Copyright © Norman Swartz 1993, 1999, 2010, 2011
This revision: July 21, 2011
Department of Philosophy
Simon Fraser University

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'The' Modal Fallacy

Note: the technical vocabulary used in this article is explained in this glossary.



The following argument appears to be valid and to have true premises; yet its conclusion is false.

If Paul has one daughter and two sons, then Paul has to have at least one son.
Paul has one daughter and two sons.
Paul has to have at least one son.
The problem is that although Paul (my brother) does have a son (he in fact has two sons), he does not have to have any. His having any children at all, as well as the exact number, are contingent matters, not matters of logical necessity.

The apparent form (in propositional modal logic) of this argument is:

The 'correct' form of the argument, one which does not have "S" as a conclusion, but only "S", is:

(H S)
The trouble is that, in English, and (I have been told) a number of other natural languages as well, there is a tendency (a style) of moving the modal operator – which, as it were, 'applies' to the entire conditional sentence – inside the sentence, more exactly, onto the consequent, e.g.
Paul's having one daughter and two sons necessitates [makes necessary] his having at least one son.
This is a so-called relative necessity. Paul's having at least one son is not logically necessary. It is relatively necessary, given his having one daughter and two sons.

This can be put still another way: Paul's having at least one son is a necessary condition of  his having one daughter and two sons.

The modal fallacy we are here examining can be regarded as construing or mistaking S's being a logically necessary condition for H as S's being a logically necessary truth.

Some writers prefer to call this fallacy "the modal scope fallacy" by which they mean that the fallacy consists of constricting the 'scope' of the necessity from the entire conditional (wide scope) to just its consequent (narrow scope). See e.g. the Modal Scope Fallacy.

There are a great number of fallacious arguments which, in one way or another, misuse modal concepts and thereby seem to establish conclusions which are not warranted at all. We will look at certain arguments concerning necessity and contingency, knowledge, fatalism, time travel, omnipotence, free will, and obligation. But we will begin by drawing a parallel with some lessons to be learned in inductive logic.

Parallel fallacy in inductive logic

Suppose there is a 98% recovery rate among patients with pneumonia. Thus we might suppose (where "Prob" = "the probability of"; "P" = "has pneumonia", "R" = "recovers", and "j" = "Josephine").

Prob(x)(Px Rx) = 0.98
Prob(Rj) = 0.98
(An alternative way of writing the first premise is:

(x)(Px 0.98 Rx)
We might express this, in quasi-English, this way: For any x, if x has pneumonia, then there is a 98% chance [or probability] that x will recover. Notice how it is 'natural' to attach the probability to the consequent.)

The trouble with this argument is that it may also be true that

Prob(x)([Px & Ox] Rx) = 0.23
Pj & Oj
where "O" = "is an octogenarian". Similarly, it may also be true that

Prob(x)([Px & Cx] Rx) = 0.11
Pj & Cj
where "C" = "has congestive heart failure".

Each of these three arguments has a conclusion different from the others. Which of the three conclusions shall we adopt (/believe, /subscribe to)?

Prob(Rj) = 0.98
Prob(Rj) = 0.23
Prob(Rj) = 0.11
All six premises (two each per argument) are true, each argument appears to be sound, and yet their conclusions are contraries of one another. Thus, at most, only one of the arguments can actually be sound. But since designating any one of them as sound would be arbitrary, we must conclude that none of them is sound.

The problem here is 'solved' by insisting that there are no 'detachment' rules in inductive logic (that the probability of the premises does not transfer to the conclusion if the conclusion is taken to stand 'alone'). Probabilities of conclusions are relative to one's premises and are not absolute (detachable). (Compare with Modus Ponens which is a detachment rule.)

Similarly, the 'necessity' of a necessary condition is 'relative' to the antecedent, and may not be attached to the conclusion absolutely, i.e. the necessity may not be 'detached'.

'If p is true, then p cannot be false'

"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
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
From these latter two premises, one cannot validly infer the conclusion:

P P.

'If x knows that p, then p must be true'

If x knows that P, then P must be true. But if P must be true, then P cannot be false. Any proposition which cannot be false, is not only true, but necessarily true. Thus if x knows that P, then P is necessarily true. By the transposition (or contraposition) theorem, we can restate this last conclusion in this manner: if P is not necessarily true (i.e. is contingent or necessarily false), then it is false that x knows that P. In short, the only knowable propositions are necessary truths, e.g. the truths of logic and mathematics. All other truths, although, perhaps, highly confirmable, are not, ultimately, knowable and must remain, therefore, a matter of opinion only."
Again, as in the previous case, the fallacy in this argument lies in the ambiguity of the first premise. If we take the English grammar to mirror the logical grammar, we get:

Kxp p
The correct interpretation of the first premise is:

(Kxp p)
The fallacy consists in taking a necessary condition (p) of knowledge (Kxp) to be necessary (p) 'on its own'.

