Explanation, Entailment, and Leibnizian Arguments
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Explanation, Entailment and Leibnizian Cosmological Arguments
There are new arguments for the existence of God. The newer formulations of the
cosmological argument often contain restatements of the principle of sufficient reason (PSR).
One such new formulation is defended by Richard M. Gale and Alexander R. Pruss. 1 The GalePruss argument incorporates an affirmation of both a weak and strong version of the principle of
sufficient reason.2
(1) W-PSR: For any contingently true proposition, p, and any world, w, if p is in
w’s Big Conjunctive Contingent Fact (BCCF), then there is some possible
world, w1, and proposition, q, such that w1’s BCCF contains p and q and the
proposition that q explains p.
(2) S-PSR: For every contingently true proposition, p, there is a proposition, q,
that explains p.
Both Graham Oppy and Alexander Pruss have independently shown that (1) entails (2)
by way of the so-called Oppy-Fitch proof and the Pruss deduction.3 If we accept (1), then it
follows that all contingently true propositions have an explanation for their truth.4 If (2) is true,
then the big conjunctive contingent fact ([BCCF], i.e., the aggregate of all contingently true
propositions joined together by iterative conjunctions) is thought to require an explanation
beyond any contingent fact within the BCCF. But why think that the idea of a BCCF is a
coherent one? Furthermore, isn’t there a good reductio argument against accepting (1), which
shows that explanations entail their explanandums and that therefore a necessary proposition
cannot explain a contingent one? In what follows I will seek to address these difficult questions
* I am greatly indebted to Alexander Pruss’ work. His thought is my spring board into this subject.
1
Richard M. Gale and Alexander R. Pruss, “A New Cosmological Argument,” in Religious Studies 35 (1999):
461–476. I am not attempting any kind of amelioration of the Gale-Pruss paper.
2
For a stringent defense of S-PSR, see Alexander R. Pruss, The Principle of Sufficient Reason: A Reassessment
(Cambridge: Cambridge University Press, 2006); Quentin Smith, “A Defense of a Principle of Sufficient Reason,”
Metaphilosophy, 26:1 & 2, (January/April 1995): 97-106.
3
For the Oppy-Fitch proof see Graham Oppy, “On ‘A New Cosmological Argument’,” in Religious Studies 36
(2000) 347-348; and the Pruss deduction within Richard M. Gale and Alexander R. Pruss, “A Response to Oppy,
and to Davey and Clifton,” in Religious Studies 38 (2002): 90; cf. note 42 below.
4
For the metaphysics of “books on worlds” see Plantinga, The Nature of Necessity, 45-46.
Explanation, Entailment, and Leibnizian Arguments
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head on. I will also fend off two more objections to (1), the first of which attempts to ameliorate
the summarized reductio just mentioned by appealing to relevant entailment, and a second that
tries to challenge the proponent of (1)’s ideas about the general nature of an explanation.
The BCCF?5
So why think that there is anything such as a BCCF? Surely a conjunction of this
mammoth proportion is altogether incoherent!6 Let’s assume with Kevin Davey and Rob Clifton,
that by the phrase “E(p)” we mean p has an explanation. Furthermore, assume that by T(p) we
mean p is true. Assume further that r is the conjunction of all contingently true propositions. Let
the relation (p < q) hold between propositions p and q if and only if q = E(p) or q is a conjunction
of which p is conjunct. Suppose also that ¸ denotes a transitive closure of the relation <. Kevin
Davey and Rob Clifton maintain that one should say that p is a proper sub-proposition of q if p ¸
q. If p ¸ q or p = q, we write p  q, and say that p is a sub-proposition of q. This idea of subpropositions is akin to subformulas in mathematics. Take for example the following:
(3) 3 = 3 and (3 = 3 and (3 = 3 and (3 = 3 and….))))….
The proposition 3 = 3 on the left side of the main connective has its corresponding proper
subformula on the right side of the main connective. The right conjunct just is synonymous with
the entire connective. Moreover, the same side of the conjunction is also a sub-conjunction of
(3). Let r be the conjunction of all contingently true propositions, and suppose that the entire
proposition (3) is a proper subformula of itself. Now suppose that r* is the subconjunction of all
contingently true propositions (those propositions incorporated in r) z, such that z is not a proper
subformula of itself. It follows that r* is a contingently true proposition, and therefore so is
Many thanks to Alexander Pruss’ work in this area. I follow his thought closely.
