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Some theorems of Fitch on omnipotence

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Frederic Fitch, in a fascinating article, most regrettably ignorod by philosophers of religion, proves the following theorem on omnipotence: If for each situation that is the case it is logically possible that that situation was brought about by some agent, then whatever is the case was personally brought about by that agent. This is a mightily perplexing result. It seems to say that an omnipotent agent, in this sense, must personally have brought about every actual state of affairs that obtains. Yet many theologians have held that God is omnipotent while not being a universal agent. The free will defense, ~ for example, seems to require that there should be some actual states of affairs not (personally, at any rate) brought about by God. Whether and how God acts is puzzling in its own right. But in any case it has often been assumed that God is omnipotent, at least minimally in Fiteh's sense, without being a universal personal agent.
SOME THEOREMS OF FITCH
ON OMNIPOTENCE
DOUGLAS WALTON*
Frederic Fitch, in a fascinating article 1, most regrettably
ignor6d by philosophers of religion, proves the following theorem
on omnipotence: If for each situation that is the case it is logically
possible that that situation was brought about by some agent, then
whatever is the case was personally brought about by that agent.
This is a mightily perplexing result. It seems to say that an omni-
potent agent, in this sense, must personally have brought about
every actual state of affairs that obtains. Yet many theologians
have held that God is omnipotent while not being a universal
agent. The free will defense, ~ for example, seems to require that
there should be some actual states of affairs not (personally, at
any rate) brought about by God. Whether and how God acts is
puzzling in its own right. 3 But in any case it has often been
assumed that God is omnipotent, at least minimally in Fiteh's
sense, without being a universal personal agent.
Let us therefore carefully examine Fitch's proof. We need
to establish two preliminary theorems before we can arrive at
Theorem 3, the omnipotence theorem. And in order to establish
these two lemmas, we must introduce some of Fitch's terminology.
A class of propositions is said to be closed with respect
to con-
junction
elimination if (necessarily) whenever the conjunction of
two propositions is in the class so are the two propositions them-
selves. 4 For example, the relation of believing (that obtains
between an agent and a possible state of affairs) is closed with
respect to conjunction elimination because if I believe 'p & q'
1 Frederic Fitch, 'A Logical Analysis of Some Value Concepts',
Journal
of
Symbolic Logic, Vol. 28, No. 2, June 1963, pp. 135-142.
Vide J. L. Mackie, 'Evil and Omnipotence', Mind, Vol. LXIV, 1955.
Reprinted in Nelson Pike (ed.), God and Evil, Englewood Cliffs, Pren-
Defense', Religious Studies, forthcoming.
3 Gordon D. Kaufman, 'On the Meaning of "Act of God",'
Harvard
Theological Review, Vol. 61, 1968, pp. 175-201.
4 Fitch, 136f.
SOME THEOREMS OF FITCH ON OMNIPOTENCE
then it follows that I believe p and that I believe q.a In symbols,
B (p & q) --. (B p & B q)
Thus for any class of operators L that obtain between an
agent and a state of affairs, L is closed with respect to conjunction
elimination where
L (p&q) --~ (Lp&L q)
for any p and q. Fitch also postulates that the relation of doing
(personally bringing it about that p) is closed with respect to con-
junction elimination. If I do both p and q then I do p and also I
do q.
S(p&q)--~(Sp&Sq)
Here we read 'S p' as 'the agent a personally brings it about that
p obtains'2 Fitch understands S as being a truth-entailing, that is,
Sp --~ p
Defining the general notion, Fitch postulates that a Class of proposi-
tions is said to be a truth class if (necessarily) every member of it
is true. Thus if L is a truth class, we have it that
Lp "--~ p
Hence S is such a truth class. Truth and logical necessity are also
obviously truth classes.
We can now approach Theorem 1 Which I shall quote below
in entirety with its proof.
Theorem 1. If L is a truth class which is closed with
respect to conjunction elimination, then the proposition
[P & ~ (L p)], which asserts that p is true but not a
member of (where p is any proposition), is itself neces-
sarily not a member of L.
