To expose the mistakes in the deterministic arguments, we will need some tools of modern logic. Some elementary symbols will help to illuminate the concepts at play in the deterministic arguments. However, all the formulas that will be used, which incorporate these symbols, will also be expressed in English prose.
| Symbol | Its meaning | Explanation |
| P, Q, R, … | propositions | |
| ~P | it is not the case that P | Example: It is not the case that copper conducts electricity. (Note: “P” and “~P” have opposite truth-values – whichever is true, the other is false.) |
| P ⊃ Q | if P, then Q | Example: If she is late, (then) the meeting will be delayed. |
| gKP | God knows that P | Example: God knows that the Mississippi River flows north to south. |
Next we need three concepts at the heart of modern modal logic. The symbols are:
| Symbol | Its meaning | Explanation |
| ◊P | it is (logically) possible that P | Example: It is (logically) possible that the United States was defeated in World War II. (Note: Whatever is not self-contradictory is logically possible.) |
| ☐P | It is (logically) necessary that P | Example: It is logically necessary that every number has a double. (Note: If Q is not logically possible, then ~Q is logically necessary.) |
| ∇P | It is contingent that P | Example: It is contingent that the United States purchased Alaska from Russia. (Note: A proposition, Q, is contingent if and only if ◊Q and ◊~Q.) |
These latter three concepts require further elaboration.
P is possible (symbolized “◊P”). A proposition, P, is possible if and only if it is not self-contradictory. All propositions that are true are possibly true. In addition, some false propositions are also possibly true, namely those that are false but are not self-contradictory. Some philosophers like to explicate “P is possible” in this way: “There are some possible circumstances in which P is true”. And some philosophers, adopting the terminology popularized by Leibniz (1646-1716), will substitute “worlds” for “circumstances”, yielding “P is true in some possible worlds”. Examples of possibly true propositions include:
- Ottawa, Canada, is north of Washington, DC.
- The Great Salt Lake is saltier than the Dead Sea.
- The Dead Sea is saltier than the Great Salt Lake.
- John Lennon was the first songwriter to travel in a space capsule.
- There are three times as many species of insect as there are species of mollusk.
- 2 + 2 = 4
- All aunts are female.
- Some pigs can levitate.
Understand that prefacing a proposition, P, with “◊” does not ‘make’ P possible. What it does is to create a new, different, proposition, namely ◊P, which, in effect, says that P is possible. If P is possible (for example, suppose “P” stands for “Gold was first discovered in California in 1990″), then (although P is false), ◊P is true. Or, suppose “Q” stands for “2 + 2 = 7″. Then prefacing “Q” with “◊” does not ‘make’ Q possible. It produces a new proposition, “◊Q”, which is false. Q is, and remains, impossible whether or not it is prefaced with “◊”.
Everything that is actual (or actually true) is possible (that is, possibly true). But if a proposition is actually false, then it is impossible only if it is self-contradictory; otherwise it is a false contingency, and all contingencies, whether true or false, are possible.
We may ask “What color did Sylvia paint the lawn chair?” We look at the chair and see that she has painted it yellow. Thus it is demonstrable that it is possible that she painted the chair yellow. And its being yellow implies it is false that she painted the chair blue. But the falsity of the proposition that she painted the lawn chair blue in no way precludes that she could have done so. Even though false, it still remains possible that she painted the chair blue.
P is necessary (symbolized “☐P”). Necessarily true propositions are those that are true in all possible circumstances (/worlds)—that is, are not false in any. Necessary truth can be defined in terms of possibility, namely P is necessary if and only if its negation (that is, “~P”) is impossible. In symbols (where “=df” stands for “is by definition”):
☐P =df ~◊~P
Examples of necessarily true propositions:
- 2 + 2 = 4
- All aunts are female.
- Whatever is blue is colored.
- There are either fewer than 20 million stars or there are more than 12 million. (This statement may be unobvious; but if you think about it you may come to see that it cannot be false.)
- It is false that some triangle has exactly four sides.