Kevin Davey and Rob Clifton, “Insufficient Reason in the ‘New Cosmological Argument,” in Religious
Studies 37(2001): 486-487; cf. Jordan Howard Sobel, Logic and Theism: Arguments for and Against Belief in God
(Cambridge: Cambridge University Press, 2004), 389-393.
5
6
Explanation, Entailment, and Leibnizian Arguments
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T(r*). Now as Davey and Clifton, seem to point out if T(r*) is a proper subformula of itself, then
T(r*) is a subformula of r*, since it is the case that only proper subformulas of T(r*) are the
subformulas of r*. Since T(r*) is not a conjunction, it must be a subformula of a conjunct in z, of
r*. It follows then that z is a subformula of r* and hence a proper subformula of T(r*).
Proposition z is therefore a subformula of T(r*), which is also a subformula of z, and oddly it
follows then that z is a subformula of z, reductio ad absurdum.7 I have symbolized this reasoning
as follows:
(4) Proposition (17)  (17)
(5) [(r* < r) & (z  z)]
(6) (r* & ◊~r*)
(7) ï [T(r*) and ◊~T(r*)]
(8) All subformulas of T(r*) are subformulas of r*.
(9) ï {[T(r*)  T(r*)] → [T(r*)  r]}
(10) It is not the case that T(r*) is a conjunction.
(11) ï T(r*) is a subformula of a conjunct in z.
(12) ï {[z  r*] & [z  T(r*)]}
(13) ï (z  z)
This argument is invalid if it is an attempt to trap the idea of a BCCF in a reductio. We
need the extra premise: “for every conjunctive proposition any subconjunction of that
proposition is also a proposition.”8 With this premise (4)-(13) looks more like a reductio for the
BCCF. However, this follows only if the BCCF can be properly thought of as a set, which it
cannot.9 There can be no set of all cardinalities, and this is affirmed by the consensus of
mathematicians and philosophers of mathematics today. Why is this relevant? Suppose that c is
an infinite cardinality, and let p(c) be the proposition that exactly c amount of photons exist, but
then it will also be true that ~p(c). If the collection of all positive contingently true propositions
about photons is a set, then so is ~p(c). However, there simply cannot be a set of all the ~p(c),
7
See also Alexander Pruss, The Principle of Sufficient Reason 99-100.
As Pruss has pointed out.
9
Alexander Pruss, “The Cardinality Objection to David Lewis’ Modal Realism,” in Philosophical Studies
104(2001): 167-176.
8
Explanation, Entailment, and Leibnizian Arguments
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since it is mathematically impossible to have a set of all cardinalities. Therefore, given the
preceding argumentation, Davey and Clifton are simply mistaken. One cannot talk meaningfully
about arbitrary sub-collections or subformulas of the BCCF. I think that a candidate for a
proposition is innocent until proven guilty and the argumentation above does not show the BCCF
to be guilty.
The Rowe/van Inwagen Reductio
Philosophers who do not quibble over the BCCF have deep seated problems with the PSR
itself. Peter van Inwagen and William Rowe believe that the PSR entails the collapse of all
modal distinctions. Rowe and van Inwagen assumes that if x is a sufficient reason (an
explanation) of y, then x must entail y.10 He asks if x did not entail y, and it were possible that x
obtain and y not obtain, then how could x be a sufficient reason for y? The argument somewhere
in the neighborhood of van Inwagen’s worry can be represented as follows: [qEp = q explains p]
(14) If the PSR holds, then every true contingent proposition has an explanation.
(15) No necessary proposition explains a contingent proposition.
(16) No contingent proposition explains itself.
(17) If a proposition explains a conjunction, it explains every conjunct.
(18) A proposition q only explains a proposition p if q is true.
(19) There is a BCCF that is the conjunction of all true contingent propositions, perhaps with
logical redundancies removed, and the BCCF is a contingent conjunction.
(20) The PSR holds.
[Assumption for Indirect Proof]
(21) Then, the BCCF has an explanation, viz. q.