Proof. Suppose, on the contrary, that [p & ~ (L p)] is
a member of L; that is, suppose (L [P & ~ (L p)])-
Since L is closed with respect to conjunction elimination,
the propositions p and ~ (L p) must accordingly both
be members of L, so that the propositions (L p) and
(L ("~ (L p))) must both be true. But from the fact
that L is a truth class and has ~ (L p) as a member, we
conclude that ~ (L p) is true, and this contradicts the
result that (L p) is true. Thus from the assumption that
Vide Jaakko Hintikka, 'Knowledge, Belief, and Logical Consequence',
Ajatus, Vol. 32, 1970, pp. 32-47.
6 In Fitch's notation, the usual subscript denoting an individual is dropped.
Instead of 'S a p' we simply have 'Sp' meaning 'a brings it about that p'.
21
SOPHIA
[P & ~ (L p)] is a member of L we have derived con-
tradictory results. Hence that assumption is necessarily
false. 7
Formally, the proof of Theorem 1 can be exhibited in five
steps as follows:
(1) L [p & ~ (L p)] Assumption
(2) L p (1), Conj. Elim.
(3) L ~ (L p) (1), Conj. Elim.
(4) ~ (L p) (3), L p ~ p
(5) L p & "~ (L p) (2), (4) Conj. Intro.
Thus by reduetio we can conclude that:
(6) ~ML [p&~-(Lp)]
where 'M' is read 'it is logically possible that'.
The second theorem reads as follows:
Theorem 2. If L is a truth class which is closed with
respect to conjunction elimination, and if p is any true
proposition which is not a member of L, then the pro-
position [p & ~ (L p)], is a true proposition which is
necessarily not a member of L.
Proof. The proposition [p & ~ (L p)] is clearly true,
and by Theorem 1 it is necessarily not a member of L. ~
We can perhaps see a little clearer how this proof works if
we write a 'T' in the right upper corner of a proposition (state of
affairs) that is true (obtains). 9 The theorem then reads:
T2: ~LpT---~ "~ M L [p & ~ (Lp)] T
That the consequent obtains (disregarding the 'T') simply follows
from the fact that it is an instance of (6), that is, Theorem 1.
That the part in brackets of the consequent is true is apparent in
that both conjuncts of it follow trivially from the antecedent. By
the antecedent, we have it that ~ L p, and since p, according to
the antecedent, is true, we have it that p.
Now let us proceed to Theorem 3.
Theorem 3. If an agent is all-powerful in the sense that
for each situation that is the case, it is logically possible
that that situation was brought about by that agent,
r Fitch, p. 138.
8 Fitch, p. 138.
a This is not used by Fitch.
22
SOME THEOREMS OF FITCtt ON OMNIPOTENCE
then whatever is the case was brought about (done) by
that agent.
Proof. Suppose that p is the case but was not brought
about by the agent in question. Then, since doing is a
truth class closed with respect to conjunction elimination,
we conclude from Theorem 2 that there is some actual
situation which could not have been brought about by
that agent, and hence that the agent is not all-powerful
in the sense described, t~
Theorem 3 is proven simply by substituting the operator S
(bringing it about) for the general operator L. If we assume:
(1) ,~ Sp T
namely that p obtains but was not brought about by the agent in
question, then it follows simply by substitution in T 2 that we have
(2) ~MS [P&~ (Sp)]T
This asserts that there is an actual state of affairs, represented by
the proposition in the square brackets that obtains (note the T
in the right comer) but that it is logically impossible for the agent
to bring about. Fitch concludes from (2) that
(3) --~M S p T
For the agent in question, it is not logically possible that he should
fail to bring about any actual state of affairs. Necessarily, every-
thing that obtains has been brought about by him. Thus any
omnipotent agent (in this sense) is also thereby necessarily a
universal agent of actual states of affairs. Formally, this result
(4) ~ SpT ~ ,.~ M SpT (1), (3), Conditionalization
(5) M SpT ~ SpT (4), Contraposition
(5) states that for a given agent, if it is possible that he brings
about any actual state of affairs then it follows that he personally
brings about that actual state of affairs. Any omnipotent being
is also a universal agent.
Fitch's theorem may not seem too significant because it is
generally felt that God's omnipotence consists not merely in the
logical possibility of his doing anything but in the physical possi-
bility of his doing anything. It is not enough that it should be
logically possible for God to do anything, but it should be required
lo Fitch, p. 138.