'If you really know that p, then you cannot be mistaken'

"If you genuinely (really) know that P, then you cannot be mistaken. Thus if there is any possibility of your being mistaken, however remote, then you do not know that P. Since we can never rule out the possibility of being mistaken (you might be drugged; have poor eyesight, hearing, etc.; the instruments you use may be defective; someone may have lied to you; etc., etc.), you can never (really) know anything."
This fallacious argument, like both of its predecessors, has an ambiguous first premise (where "Mxp" = "x is mistaken in believing that p"):

Incorrect interpretation:   Kxp ~Mxp
Correct interpretation:
(or equivalently)
  ~(Kxp & Mxp)
(Kxp ~Mxp)

Logical Determinism

Aristotle's problem of tomorrow's sea battle (here reconstructed and considerably embellished).
"Two admirals, A and B, are preparing their navies for a sea battle tomorrow. The battle will be fought until one side is victorious. But the 'laws' of the excluded middle (no third truth-value) and of noncontradiction (not both truth-values), mandate that one of the propositions, 'A wins' and 'B wins', is true (always has been and ever will be) and the other is false (always has been and ever will be). Suppose 'A wins' is today true. Then whatever A does (or fails to do) today will make no difference; similarly, whatever B does (or fails to do) today will make no difference: the outcome is already settled. Or again, suppose 'A wins' is today false. Then no matter what A does today (or fails to do), it will make no difference; similarly, no matter what B does (or fails to do), it will make no difference: the outcome is already settled. Thus, if propositions bear their truth-values timelessly (or unchangingly and eternally), then planning, or as Aristotle put it 'taking care', is illusory in its efficacy. The future will be what it will be, irrespective of our planning, intentions, etc."
(If the error is unobvious to you, click here.)

Time travel

"Suppose it were possible to travel backwards in time. Murder, we know, is logically possible. Indeed it is much more than merely logically possible. It is physically possible, and indeed there are far more incidents of it in this (the actual) world than persons with moral sensibility can abide. But given that murder is both physically and technologically possible (even if not morally permissible), if you could travel backwards in time, it would be possible for you to meet and murder one of your (four) great-grandfathers, indeed you could do so when he was only six years old (shame and horror!), years before he fathered the child who was to be one of your grandparents. You would thus, in committing this murder, prevent the birth of one of your grandparents, and thus one of your parents, and thus of you yourself. But if you are never born, then you do not travel back in time, and a fortiori do not murder your great-grandfather. Since traveling back in time involves both your being born and your not being born, time travel into the past is self-contradictory, and hence logically impossible."
Expressed formally the reasoning is of this sort (where "T" = "You travel backwards in time"; "K" = "You kill your great-grandfather"):

~(T & K)
The argument (expressed in the symbols of modal propositional logic) is nonvalid.1 ] To see the nonvalidity of the form of the argument, compare:

~(Robby is exactly 5 feet tall and Robby is exactly 6 feet tall)
Robby is exactly 6 feet tall
~Robby is exactly 5 feet tall.
What does follow from the premises of the (reconstructed) argument? (There are, of course, an infinite number of consequences, but we'll single out one in particular.)

Incorrect   Correct
~(T & K)
  ~(T & K)
(T ~K)
A curious fact: this latter (correct) conclusion follows directly from the first premise; the second premise is superfluous for this particular inference. And what does this conclusion say? Simply, (it is logically necessary) that if you travel back in time you do not kill your grandfather; it does not say that you cannot, only that you do not.

Oddly, the talk of time travel masks the logic involved. In precisely the way you cannot murder your great-grandfather, given that he was not murdered, you cannot alter the present, either, from the way it is. Try it. Look at your hands the very way they are now. Try to make them some other way now (not a fraction of a second from now). For example, you might touch the tip of your thumb with the tip of your index finger. But in doing whatever it is you do, you do not succeed in changing your hand from the was it is. All you can do is to change your hands (a moment from now) from the way they would have been; not from the way they are. We often talk of the past as being fixed. But in just the way that the past is fixed, so are the present and the future. True, one cannot change the past; but neither can one change the present, and neither can one change the future.

Now what about the second premise, viz. that it is logically possible to kill your grandfather (i.e. K)? Is it true? Is it logically possible to kill your great-grandfather? We have to be very careful how we answer this question.

Suppose we select one of your four great-grandfathers, and just to have a way of referring to him, we will (arbitrarily) call him "Orion". Is it logically possible that Orion should be murdered? Seems to me the answer is yes. The statement that Orion is murdered is contingent, hence is logically possible.

But what if we talk not about Orion, but "your great-grandfather", and ask "Is it possible that someone should have murdered your great-grandfather before he fathered your grandparent?" In this instance I think the answer is this: "Yes, it is possible, if you use 'great-grandfather' in a purely referring way, i.e. to refer to the person, Orion; No, if by 'great-grandfather' you intend that description to hold." It is logically impossible for someone – anyone at all (you or someone else), and quite independent of that latter person's traveling through time – to murder someone who is (to be) a great-grandfather and who has not (at the time of the killing) fathered a child.

(For more on time travel, click here.)