(22) The proposition q is not necessary.
(23) Therefore, q is a contingently true proposition.
(24) Thus, q is a conjunct of the BCCF.
(25) Therefore, q is self-explanatory.
(26) But q is not self-explanatory.
(27) Therefore, it is not the case that the PSR holds. [Indirect proof and Reductio]11
10
See Peter van Inwagen, An Essay on Free Will (Oxford: Clarendon Press, 1983), 202-204.
cf. William Rowe, “Rationalistic Theology and Some Principles of explanation,” in Faith and Philosophy
1(1984): 357-369; James Ross, Philosophical Theology (Indianapolis, IN: Bobbs-Merrill, 1969), 295-304; Patrick
Francken and Heimir Geirsson, “Regresses, Sufficient Reasons, and Cosmological Arguments,” in Journal of
Philosophical Research 24(1999): 285-304; Peter van Inwagen, An Essay on Free Will (Oxford: Clarendon Press,
11
Explanation, Entailment, and Leibnizian Arguments
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Christopher Hill has sought to escape (27) by giving up premise (14), and restricting his PSR to
particular kinds of contingently true propositions (not all in the BCCF).12 However, this is
problematic. Partition the BCCF into two sub-conjunctions. Call one such sub-conjunction p, and
call the other q. Suppose further that p is the set of contingently true propositions which together
entail the existence of bears. Suppose that q is unbearly, in that it contains no facts which entail
the existence of bears. Now, given Hill’s restriction both p and q have an explanation e1-2 such
that e1 explains p, and e2 explains q, but then our Rowe/van Inwagen reductio can be applied to
e1 and p, and therefore the e1Ep relation leads to absurdity. The same can be said about e2 and q.
In order to escape this extension of the reductio, Hill must believe that e1 is completely
“unbearly”, and e2 is “bearly”. Explanation e2 could be thought of as “bearly” i.e., involving the
existence of bears if it contained a contingently true proposition about the marks of bear claws on
trees, or something of this nature. However, with respect to e1’s unbearlyness, we have a
problem. Given that the main argument for (15) is the contention that an explanans always
entails its corresponding explanandum, postulating that e1 is unbearly and that p itself is bearly
implies that the relevant explanation does not entail its corresponding explanandum (viz. e1 and
its relationship to p). Thus, e1 would not entail p, but p would entail the existence of bears.
Suppose though that e1 did explain p, in such a way that it did not entail p, when we reflect upon
the content of q, we realize that q contains e1. It follows then that e2 explains e1, but then e1
would also explain e2 since e2 would be in p just as e1 was in q. Thus, we have a circular
explanation on Hill’s attempt to dodge the Rowe/van Inwagen argument.
1983), 202-204; Alexander Pruss, The Principle of Sufficient Reason: A Reassessment (Cambridge: Cambridge
University Press, 2006), 97.
12
Christopher Hill, “On a Revised Version of the Principle of Sufficient Reason,” in Pacific Philosophical
Quarterly 63(1982): 236-242.
Explanation, Entailment, and Leibnizian Arguments
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What we learn from Hill’s response is that essential to the argument for (16) is the claim
that an explanans must entail its corresponding explanandum. It is said to follow from this that if
the explanation of the BCCF were a necessary one, then the BCCF would no longer be
contingent, but in actuality a necessarily true conjunction. This reasoning can be represented as
follows13: [qEp = q explains p]
(28) [qEp → □(q → p)]
(29) qEp
(30) □(q → p)
(31) □ q
(32) ï □ p
Many find this argumentation very convincing. For example, theist Timothy O’Connor stated,
“[t]his argument from the Principle of Sufficient Reason (PSR) to the necessity of all truths is, I
believe, unanswerable.”14 Well, I think I’ve found a way out. The assumption of (28)-(32) is that
the type of explanans the PSR demands is one which is logically sufficient for its explanandum.
One can question the truth of (28) by a positive appeal to agent causation.15 I am quite aware of
this positions unpopularity, but all I need affirm here is its possible truth.