23
SOPHIA
that there can be no physical obstruction to his action either. 11
For any given state of affairs it could never be the case that a set
of antecedent conditions of God's bringing about that state of
affairs, taken together with the set of nomic universals, entails the
proposition that God does not bring about the state of affairs. There
are no physical obstacles to God's action. This sense of 'physical
possibility' has never been very clearly explained, '2 but what is
clear is that something like it, rather than merely logical possi-
bility, is the true sense in which it is possible for God to do
anything. Whatever 'physical possibility' is, minimally, we can
say that it includes logical possibility. That is for this sense of
possibility, M*,
(1) M* p--~ Mp
And by substitution in (i) we have
(2) M* S p T ~ M S P T Sub. [S P T/p]
Then from Fitch's theorem
(3) M
Sp T ~ Sp T
and (2), the transitivity of --> yields
(4) M* S PT ~ S P T (2), (3), H.S.
Thus if it is physically possible that an agent brings about every
actual state of affairs, that agent has brought about every actual
state of affairs. The import of this theorem for philosophical
theology is even more evident than Fitch's. If God is omnipotent,
in perhaps the most usual or standard sense, then Aquinas' tradi-
tional solution to the problem of evil, that God permits rather
than brings about evil, cannot be defended. Indeed, if (4) obtains,
it is hard to see how any free will defense could possibly succeed.
I cannot pretend that I know definitively what all the implica-
tions for philosophical theology are of Fitch's theorem and my
corollary of it, although it seems to me that these results must be
importantly relevant to the problem of evil and related problems
of the notion of omnipotence in a number of ways. I will content
myself with adumbrating two primary implications here.
First, Fitch's Theorem 1 provides interesting new grounds for
rejecting a claim of Aquinas that God can bring about any state
la For an excellent recent discussion and review of various traditional
conceptions of omnipotence, see Peter Geach, 'Omnipotence', Philosophy,
January 1973, Vol. 48, pp. 7-20, and Peter Geach, 'An Irrelevance of
Omnipotence', Philosophy, October 1973, Vol. 48, pp. 327-333. Refer-
ences in the sequel are to the former article.
12 For a helpful discussion, see Myles Brand, 'On Having the Opportunity',
Theory and Decision, Vol. 2, 1972, pp. 307-313.
24
SOME THEOREMS OF FITCH ON OMNIPOTENCE
of affairs that is logically possible. 13 Fitch has shown that
~-- M S [p & -- (Sp)]
even though the state of affairs represented within the brackets is
logically possible. It is logically possible that the door is open
where God has not brought it about that the door is open (unless
it is assumed at the outset that God does everything). Yet it is
logically impossible that God personally brings it about both that
the door is open and that God does not bring it about that the
door is open. Of course this Thomist doctrine has often been
rejected, 1. perhaps even by Aquinas himself elsewhere, 1"~ but so
far as I know it has never been rejected on the grounds of any-
thing like Theorem 1.
Second, Theorem 1 and its corollary expose the absurdity of
a common but spurious notion of omnipotence. In this sense, to
say that God is omnipotent is to say that for any actual state of
affairs, God could have been the agent of that state of affairs. To
put it another way, let us say that every actual state of affairs is
such that either it was brought about by some agent or it just
happened. In the latter case, we might say that it was brought
about by the null agent. Now to say that God is omnipotent, in
this sense, is to say that God is an alternate for any agent. Given
any agent of an actual event, even the null agent, God can be put
in for this agent. He is the universal agent-substitute. For any-
thing that happened, God can be conceived as being the agent of it.
This notion is demonstrably absurd because if God is omni-
potent in this sense, there are no Other agents, nor can anything
just happen. It follows from the conception of omnipotence that
God is the author of literally every state of affairs. Thus the notion
that GQd's power can consist in such agent-substitutability is
absurd.
This result, however, by no means implies that all conceptions
of omnipotence are inchoate or contradictory. Indeed, I would
like to suggest that it indicates that we should concentrate our
attention on explicating another quite distinct model of omni-
potence while rejecting the above spurious conception. I suggest
that we need to think of God's omnipotence not as directed to
actual (past) states of affairs, but as directed to unactualized
(future) states of affairs. Given any future state of affairs p, God
can bring it about that p or bring it about that ~ p. There are no
obstacles to his actualization of any possible (that is, unactualized)
1:~ Summa Theologica, Ia q. xxv art. 3. Vide Geach, p. 12f. What I take
to be the same view has been held by Alvin Plantinga (see note 16).
