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
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.
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.
Paul has one daughter and two sons.
Paul has to have at least one son.
The apparent form (in propositional modal logic) of this
The 'correct' form of the argument, one which does not have
"S" as a conclusion, but only "S", is:
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
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"
Prob(x)(Px Rx) = 0.98
(An alternative way of writing the first premise is:
Prob(Rj) = 0.98
(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
where "O" = "is an octogenarian". Similarly, it may also be true
Pj & Oj
Prob(x)([Px & Cx] Rx) = 0.11
where "C" = "has congestive heart failure".
Pj & Cj
Each of these three arguments has a conclusion different from the
others. Which of the three conclusions shall we adopt (/believe,
Prob(Rj) = 0.98
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
Prob(Rj) = 0.23
Prob(Rj) = 0.11
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
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
The fallacy arises in the ambiguity of the first premise. If we
interpret it close to the English, we get:
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)]
From these latter two premises, one cannot validly infer the
'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:
The correct interpretation of the first premise is:
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"):
||~(Kxp & Mxp)
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,
(If the error is unobvious to you, click here.)
"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
~(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
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.)
Robby is exactly 6 feet tall
~Robby is exactly 5 feet tall.
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.
~(T & K)
~(T & K)
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
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
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"):
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:
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:
which, in turn, is logically equivalent (by the Equivalence Rule of Transposition [or Contraposition]) to:
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
(For more on this topic, see
"Foreknowledge and Free Will".)
"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
||If Smith is going to murder Jones, then Oblig(Smith
murders Jones gently).
||Oblig(If Smith is going to murder Jones, then Smith murders
- 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
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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