The argument is as follows.

[1] Assume that two-valued logic is a universally valid method of reasoning.

[2] In that case, every proposition must be either true or false -- no other alternatives are allowed.

[3] Now consider the proposition «Free will exists» (or, synonymously, «Choice exists»).

[4] Under two-valued logic, this proposition must be either true or false.

[5] Assume now that the proposition «Free will exists» is false.

[6] In that case, free will (or, synonymously, choice) cannot exist.

[7] This means that whatever is believed would be believed simply because there is no choice in the matter as to what is believed.

[8] If everything that is believed is believed simply because there is no choice in the matter as to what is believed, it can never be known (or proved) whether any belief is true.

[9] As a consequence of all the above, it can never be known that free will does not exist.

[10] Under two valued logic, if it cannot be known that free will does not exist, then its opposite, namely that free will does exist, can be known or proved to be true -- there is no other alternative.

[11] Thus the assumption made at [5] above -- namely that the proposition «Free will exists» is false -- is itself false. Or, in other words, under two-valued logic free will must exist.

[12] If free will exists, any proposition that deals with the future must be neither true nor untrue: for what the future will turn out to be will depend on how free will is exercised.

[13] This contradicts [2] above. Or in other words, [2] above must be false.

[14] If [2] above is false, then [1] above must also be false.

[15] Therefore two-valued logic cannot be a universally valid method of reasoning.

Q. E. D.

It is to be noted that the above argument was to some extent foreseen by Aristotle himself -- the «founder», if one may so call him, of two-valued logic. He wrote words to the effect that the proposition «There will be a sea-battle tomorrow» can be neither true nor untrue.

However, the venerable Stagirite never argued his case by showing that under the assumption that two-valued logic is universally valid, free will must exist, and that is why it is impossible to speak of the truth or falsehood of any proposition which speaks of events in the future. Indeed, as far as is known to the author of this paper, no one has made such an argument before.

It has been argued, in an attempt at countering the argument given at the beginning of this paper, that it is possible for two-valued logic to be rendered universally valid by using «If ... then ...» statements: such as «If I were rich I'd be driving a Porsche Carrera» (but I'm not rich, so I drive a Honda Civic ... which is by no means a bad car, but is by no means a Porsche Carrera either!). Under such conditions, although it is not true that as things stand I drive a Porsche Carrera, it is true that if I were rich I'd be driving one. In this way, future contingencies can be dealt with in the present by disjoining all contingencies and reasoning separately about each.

However, this counter-argument can itself be countered using the following counter-counter-argument:

Let a proposition p be enunciated as follows:

p: «I will pick up this pen within the next ten seconds».

Then

~p: «I will not pick up this pen within the next ten seconds».

However, since by two-valued logic, free will must exist, neither p nor ~p can possibly be true: or in other words, we get

~(p v ~p)

-- which goes directly counter to the axioms of two-valued logic.

Of course one could always enunciate another proposition -- let it be called q -- as follows:

q: «If I choose to do so, I will pick up this pen within the next ten seconds».

In this case q might be regarded as true and ~q as false. however, the proposition q is not the proposition p! The propositions p and q are two very different propositions.

Essentially, if two-valued logic is to be universally valid, it has got to apply to all propositions, without a single exception. But it doesn't, so it isn't.

Besides -- and to elaborate further on the above argument -- even if I do choose to pick up the pen I might not actually pick it up: someone might decide to prevent me from doing so, or something else might come in the way: for countless reasons during the next ten seconds events might come to pass which would result in my not picking up that pen, and it would be impossible to foresee them all. For that matter, I might choose not to pick it up but might be compelled to pick it up nevertheless (say, by a threat, or as a result of an epileptic seizure), or I might change my mind at the last moment. The pen might even slip my fingers against my will.

Whatever the case, even the answer:

«If you choose to pick it up you will, otherwise you won't»

is untrue. And it doesn't seem possible to see how one could account for and reason separately about every possible future contingency, which is what would really be needed for an «If ... then ...» statement to be true. How could one ever be sure that absolutely no contingency has been missed? Obviously one couldn't. Thus the only true answer would be:

«If in the end you do pick it up you will have picked it up»

which is just two ways of saying the same thing, and thus is really no answer.

And as a clincher, the following proposition p' may be considered:

p': «I will choose to pick up this pen within the next ten seconds».

Of course it will be agreed that such a proposition can be neither true nor false: indeed, even I don't know whether I will choose to do something or not in the next ten seconds. But it is to be noted that in addition, p' is also incapable of being disjoined into yet further contingencies! No further «If ... then ...» statements about the matter can be made at this stage.