Vol. LXXVI, No. 1, January 1967, 74-79.
1.~ See the discussion in Geach, p. 12f.
25
SOPHIA
state of affairs. It should be a desideratum of this concept of
omnipotence (which we might call the nihil obstat conception) that
it not be true of every state of affairs that God personally brings
it about (or that God will bring it about), except perhaps in some
suitably vicarious sense. Importantly, God's power is viewed as
being directed to an unactualized future (though he himself is time-
less) rather than to the actualized past. Equally importantly, God's
power is not the power to substitute for (even future) agents, but
the power to bring it about either that p or that ~ p. Even where
p has actually obtained at t, God's power is such that only at a
previous time, t-A, could he have brought it about that ~ p.
It is important to emphasise that the power of an omnipotent
agent must not be thought of as extending over actual states of
affairs that have occurred, since in this instance otherwise viable
accounts of omnipotence are driven to absurdity. Consider Plan-
tinga's view that God is omnipotent just in case God can create
any state of affairs p such that God brings it about that p is
consistent? 6 On this view, an agent is omnipotent where for
that agent
(P) M S p
But we remember that Fitch's theorem reads
M Spr---~Sp T
Obviously if we allow
pT as
a legitimate substitution instance of
p in (P), we get the undesirable result
Sp T
namely that the agent in question has brought about everything that
has happened. The same result is forthcoming.on the nikU olxstal
account of omnipotence which asserts that an agent is omnipotent
where
M* S p
By the corollary to Fitch's theorem we have it that
M* SpT--~Sp T
Thus substituting
pT
for p yields the same result. The same result
is a consequence of a third conception of omnipotence enter-
tained by Duns Scotus, who was concerned to determine whether
we can prove that God can produce directly whatever can be
caused, lr We might reconstruct this conception by ruling that an
agent is omnipotent in this sense where
M* p --~ M* Sp
An agent is omnipotent where if a state of affairs is physically
possible then it is physically possible that it be brought about by
16 Alvin Plantinga, 'The Free Will Defense', Philosophy in
America, ed.
Max Black, London, Allen and Unwin, 1965, p. 209.
26
SOME THEOREMS OF FITCH ON OMNIPOTENCE
the agent. Here again, if we allow pT as an instance of p, we
have it that
M* pT ~ M* Sp T
But assuming our corollary to Fitch's theorem
M* Sp T --~ Sp T
it follows by hypothetical syllogism that
M* pT __> Spa"
Since we also have it that any actual state of affairs must be
physically possible
pT ~ M* pT
it also follows that every actual state of affairs is an action of the
agent in question
pZ ._~ SpT
Or to put it more simply,
Sp T
Thus even these otherwise quite plausible accounts of omnipotence
are vitiated if we allow the substitution in question, that is, if we
allow the agent's power to extend over actual states of affairs.
Our conclusion from Fitch's theorem that omnipotence is
best thought of as directed only to unactualized states of affairs
is thus doubly reinforced. Even otherwise promising explications
of omnipotence, such as the nihil obstat conception, admit of
Fitch-like consequences if actualized states of affairs are included
in the domain of the variables. An omnipotent being is better
thought of as having unlimited power over which unactualized
states of affairs he will bring about, is At least, I would add, that
is the conclusion that Fitch's theorems suggest to me. It is not clear
to me that this conclusion is the only one that can be unequivocally
and finally drawn from Fitch's results. What I would principally
like to leave in the reader's mind is the importance of Fitch's
theorems in any serious attempt to understand the concept of
omnipotence and the problem of evil. Here I leave off and com-
mend them to the attention of philosophical theologians and
atheologians.
17 Felix Alluntis and Allan B. Wolter, 'Duns Scotus on the Omnipotence of
God', Studies in Philosophy and the History of Philosophy, Vol. 5, 1970,
pp. 178-222.
is For an illuminating discussion of dwme control of past events, see Geach,
p. 16f. I take it that our conclusions here strongly support Geach's
view that what is past ceases to be alterable even by God (Geach, p. 17).