When utilizing agent causation, we should suppose with that view an intensional account
of explanation. This is akin to the very account proffered by none other than O’Connor himself.16
We ought to understand particular kinds of explanations in terms of desires, intentions, and
beliefs. So let us entertain the idea that human action be thought of as a type of causal action, and
this should be thought about in terms of a conceptual truth about acting for reasons. This causal
theory of action holds that some bodily behavior of mine is an action only if it is a causal
cf. William Rowe, “Rationalistic Theology and Some Principles of Explanation,” in Faith and Philosophy
(1984): 357-369.
14
Timothy O’Connor, Theism and Ultimate Explanation: The Necessary Shape of Contingency (Malden, MA:
Blackwell Publishers, 2008), 80.
15
This “possible” way out vexes me greatly when I think about O’Connor’s work. O’Connor actually affirms
the possible truth of agent causation and should therefore lend his support to the rejection of premise (26) I’m about
to explicate.
16
Timothy O’Connor, Theism and Ultimate Explanation: The Necessary Shape of Contingency 81
13
Explanation, Entailment, and Leibnizian Arguments
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consequence, in an appropriate manner, of factors prominently including my having a reason to
do to perform that action.17 Movements which are not grounded in reasons are mere
unintentional movements. An agents having control over his activity is taken to reside in the
causal efficacy of his/her reasons. We need not think that the reasons determine the action; rather
we could think that the act performed is formed by free intentions that result in the action while
having been free to form competing intentions that are motivated by other reasons. Again, all I
need show is that this account is coherent so as to blunt the attacks of those who would think of a
divine causal explanation as suspect. The agent causal account of action permits contingent
reasons/explanations of actions and their corresponding effects. I must assume that causal
capacities of agents are ontologically primitive and basic. A personal necessary being’s activity
in generating a contingent order is to be thought of, in the first instance, as the direct causing of
an internal state of intention that a particular determinate state of affairs obtains.18 The intention
is irreducibly the product of the agent qua agent. Let it be supposed that our necessary being has
some purpose whose content is p and recognizes that creating a contingent order c would satisfy
p. Suppose further that he generates an intention whose content is that c obtain in order to attain
p, and that c’s obtaining is itself an immediate product of that intention. In this circumstance, the
core activity and its product are perfectly well explained by reference to the agent’s purpose and
his belief that c would satisfy it, without the idea that the activity is necessitated by the reasons.19
There is another means by which we can reject premise (28) without appealing to the
nature of agency. The formulations of explanation that demand entailment relations are
17
Timothy O’Connor, Persons and Causes: The Metaphysics of Free Will (NY: Oxford University Press,
2000), 43-66; idem “Libertarian Views: Dualist and Agent-Causal Theories,” in Robert Kane (ed.), The Oxford
Handbook of Free Will (NY: Oxford University Press, 2002), 337-354; Robert Kane, The Significance of Free Will
(Oxford: Oxford University Press, 1996).
18
Timothy O’Connor, Theism and Ultimate Explanation: The Necessary Shape of Contingency 82.
19
Timothy O’Conner, “Freedom with a Human Face,” in Midwest Studies in Philosophy 29: (2005): 207-227.
Explanation, Entailment, and Leibnizian Arguments
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themselves multiply flawed. Some philosophers of science have thought that the nature of
scientific explanation had been solved by the deductive-nomological account of explanation
promulgated by R.B. Braithwaite, Carl Hempel, Ernest Nagel, and Karl Popper.20 Wesley
Salmon explicates this view as follows:
According to that account, a D-N explanation of a particular event is a valid
deductive argument whose conclusion states that the event-to-be-explained did
occur. Its premises must include essentially at least one general law. The
explanation is said to subsume the fact to be explained under these laws. Hence, it
is often called the covering law model.21
The premises in such an explanation contain at least one natural law and perhaps also some
initial conditions.
Premise 1: Natural Nomicity
Premise 2: Initial Condition
Conclusion: Explanandum
Explanans
Hempel noted how not all scientific explanations are of the form explicated above. In his
“Deductive vs. Statistical explanation” paper he offered a treatment of statistical explanation.
Inductive statistical explanation (I-S) explains occurrences by subsuming them under statistical
laws, much as D-N explanations subsume particular events under laws.22
Premise 1: Generalization “Almost all p’s do q’s when q’s are under conditions c1-c2.