P is contingent (symbolized “∇P”). A proposition, P, is contingent if and only if it is both possibly true andpossibly false. Contingent propositions are those that are true in some possible circumstances (/worlds) and are false in some possible circumstances (/worlds). Contingency can be defined in terms of possibility, namely:
∇P =df ◊P & ◊~P
It is essential to understand that “◊P & ◊~P” does not mean “P is true and false in some possible circumstances (worlds)”. No proposition whatsoever is both true and false in the same set of circumstances (law of non-contradiction). To say that a proposition is contingent is to say that it is true in some possible circumstances and is false in some (other!) circumstances.
Examples:
- The Boston Red Sox won the World Series in 2002.
- It is false that the Boston Red Sox won the World Series in 2002.
- Steel-clad ships can float in the ocean.
- It is false that steel-clad ships can float in the ocean.
Modal terms and modal status
Terms such as “must”, “has to”, “cannot”, “is necessary”, “is impossible”, “could not be otherwise”, “has to be”, “might”, “could be”, “contingent”, and the like, are known as “modal” terms. All of these are definable in terms of “possibility”.
Every proposition is either logically possible or logically impossible. And no proposition is both.
Drawing the net a bit finer, and dividing the class of logically possible propositions into those that are necessarily true and those that are contingent, we have three exclusive categories. Every proposition is exclusively either necessarily true, necessarily false, or contingent. That is, every proposition falls into one of these latter three categories, and no proposition falls into more than one.
Just as the expression “truth-value” is a generic term encompassing “truth” and “falsity”, the expression “modal status” is a generic term encompassing “contingent”, “necessarily true”, and “necessarily false”.
Finally, no proposition ever changes its modal status. We will call this principle “The Principle of the Fixity of Modal Status“. And for the purposes of assessing the deterministic arguments we note especially: no contingent proposition ever ‘becomes’ necessary or impossible.
6. The Modal Fallacy
From a mathematical point of view, if we arbitrarily pick any two propositions, truth and falsity can be attributed to them in four different combinations, specifically
- the first is true, and the second is true
- the first is true, and the second is false
- the first is false, and the second is true
- the first is false, and the second is false
However, it sometimes happens that two propositions will have certain logical relationships between them such as to make one or more of these four combinations impossible. For example, consider the two propositions α and β.
α: Diane planted only six rosebushes.β: Diane planted fewer than eight rosebushes.
While each of these propositions, by itself, could be true and could be false, there are – as it turns out – only three, not four, possible combinations of truth and falsity that can be attributed to this particular pair of propositions. On careful thought, we can see that the second combination – that is, the one which attributes truth to α and falsity to β – is impossible. For if α is true (that is, if it is true that Diane has planted only six rosebushes) then β is also true. Put another way: the truth of α guarantees the truth of β. This is to say
(1) It is impossible (for α to be true and for β to be false).
Unfortunately, ordinary English does not lend itself easily to express the quasi-symbolic sentence (1). In symbols we can express the sentence this way:
(1a) ~◊(α & ~β)
About the best we can do in English is to create the following unidiomatic, extremely clumsy sentence:
(1b) The compound sentence, α and not-β, is impossible (that is, is necessarily false).
English prose is a poor tool for expressing fine logical distinctions (just as it is an unsuitable tool for expressing fine mathematical distinctions[3] ). But, as it turns out, the situation is worse than just having to make do with awkward sentences. For it is a curious fact about most natural languages – English, French, Hebrew, etc. – that when we use modal terms in ordinary speech, we often do so in logically misleading ways. Just see how natural it is to try to formulate the preceding point [namely proposition (1)] in this fashion:
(2) If α is true, then it is impossible for β to be false.