27
Article
Full-text available
1. The contents of this manuscript were first described and printed in F. S. Schmitt, Ein neues unvollendetes Werk des Ll. Anselm von Canterbury, Beiträge zur Geschichte der Philosophie und Theologie des Mittelalters, 3 (1936). The manuscript is reprinted in F. S. Schmitt and R. W. Southern, Memorials of St. Anselm (London: Oxford University Press, 1969), pp. 333-354. A very helpful commentary and partial translation is to be found in Desmond Paul Henry, The Logic of St. Anselm (Oxford: Oxford University Press, 1967), § 4. A more detailed analysis, also very helpful, is D. P. Henry, "Saint Anselm on the Varities of 'Doing,' " Theoria, 19 (1953), 178-183. 2. A good general source of material here is Myles Brand (ed.), The Nature of Human Action (Glenview, Illinois: Scott Foresman, 1970). 3. See G. H. von Wright, An Essay in Deontic Logic and the General Theory of Action (Amsterdam, North-Holland, 1968). 4. See John Kimball (ed.), Syntax and Semantics, Vol. 1 (New York and London: Seminar Press, 1972). 5. Memorials, pp. 342 f. 6. I will not try to define 'state of affairs,' but refer the reader to G. H. von Wright, "The Logic of Action — A Sketch," The Logic of Decision and Action (Pittsburgh; University of Pittsburgh Press, 1966), pp. 121 f. 7. Donald Davidson, in "The Logical Form of Action-Sentences," in The Logic of Decision and Action (op. cit.), has suggested various significant difficulties for the Anselmian mode of analysis here reconstructed. Davidson cites various action-sentences such as 'I coughed' and 'He walked to the corner' where there seems to be no automatic way to produce the right description of the purported state of affairs that is said to be brought about. 8. Square corners mark off expressions constructed in the manner indicated, following the usage of W. V. Quine, Mathematical Logic, Rev. ed. (Cambridge, Mass.: Harvard University Press, 1951). 9. Memorials, p. 343. 10. The Logic of St. Anselm, p. 124. 11. Some recent theories of agency that more or less take the Anselmian approach are the following: Anthony Kenny, Action, Emotion and Will (London: Routledge and Kegan Paul, 1963), ch. 7; Roderick Chisholm, "Some Puzzles about Agency," The Logical Way of Doing Things, ed. Karel Lambert (Yale University Press, 1969), pp. 199-217; and Roderick Chisholm, "The Descriptive Element in the Concept of Action," The Journal of Philosophy, Vol. LXI, No. 20, 613-625. 12. Memorials, p. 337. 13. The expression ┎~δ~p┒ may require some elucidation. A paraphrase would read, 'a fails to bring it about that not-p obtains,' or 'a allows p to happen.' The resultant sense of agency is weaker than ┎δa p┒, i.e., if we have it that ┎δa p┒ then we have it that ┎~δa~p┒, but not conversely. Hart and Honoré discuss the case of Hardcastle v. Bielby, I Q.B. 709, 1892, where a distinction is made "between 'causing' a heap of stones to be laid upon the highway and 'allowing' it to remain there at night, to the danger of persons passing thereon." H. L. A. Hart and A. M. Honoré, Causation in the Law (Oxford: Oxford University Press, 1969), p. 330. The first case requires proof that the stones were laid by the accused, whereas allowing the stones to remain, it was ruled, required no positive act. 14. James D. McCawley, "English as a VSO Language," Language, Vol. 46, No. 2 (June 1970), 286-299. See also Dieter Kastovsky, "Causatives," Foundations of Language, Vol. 10, No. 2 (July 1973), 255-325. 15. Frederic B. Fitch, "A Logical Analysis of Some Value Concepts," Journal of Symbolic Logic, Vol. 28, No. 2 (June 1963), 135-142. 16. Fitch, ibid., p. 138. 17. See G. E. Hughes and M. J. Cresswell, An Introduction to Modal Logic (London: Methuen, 1968). Hughes and Cresswell have shown that T and T' are deductively equivalent. T consists of the following axioms and rules: A1-A4 for PM, plus: A5: Lp ⊃ p A6: L (p ⊃ q) ⊃ (Lp ⊃ Lq) TR1 and TR2: Uniform Substitution and Modus Ponens TR3: |— Ø, to infer |— L Ø T' consists of A1-A4 for PM, TR1 and TR2, plus: A5: Lp ⊃ p A9: (Lp & Lq) ≡ L (p & q...
... 6 See also Walton (1976), who uses Fitch's theorems 1 and 3 to show, paradoxically, that there is something logically possible that an omnipotent being could not bring about. ...
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