Premise 2: q’s are under c1-c2.
“S”
20
R. B. Braithwaite, Scientific Explanation (Cambridge: Cambridge University Press, 1953); Carl G. Hempel
and Paul Oppenheim, “Studies in the Logic of Explanation,” in Aspects of Scientific Explanation and Other Essays
in the Philosophy of Science (NY: Free Press, 1965), 245-296; Ernest Nagel, The Structure of Science; Problems in
the Logic of Scientific Explanation (New York: Harcourt, Brace, and World, 1961); Karl Popper, The Logic of
Scientific Discovery (New York: Basic Books: 1959); cf. Richard H. Jones, Reductionism: Analysis and the Fullness
of Reality (London: Associated University Press, 2000), 145ff.
21
Wesley C. Salmon, “Scientific Explanation: How We Got from There to Here,” in Wesley C. Salmon,
Causality and Explanation (NY: Oxford University Press, 1998), 303.
22
Carl Hempel, “Deductive-Nomological vs. Statistical Explanation,” in Herbert Feigl and Grover Maxwell
(eds.), Scientific Explanation, Space, and Time vol.3 Minnesota Studies in the Philosophy of Science (Minneapolis:
University of Minnesota Press, 1962), 98-169.
Explanation, Entailment, and Leibnizian Arguments
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Conclusion: Therefore, p’s do q’s.
The first premise states a statistical regularity (a statistical law of sorts), while premise 2
states an initial condition. On I-S the premises do not entail the conclusion, since the logical
form of the argumentation is inductive. So the conclusion follows from the premises with a
certain inductive strength/cogency or lack thereof. Call such inductive strength “S”. It is not clear
how this strength ought to be expressed exactly. Some philosophers such as Keith Lehrer and J.
Cornman have argued that explanatory inferences cannot do their justifying intrinsically and that,
they must therefore be derived from a more basic ampliative inference rooted in probability
theory.23 Even if Lehrer and Cornman are right, we could still reject (28) for we could then have
legitimate explanations that do not entail their corresponding explanandums.
There are however probabilistic accounts of entailment, though arguably such
probabilistic accounts of entailment do not force constraints on proofs in an effective enough
way. Such accounts might affirm an overextension of Adam’s thesis:
(33) □ (q  p) iff Pr(p/q) > .5
Modified Adam’s Thesis
Such a transmutation is flawed. There are cases in which the consequent of a strict implication
relation between q and p (where p is the consequent), has a high probability independently of the
probability of the antecedent.24 It seems then that the categorical probabilities of p and q need to
be weighed and assessed logically prior to assessing (33).
Why should we think that D-N/I-S models are altogether inaccurate and false? There are
well-known counter-examples which pass the stated criteria of a D-N or I-S explanations, but
23
Keith Lehrer, Knowledge (Oxford: Oxford University Press, 1974); J. Cornman, Skepticism, Justification and
Explanation (Dordrecht: Reidel, 1980).
24
Edwin D. Mares, “Relevance Logic,” in Dale Jacquette (eds.), A Companion to Philosophical Logic (Malden,
Blackwell Publishers, 2002), 622. I should also add that this thesis denies the closure assumptions of the probability
calculus; “p” in this case is not strictly a probability function.
Explanation, Entailment, and Leibnizian Arguments
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clearly do not actually function as explanations proper. Thus, the requirements for an explanation
as demanded by D-N and I-S are not sufficient for informing us about what is and is not an
actual explanation. Numerous counter-examples to both D-N and I-S models of explanation
support this admission. These examples posit an intuitively obvious scientific explanation as an
explanans of an explanandum, and upon reflection we see that this obvious scientific explanation
does not meet Hempel’s criteria.25 Looking to I-S explanation does not help the reductio since
such explanations do not entail what they explain.26 So there is no defense of the explanationentailment idea in the debates about scientific explanation.