In other words, the «free will buck stops here», as it were. (And there has to be a point where it stops, because otherwise free will would not really be free, now would it.)

This should clinch the above argument that two-valued logic cannot be universally valid.

It is to be noted that the above reasoning implies some very significant philosophical conclusions. I will outline two of them here below. I am sure others will occur to my readers as well.

1. One conclusion is, that since all science depends on the results of experiments, and since at the beginning of any experiment the results thereof can be available only in the future, and since by the above reasoning the future can never be predicted with 100% certainty, one can never be one hundred per cent sure that any experiment will turn out as predicted, no matter how scrupulously or carefully it is performed! There must always remain a small but finite possibility that the outcome of any experiment will be the result, not exclusively of the laws of science as they are known to be at any given time, but of the action of free will interfering with those laws.

As a result, no scientific experiment can establish its results absolutely conclusively.

It is to be noted, by the way, that clause [12] of the argument given at the beginning of this paper refers to all propositions that deal with the future. This is because it is impossible to predict with 100% certainty that free will will not be able to come into play in any given realm. It is of course normally accepted that where the human influence is negligible -- such as in the movements of large astronomical bodies -- free will does not come into play. This is why it is possible to predict the movements of planets years, decades, centuries and even millennia in advance. However, from a purely technical point of view, it is possible even to jiggle the orbit of Jupiter from here on Earth, albeit by an imperceptible amount, by simply shining a flashlight in the direction of that giant planet: the slight push imparted to Jupiter by the beam of light would imperceptibly push that planet farther away from the Earth, whence the light beam originates. (Indeed by Newton's Third Law of Motion, such an action would jiggle the orbits of both Jupiter and the Earth.)

And of course, with the passage of time, human technology is likely to advance to such an extent that it will likely be possible for us to jiggle the orbit of Jupiter quite perceptibly ... or even break it up altogether, along with all the other planets, and make a giant «Dyson Sphere» out of the raw material so obtained: as is foreseen by the eminent astrophysicist Freeman Dyson of the Institute for Advanced Study, Princeton!

Thus to be absolutely precise, it is possible to bring free will into play in any experiment, theoretically involving even the most remote and most massive quasars ever detected.

And of course, here on Earth itself, it is recognised that free will cannot entirely be ruled out in any system, even one as impersonal as the global weather system: for as the well-known «butterfly effect» of climatology asserts, it is possible for a butterfly to choose to flap its wings in Hong Kong, and for a typhoon to result therefrom in California.

2. Mathematics is based entirely on two-valued logic, in the sense that every theorem of mathematics uses two-valued logic to attain its proof. There is only one true answer to any mathematical question, and all other answers are false: and there is no other alternative. (For example, the sum of two plus two must be four, and no other number.)

Thus if two-valued logic is not universally valid, it is impossible for mathematics to be so either.

However, mathematics is the basis of all of modern physics; and modern physics is the basis of all the other physical sciences: chemistry, biology, geology, astronomy, etc., etc.

This implies that if two-valued logic cannot be universally valid, then neither can any of the physical sciences be universally valid: indeed, not even all the physical sciences taken together -- with mathematics thrown in for good measure -- can be universally valid!

Philosophically this conclusion gives rise to a most interesting question: if mathematics and the physical sciences cannot be universally valid, then what mental disciple -- or combination of mental disciplines -- can be universally valid? At present there does not seem to be a clear and unequivocal answer to this question. Of course there are many separate and sometimes conflicting assertions in this regard. Some say it is Religion -- or a particular Religion -- that is universally valid; others assert that it is Reason (using that term in the broadest possible sense, as encompassing the underlying principles common to all possible logics, if any such may be found) that is universally valid; yet others affirm that it is Divine Revelation, not such as is contained in any scripture, but as is Revealed from time to time by the Supreme Mind Itself to a living, breathing human being, that is universally valid. Of course there always remains yet another alternative -- that it is Analytical Philosophy that is universally valid: or at least we who read and contribute to Sorites may hope it is. But there is no universal agreement as to the answer to the question: on the contrary, there is almost universal disagreement -- to the extent that there is a saying in India that it is impossible to find two gurus who will agree with one another. (Perhaps the same thing can be said about analytical philosophers! ... just kidding.)

It seems clear that two-valued logic cannot be universally valid. As a result, it also seems clear that neither mathematics nor the physical sciences -- nor both of them taken together -- can be universally valid. However, it is a question yet to be answered as to what mental discipline is universally valid. Perhaps there is none.

The author would appreciate comments to this paper: whether adverse or sympathetic. Indeed cogent and constructive criticism is more welcome than uncritical accolade! He can be reached via e-mail at either one of his e-mail addresses: