classical values {T, F} it behaves classical (and, so, it is called "normal"), but it is a different connective from the classical or standard "not" still, The logic we now have is extensional, it is not modal. It may appear that we could match modal logics with corresponding many-valued logics but it has been proven rigorously that this is not the case. This enforces a choice as to what claim is made: is the logic of an argument modal or extensional but many-valued? Aristotle does not detect this and shifts between the two. He starts with a construction of his alarming argument in modal and then considers a solution that can be sufficiently conceived as extensional but many-valued. Moreover, as we indicated, Aristotle can only consider logical necessity and possibility - he cannot consider modalizations as being possibly temporal in character.

Here is the famous argument about future contingents. The example of a possible future event which, as such, may or may not take place, is a sea-battle. Thus, the sentences are "there will be a sea-battle tomorrow" and "there will not be a sea-battle tomorrow." We can symbolize them respectively as "M S" and "M - S". "M" stands for "possibly" in the logical sense and "-" is the symbol for negation. It appears that Aristotle is thinking along modal lines since he is not bothered by the possibility of a contradiction in "M S & M - S" where "&" symbolizes the standard "and." Indeed, "S & - S" is a contradiction but "M S & M - S" is not. Modally, we have access to variable contexts: so, "M S" compels "S, w1" and "M - S" compels "- S, w2" where the contexts are "w1" and "w2". In common sense, we have S in one possible context and not-S in some other possible context; this shows that we may have simultaneous possibilification of a proposition and of its negation. This should not be possible only when the compound proposition possibilified is itself a logically necessary proposition. On the other hand, no atomic or simple proposition is to be considered as logically necessary; thus, "possibly p and possibly not-p" for any simple proposition p ought to be forthcoming - and it is. This is actually the definition of what we mean by a contingent proposition. This is how we have to understand logically contingent propositions anyway and Aristotle is aware of this since he himself defined logical contingency along these lines (without using a symbolic language.)

The proposition "S v - S", where "v" stands for the inclusive either-or of language, expresses a law of two-valued logic. It is true under any assignment of truth-values. If S is true, then "S v - S" is true. But if S is false, then "S v - S" is true again. The law instantiated by this is called the Law of Excluded Middle and this actually reflects that there are exactly two truth-values, T and F, in the language. Aristotle is not clear on this underlying connection - or on the fact that the law should not hold if there are more than two truth-values. Or, if Aristotle realizes this, he is convinced that the Law of Excluded Middle must hold absolutely as a law of logic and cannot be given up no matter what. Another possibility is that one could confuse a logic with the logic of the language in which we speak about this logic: suppose that we have three truth values in a logic L; still, in the metalanguage in which we speak about L, we want to say that any proposition p either receives or does not receives any one of the three truth values in L. Thus, while L should not have the Law of Excluded Middle, we see that the metalanguage of L does have this law. This, however, does not seem to establish that anything is wrong with L - even though critics of multivalent logics have actually made this point! It is obvious that the Law of Excluded Middle should fail for the three-valued logic because the third value - call it "n" - is neither T nor not-T! The Law of the Excluded Middle is replaced by another law, which we can call the Law of the Excluded Fourth: either p, or not-p, or n. ( We are assuming that T and F negate each other still in the three-valued logic.) In classical logic, this is equivalent with: if p-and-not-p, then n. Nevertheless, the meanings of "if-then" and the meanings of "either or" and "not" are different in the three-valued logic; this option of deriving the third value simply from a classical contradiction is thus not available - and it should not be! We see that a three-valued logic has its own rules - something Aristotle misses, as we will see.

Let us return to the modal setup. Modal logics are extensions of classical logic - and, thus, they have two truth values. Because "S v - S" is a bivalent logical law, then it must be true in every context as we would say. Hence, it is logically necessarily true that "S v - S" or "N(S v - S)", where "N" stands for "logically necessarily." Aristotle starts with this premise. Here comes the problem. Aristotle thinks that the modal operator N distributes over inclusive disjunction: this means that from "N(S v - S)" we can validly derive "N S v N - S". This, however, is an invalid inference. It is a formal modal fallacy for the modal system considered. This is one quick way to criticize and dispose of Aristotle's argument but we will need to apply further scrutiny as it is.

What would happen if we had a proof of "N S v N - S"? Aristotle thinks that cannot be the case because the truth-making event, the sea-battle, has neither taken place nor failed to take place yet. Moreover, Aristotle is alarmed that, if the above were the case, then we would have proved that we live in a universe in which no matter what we do we cannot alter whether future events happens or do not happen. The two propositions, "NS" and "N - S", are both future-contingent propositions -- they are about the future. They are meaningful and, as such, Aristotle thinks, they should be either true or false. This, however, gets us into the problem. Even without the inference to "NS v N - S" from "N(S v - S)" we are facing the problem that truth-values seem to be pre-assignable to propositions - since propositions about the future are treated as meaningful and meaningful propositions have truth-values!

Aristotle's worry about a pre-determined future can be assuaged in some other way perhaps. A similar problem, the problem of God's middle knowledge, was discussed in the Middle Ages. To resolve the issue, Aristotle abandons the modal presentation of the problem and takes it up as an extensional subject: he decides that there may be a third type of meaningful proposition - besides the true and the false there may be the future-contingent type which we can call "indeterminate." This adds a third truth-value. We may wonder if, instead, we should take this third type of meaningful proposition to correspond to no truth-value at all. Nevertheless, we have seen that Aristotle insists that meaningful propositions have to have truth-values. Moreover, he clearly goes on to make truth-functional computations with all three truth-values. Here are such computations. Aristotle realizes that for the third type, the law of excluded middle fails: if "S" takes as value not T and not F but I, then "S & - S" yields for computations: I & - I = I & I = I [The value of " - I" is "I" and the value of "I & I" is again "I": not-indeterminate should be indeterminate, and indeterminate-and-indeterminate should be indeterminate.]

Since excluded middle fails for "I", excluded middle fails as such - it is not a law of the three-valued language. This is exactly what should be expected but Aristotle, who takes an absolute view of logic and its laws, rejects this possibility as absurd. Disregarding the necessity of a non-bivalent logic having different logical laws, he throws out his three-valued solution to the problem. This solution was revived by the Polish logician Jan Lukasciewicz in the early 20th century and served as a philosophic motivation for ushering in a new era in the study of multivalent logical systems.

## Modal Logic

**© 2015 Odysseus Makridis**

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