Entailment and Relevance: Amelioration
Quentin Smith sympathesizes with the thesis that explanations entail their explanandums,
and he tries to better the thesis by understanding the entailment relation in a manner consistent
with relevance logic:27 According to Smith:
(34) qEp iff q explains why p is true and □ (q  p) where the entailment relation is
understood as entailment in the sense of relevance logic.28
Now, ashamedly, I have intuited at times that the standard Moorean single premise deduction
theorem is false. This seeming is essential to relevance logics, and by virtue of this rejection,
Sylvain Bromberger, “Why-Question,” in Robert G. Colodny (ed.), Mind and Cosmos (Pittsburgh, PN:
University of Pittsburgh Pres, 1966), 86-111.
26
J. Alberto Coffa, Foundations of Inductive Explanation PhD Dissertation (Pittsburg, PN: University of
Pittsburgh Press, 1973); Also see N.R. Hanson, “On the Symmetry Between Explanation and Prediction,” in
Philosophical Review 68 (1959): 349-358; Michael Scriven, “Explanations, Predictions, and Laws,” in Herbert Feigl
and Grover Maxwell (eds.), Scientific Explanation, Space, and Time vol.3 Minnesota Studies in the Philosophy of
Science (Minneapolis: University of Minnesota Press, 1962), 170-230.
27
Quentin Smith, “A Defense of a Principle of Sufficient Reason,” Metaphilosophy, 26:1 & 2, (January/April
1995): 98; cf. Garrett J. DeWeese and Joshua Rasmussen, “Hume and the Kalam Cosmological Argument,” in
James F. Sennett and Douglas Groothuis (eds.), In Defense of Natural Theology: A Post-Humean Assessment
(Downers Grove, IL: InterVarsity Press, 2005), 140-141.
28
Edwin D. Mares, “Relevance Logic,” in Dale Jacquette (ed.), A Companion to Philosophical Logic (Malden,
MA: Blackwell Publishers, 2002), 609. Remember that the semantics for relevant logics are world based as with
modal logics, although on the truth of the Routley-Meyer semantics to be discussed below, the accessibility relation
deals singularly with a unary operator, necessity. Cf. Edwin D. Mares & Robert K. Meyer, “Relevant Logics,” in
Lou Goble (ed.), The Blackwell Guide to Philosophical Logic (Malden, MA: Blackwell Publishers, 2001), 287.
25
Explanation, Entailment, and Leibnizian Arguments
11
there does appear to be non-classical tendencies in relevance logical systems. For example,
relevance logicians reject the following proposition as a valid deduction:
(PP) p, q ⊢ p
Relevance logicians think that an affirmation of (PP) involves an affirmation of ⊢ [p → (q → p)],
and this affirmation is hailed by such logicians as a well known paradox of implication. Due to
the work of A. Church, relevance logicians have worked out a conception of deducibility that
supposedly avoids such paradoxes.29 As a consequence, relevance logics are thought to be
extremely important for performing deductions.30
In order to properly assess the truth of (34) we will need a robust definition of what it
means for q to entail p in the sense of relevance logic. According to the Routley-Meyer
semantics,31 if one has a set of worlds K, and more specifically, some of these worlds are
distinguished and represented by the variable N (i.e., call these “normal” worlds). And suppose
that we agree with Kripke and standard modal semantics (k-semantics)32 by positing a relation R
which is an accessibility relation remembering that this relation deals only with a unary operator,
necessity. These semantics are therefore a relational semantics which defines a relation on
A. Church, “The Weak Theory of Implication,” in A. Menne, A. Wilhelmy, and H. Angsil (eds),
Kontrolliertes Denken, Untersuchungen zum Logikkalkül und zur Logik der Einzelwissenschaften (KommissionsVerlag Karl Alber, Munich, 1951), 22-37.
30
For a list of the rules of inference for relevant logic, see Ibid., 613-619, 622-623.
31
See Robert K. Meyer and R. Routley, “The Semantics of Entailment,” in H. Leblanc (ed.), Truth, Syntax and
Modality (North Holland, Amsterdam, 1973); Robert K. Meyer and R. Routley, “The Semantics of Entailment II,”
in Journal of Philosophical Logic 1 (1972): 53-73.