Theology – The argument against God's omnipotence

"God is omnipotent, i.e. God can do anything which is logically possible. Making a stone which is so heavy that it cannot be moved is logically possible. Therefore God, being omnipotent, can make a stone so heavy that it cannot be moved. But if God makes a stone so heavy that it cannot be moved, then God cannot move it. But if God cannot move that stone, then there is something God cannot do, and hence God is not omnipotent. Thus if God is omnipotent, then God is not omnipotent. But any property which implies its contradictory is self-contradictory. Thus the very notion of God's (or anyone's) being omnipotent is logically impossible (self-contradictory)."
The argument, as presented just above, is an unholy amalgam of two different arguments, one valid, the other invalid. The valid argument is this (where "G" = "God is omnipotent" and "M" = "God makes an immovable stone"):

M ~G
G ~G
Although the immediately preceding argument is valid, its second premise is false. The true premise is used in this next argument, but this next argument is invalid:

M ~G
To derive ~G from the latter pair of premises, one would have to add the further premise, M. But so long as M is false, the conclusion ~G remains underivable. God, thus, remains omnipotent provided that God does nothing, e.g. making an immovable stone, which destroys His/Her omnipotence.
Question [modified 21 July 2011]: Now, what if God does make an immovable stone? Suppose there is a possible world in which God does make an immovable stone, i.e. suppose M. As just explained, in that possible world, God has relinquished His/Her omniscience, i.e.
M ~G    (supposition)
But by the very definition of "P", viz.
"P" =df "~~P",
the supposition is logically equivalent to:
M ~G
which, in turn, is logically equivalent (by the Equivalence Rule of Transposition [or Contraposition]) to:
G ~M
This latter result, although sound, will doubtless strike many persons as paradoxical. For, in effect, it says that if God is omnipotent of logical necessity – as many theologians have claimed – then it is impossible for God to make an immovable stone. In short, if omnipotence is an essential property of God, then not even God Him/Herself can relinquish that property. (Thanks to Seth Kurtenbach [ Necessary omnipotence ] who motivated me to write this addendum.)

Theology – The argument against God's omniscience

Modal argument against the possibility of both God's omniscience and human free will (the theological version of the argument of epistemic determinism):
"God knows everything (knowable), past, present, and future. God has given human beings free will so that human beings can choose between good and evil. But if God knows beforehand what you are going to choose, then you must choose what God knows you are going to choose. If you must choose what God knows you are going to choose, then you are not truly choosing; you may deliberate, but eventually you are going to choose exactly as God knew you would. There is only one possible upshot of your deliberating. Thus if God has foreknowledge, then you do not have free will; or, equivalently, if you have free will, then God does not have foreknowledge."
(For more on this topic, see "Foreknowledge and Free Will".)

Forrester's Paradox

"Suppose Smith is going to murder Jones. It is obligatory that if he murders Jones, he should do so gently. This appears to imply that if Smith murders Jones, it is obligatory that he should do so gently. However he cannot murder Jones gently without murdering him. Hence, given that Smith is going to murder Jones, it is obligatory that he do so." (reported in Paradoxes, by R.M. Stanley, Cambridge University Press, 1988, p. 149) [Original source: "Gentle Murder and the Adverbial Samaritan", by James William Forrester, in Journal of Philosophy, vol. 81 (1984), pp. 193-197.]
The argument seems to be missing some premises. Elaborated, the argument might emerge thus:
"Suppose Smith is going to murder Jones. It is obligatory that if he murders Jones, he should do so gently (presumably on the principle that one ought not to cause another person unnecessary suffering). This appears to imply that if Smith murders Jones, it is obligatory that he should do so gently. But to be obliged to do x in a certain manner (e.g. to sing loudly, walk briskly, murder gently) is to be obliged to do x (e.g. to sing, to walk, to murder). Thus if Smith is going to murder Jones, he is obliged to do so."
As in several of the previous cases, the error (it seems to me) is moving the operator – in this case "it is obligatory that" – which applies to the entire conditional sentence, inside that sentence, again, as we have seen, onto the consequent.

Suppose we let "Oblig" stand for "it is obligatory that". Then we may say:
Incorrect: If Smith is going to murder Jones, then Oblig(Smith murders Jones gently).
Correct: Oblig(If Smith is going to murder Jones, then Smith murders Jones gently.)


  1. Translating arguments originally expressed in a natural language into the symbolism of some artificial language (as we are doing here) is a risky business. Here I show that the form (reconstructed in this latter artificial language [modal propositional logic]) is nonvalid. But this does not prove that the original argument is invalid. Having a valid form in some artificial language is a sufficient condition for the original argument's being valid. But having a nonvalid form is not a sufficient condition for demonstrating that the original argument was invalid: it may be that the reconstruction in the artificial language 'loses' important informational content, content crucial to the validity of the argument.

    Many introductory logic texts manage (inadvertently or, worse, because their authors are confused) to convey the (mistaken) thesis that if an argument has a nonvalid form then that argument is invalid. Some such arguments are invalid; but not all are: some arguments having a nonvalid form (in some particular artificial language) are, nonetheless, valid. For a more thorough discussion of these points, see Possible Worlds: An Introduction to Logic and Its Philosophy, by Raymond Bradley and Norman Swartz (Indianapolis: Hackett), pp. 301-313.  [ Return ]

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