Or, in symbols:
(2a) α ⊃ ~◊~β
In ordinary speech, the latter sentence, (2), is natural and idiomatic; the former sentence (1b) is unnatural and unidiomatic. But – and this is the crucial point – the propositions expressed by (1)-(1b) are not equivalent to the propositions expressed by sentences (2)-(2a). The former set, that is (1)-(1b), are all true. The latter, (2)-(2a)are false and commit the modal fallacy. The fallacy occurs in its assigning the modality of impossibility, not to the relationship between the truth of α and falsity of β as is done in (1)-(1b), but to the falsity of β alone. Ordinary grammar beguiles us and misleads us. It makes us believe that if α is true, then it is impossible for β to be false. But it is possible for β to be false. β is a contingent proposition. Recall the principle of the fixity of modal status. Even if the falsity of β is guaranteed by the truth of some other proposition [in this case α], β doesnot ‘become’ impossible: it ‘remains’ contingent, and thereby possible.
Whatever impossibility there is lies in jointly asserting α and denying β. (See (1b) above.) The proposition “it is false that β” does not ‘become’ impossible if one asserts α.[4]
a. The Modal Fallacy in Logical Determinism
Some persons have been deceived by the following (fallacious) argument to the effect that there are no contingent propositions:
“(By the Law of Non-contradiction), if a proposition is true (/false), then it cannot be false (/true). If a proposition cannot be false (/true), then it is necessarily true (/false). Therefore if a proposition is true (/false), it is necessarily true (/false). That is, there are no contingent propositions. Every proposition is either necessarily true or necessarily false. (If we could see the world from God’s viewpoint, we would see the necessity of everything. Contingency is simply an artifact of ignorance. Contingency disappears with complete knowledge.)”
The fallacy arises in the ambiguity of the first premise. If we interpret it close to the English, we get:
P ⊃ ~◊~P
~◊~P ⊃ ☐P
~◊~P ⊃ ☐P
∴ P ⊃ ☐ P
However, if we regard the English as misleading, as assigning a necessity to what is simply nothing more than a necessary condition, then we get instead as our premises:
~◊(P & ~P) [equivalently: ☐(P ⊃ P)]
~◊~P ⊃ ☐P
~◊~P ⊃ ☐P
From these latter two premises, one cannot validly infer the conclusion:
P ⊃ ☐P.
In short, the argument to the effect that there are no contingent propositions is unsound. Its very first premise commits the
modal fallacy.
modal fallacy.
The identical error occurs in the argument for logical determinism. Recall (the expanded version of) Aristotle’s sea battle:
Two warring admirals, A and B, are preparing their fleets for a decisive sea battle tomorrow. The battle will be fought until one side is victorious. But the “logical laws (or principles)” of the excluded middle (every proposition is either true or false) and of noncontradiction (no proposition is both true and false), require that one of the propositions, “A wins” and “it is false that A wins,” is true and the other is false. Suppose “A wins” is (today) true. Then whatever A does (or fails to do) today will make no difference: A must win; similarly, whatever B does (or fails to do) today will make no difference: the outcome is already settled (that is, A must win). Or again, suppose “A wins” is (today) false. Then no matter what A does today (or fails to do), it will make no difference: A must lose; similarly, no matter what B does (or fails to do), it will make no difference: the outcome is already settled (that is, A must lose). Thus, if every proposition is either true or false (and not both), then planning, or as Aristotle put it “taking trouble,” is futile. The future will be what it will be, irrespective of our planning, intentions, etc.