32
Let it be said that I assume k-semantics for modal logic. This is opposed to C-semantics according to which, a
fixed set of worlds W just is all worlds, or interpretations, which are logically possible in a given language. The
necessity operator on this semantics is a logical constant in that it is logically fixed. A sentence on this semantics is
necessarily true, iff it is true in W (or all worlds). See Rudolf Carnap, “Modality and Quantification,” in Journal of
Symbolic Logic 11 (1946): 34. On K-semantics the necessity operator is not constant in this way. The necessity
operator (in my argumentation) is localized around metaphysically possible worlds, or what Alvin Plantinga calls,
worlds that are broadly logically possible. See G. Schurz, “Tarski and Carnap on Logical Truth—or: What is
Genuine Logic?” in J. Wolenski and E. Köhler (ds.), Alfred Tarski and Vienna Circle (Dordrecht: Kluwer, 1999),
77-94.
29
Explanation, Entailment, and Leibnizian Arguments
12
worlds ≤ such that t ≤ s if and only if there is a world n in N such that Rnts (t and s are normal
worlds accessible to one another by virtue of the relation ≤ they stand in). Furthermore, on this
semantics the relation ≤ is transitive and reflexive (as is commonly assumed in S5). In an article
which further explicated the Routley-Meyer semantics, Edwin Mares and Robert Meyer stated
that interpretations be constrained in such a way that hereditariness holds.33 So given an
interpretation I:
(35) t normally implies s iff for all n in N, I (t ¸ s, n)
And it now follows given hereditariness and reflexivity of ≤ that:
(36) On a given interpretation I (t entails s iff t normally implies s)34
If we replace the variables t and s with q and p, is (34) true? Let me say that the relevance
entailment relation is stronger than strict implication or standard entailment relations. Relevance
entailment is cashed out in terms of normal implication, which attempts to narrow the galaxy of
worlds in which a particular implication relation between q and p holds, much like a K-semantic
account of entailment would be narrower (and therefore stronger) than a C-semantic account of
entailment, since the relevant implication relation, with respect to C-semantics, would hold for q
and p in more worlds (all logically possible worlds) rather than just broadly logically possible
worlds. Thus if we can successfully argue that there are instances in which an explanans does not
entail its corresponding explanandum in the sense of strict implication or classical entailment, we
have mutatis mutandis a reason to reject the idea that the relation between explanans and
So they say: If I (T, t) = true and t ≤ s, then I (T, s) = true. See Edwin D. Mares & Robert K. Meyer,
“Relevant Logics,” 288.
34
Heavily dependent upon Ibid.
33
Explanation, Entailment, and Leibnizian Arguments
13
explanandum is one of relevance entailment. Furthermore, I believe that whether it is or not is
irrelevant since the employment of relevance logic itself is suspect.35
It is well known that relevance logics are paraconsistent logics, and they therefore try to
tolerate contradictions while also attempting to eliminate explosion. Such non-classical systems
also typically deny bivalence and seek to only apply the law of non-contradiction to particular
sentential sentences, while acknowledging exceptions to its application. This is dialetheism.36
My main worry about such systems is related to the fact that such systems cannot express the
idea that a given proposition p is solely true, or that another proposition q is solely false.37 The
standard explanations of what it means for p, and q to be solely true or solely false on
paraconsistent logics commits the dialethiest to mutual exclusivity, which holds in its bosom
bivalence. This follows from explanations of “sole truth” and “sole falsehood” because the
standard dialetheistic explanations of such terms treat paraconsisentency as if it were a standard
glut theory.38 So Smith’s amelioration of the entailment/explanation thesis fails because of its
dubious assumptions in the philosophy of logic.
Agglomerative Explanation
Bob Hale and Crispin Wright have stated that the mere reality of something like the
BCCF suggests most naturally that it in entirety must have an explanation. They champion this
35
There are other interesting ways one might argue for the truth of (13). Suppose that L = the conjunction of all
nomic propositions of the form “it is a law of nature that …” (Assume also that these laws are deterministic); I = the
conjunction of all initial conditions prior to the event/occurrence of the phenomenon in question. Suppose that one
event e was the occurrence of an apple falling. Is it the case that:
(1) (L & I)Ee ¸ □ [(L & I) ¸ e]
I believe that (1) is false. The conjunction of initial conditions and nomic laws does not entail e, if nomic laws
have a ceteris paribus clause “in the absence of other causes.” God’s miraculous intervention or prevention of e
should not be precluded. C. S. Lewis, Miracles: A Preliminary Study (New York: Collier, 1960); cf. Alexander
Pruss, The Principle of Sufficient Reason, 113.
36
See Graham Priest, J.C. Beall and B. Armour-Garb (eds.), The Law of Non-Contradiction (Oxford: Oxford
University Press, 2004).
37
Hartry Field, Saving Truth from Paradox (NY: Oxford University Press, 2008), 386.
38
Keith Simmons, Universality and the Liar (Cambridge: Cambridge University Press, 1993).
Explanation, Entailment, and Leibnizian Arguments
14
suggestion as the default position.39 Likewise, premise (17) of the Rowe/van Inwagen reductio
assumes that explanation is dissective i.e., explanations of conjunctions must explain the entire
conjunction, and cannot be taken to be mere explanations of every conjunct. However, both
David Hume and more recently Quentin Smith have argued that the PSR is inconsistent with an
infinite chain of causes, musing that explanation is agglomerative, i.e., one has an explanation of
a conjunction just as soon as one has explanations of each respective conjunct.40 Many have
responded to Hume on this point.41 I only need point out here that the plausibility of viewing
explanation as agglomerative rests upon the success of examples which show that dissective
views of explanation are viciously regressive or viciously circular, and that supposedly taking
explanation to be agglomerative rids one of these problems.
Just how a dissective account of explanation can be shown to be viciously regressive is
when one supposes that □ q and qE(BCCF). Assume further that (BCCF) = p. The question is
then, “what explains qEp?” The proposition qEp is itself contingent and is therefore a conjunct in
the BCCF. q must explain qEp since explanation is dissective and consequently explains every
conjunct of the BCCF. But, how can this be, when qEp is itself contingent? The answer to this
perplexing question comes subsequent to reflection upon what we originally meant by
“explanation.” Explanations remove puzzlement, thus subsequent to discovering that qEp, there
just simply is no further puzzlement about why q gives us a puzzlement removing explanation
about p. This is not to say that the puzzlement removing explanations might possibly introduce
mystery to us cognizers, but that mystery is inherent in the explanation and is peculiar to itself. It
Bob Hale and Crispin Wright, “Nominalism and the Contingency of Abstract Objects,” in Journal of
Philosophy 89 (1992): 111-135; cf. William Lane Craig, “Sobel’s Acid Bath for Theism: A Review Essay of Jordan
Howard Sobel’s Logic and Theism,” in Philosophia Christi 8:2(2006): 484.
40
Quentin Smith, “Kalam Cosmological Arguments for Atheism,” in The Cambridge Companion to Atheism
(Cambridge: Cambridge University Press, 2007), 192-194.
41
Richard M. Gale, On the Existence and Nature of God (Cambridge: Cambridge University Press, 1991), 254255; Peter Forrest, God without the Supernatural: A Defense of Scientific Theism (Ithaca, NY: Cornell University
Press, 1996), 92; Alexander Pruss, The Principle of Sufficient Reason, 42-44.
39
Explanation, Entailment, and Leibnizian Arguments
15
is still the case that with respect to p, and qEp puzzlement is removed, it’s just that q in its own
right adds a bit of mystery. So I agree with Pruss, that “there are cases in which the same
proposition q explains both p and qEp” although it might not explain itself completely.42 So in
reflection upon why we ought to deny agglomerative accounts of explanation we are given
another reason why we should reject premise (16) of the Rowe/van Inwagen reductio.
Conclusion
I’ve tried to give reasons why the proponent of Leibnizian-style cosmological arguments
need not be troubled by objections to its assumptions about the nature of explanation and
entailment. I explicated a way that a necessary proposition could explain a contingent one. And
I’ve shown that there are no real good reasons why we should think that the BCCF is not an
actual contingent conjunction which given (1) requires an explanation.
We are therefore
warranted in rejecting (15) within the indirect proof of the Rowe/van Inwagen argument and we
can thereby escape the contradiction that is said to follow. Proponents of the PSR are therefore
well within their epistemic rights in appropriating (1) and (2).
Alexander Pruss, The Principle of Sufficient Reason, 121. q’s not explaining itself is a different worry since q
in this case would be a necessary being’s activities.
42