If we let “A” stand for “Admiral A wins” and let “B” stand for “Admiral B wins”, the core of this argument can be stated in symbols this way:
| A or B | [one or the other of these two propositions is true] | ||
| ~◊(A & B) | [it is not possible that both A and B are true] | ||
| | |||
| ∴ | A ⊃ ☐A A ⊃ ~◊~A | } | If A is true, then A must be true. If A is true, then A cannot be false. |
| A ⊃ ☐~B A ⊃ ~◊B | } | If A is true, then B must be If A is true, then B cannot be true.false. | |
| B ⊃ ☐B B ⊃ ~◊~B | } | If B is true, then B must be true. If B is true, then B cannot be false. | |
| B ⊃ ☐~A B ⊃ ~◊A | } | If B is true, then A must be If B is true, then A cannot be true.false. |
In this argument, by hypothesis, either A is true or B is true, and since they cannot both be true, the second premise may be accepted as true. But none of the conclusions is true. A is contingent, and B is contingent. Yet the conclusions state that from the assumed truth of either of (the two contingencies) A or B, it follows that A and B are each either necessarily true or necessarily false. Each of these eight conclusions violates the principle of the fixity of modal status. What, then, are the conclusions one may draw validly from the premises? These:
| ☐(A ⊃ ~B) | or, equivalently, | ~◊(A & B) |
| ☐(B ⊃ ~A) | or, equivalently, | ~◊(B & A) |
So long as we remain mindful of the fact that “~◊(P & Q)” is logically equivalent to “☐(P ⊃ ~Q)” but is not equivalent to “P ⊃ ☐~Q”, the argument for logical determinism will be seen to be invalid. Our ordinary language treats “it is impossible for both P and Q to be true” as if it were logically equivalent to “if P is true, then Q is necessarily false”. But the profound difference between these two assertions is that the former preserves the principle of the fixity of modal status, the latter violates that principle. The proposition, “Admiral A wins”, is contingent, and if true, then it “remains” true. Indeed this is a trivial logical truth:
(i) ☐(P ⊃ P) alternatively, ~◊(P & ~P)
The argument for logical determinism illicitly treats this logical truth as if it were equivalent to the false proposition
(ii) P ⊃ ☐P alternatively, P ⊃ ~◊~P
If you do not let yourself be beguiled by the invalid ‘move’ (inference) from (i) to (ii), the argument for logical determinism collapses. The truth of a proposition concerning your future behavior does not make that future behavior necessary. What you choose to do in the future was, is, and will remain contingent, even if a proposition describing that choice is timelessly true.
b. The Modal Fallacy in Epistemic Determinism
Let’s recall Maimonides’s argument:
… “Does God know or does He not know that a certain individual will be good or bad? If thou sayest ‘He knows’, then it necessarily follows that [that] man is compelled to act as God knew beforehand he would act, otherwise God’s knowledge would be imperfect.”
We can symbolize the core of this argument, using “∴” for “it necessarily follows”; and “☐” for “compelled”; and “D” for the proposition describing what some particular person does tomorrow.
gKD
∴ ☐D
There seems to be (at least) one missing premise. [In the terminology of logicians, the argument isenthymematic.] One tacit assumption of this argument is the necessary truth, “it is not possible both for God to know that D and for D to be false”, or, in symbols, “~◊(gKD & ~D)”. So the argument becomes:
gKD
~◊(gKD & ~D)
~◊(gKD & ~D)
∴ ☐D
But even with this repair, the argument remains invalid. The conclusion does not follow from the two premises. To derive the conclusion, a third premise is needed, and it is easy to see what it is. Most persons, with hardly a moment’s thought, virtually as a reflex action, will tacitly assume that the second premise is logically equivalent to:
gKD ⊃ ☐D
and will tacitly (/unconsciously) add this further premise, so as to yield, finally:
gKD
~◊(gKD & ~D)
gKD ⊃ ☐D
~◊(gKD & ~D)
gKD ⊃ ☐D
∴ ☐D
But this third premise, we have seen above, is false; it commits the modal fallacy. Without this premise, Maimonides’ argument is invalid; with it, the argument becomes valid but unsound (that is, has a false and essential premise [namely the third one]). Either way, the argument is a logical botch.
Once the logical error is detected, and removed, the argument for epistemic determinism simply collapses. If some future action/choice is known prior to its occurrence, that event does not thereby become “necessary”, “compelled”, “forced”, or what have you. Inasmuch as its description was, is, and will remain forever contingent, both it and its negation remain possible. Of course only one of the two was, is, and will remain true; while the other was, is, and will remain false. But truth and falsity, per se, do not determine a proposition’s modality. Whether true or false, each of these propositions was, is, and will remain possible. Knowing – whether by God or a human being – some future event no more forces that event to occur than our learning that dinosaurs lived in (what is now) South Dakota forced those reptiles to take up residence there.
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