A Contingency Argument Without the PSR

1. The Principle of Sufficient Reason and Cosmological Arguments

Among the classical arguments for the existence of God, the cosmological argument has long been the most appreciated. Consider Leibniz’s version (Pruss 2009):

P1. Every contingent fact has an explanation.

P2. There is a contingent fact that includes all other contingent facts.

P3. Therefore, there is an explanation of this fact.

C. This explanation must involve a necessary being — namely, God.

P1 expresses the Principle of Sufficient Reason (PSR). The PSR has often been weakened in successive stages:

Strong PSR (S-PSR) .     Every fact has a reason.

Limited PSR (L-PSR) .  Every fact with property $x$  has a reason.

Weak L-PSR (WL-PSR) .          For every fact with property $x$ , it is possible that it has a reason.

Here, reason broadly covers causes, grounds, or explanations in general. Let WL-PSR* denote an especially weak version:

$$\forall a\left(\right(C\left(a\right)\wedge In(a,w_a))\to \exists b\exists v(R(b,a)\wedge In(b,v)\left)\right)$$

(WL-PSR*)

Here $C$ predicates contingency; $In$ expresses the belonging of an entity to the domain of a world; $w_a$ is the actual world; and $R$ is an irreflexive, asymmetric, transitive reason relation. Accordingly, WL-PSR* reads “for every actual contingent entity, there exists, in some actual or non-actual world, a reason”.

Virtually all major contemporary cosmological arguments assume some version of the PSR (¹ For instance, L-PSR is adopted by Koons (1997), Craig & Sinclair (2009: 101, 192), Sandsmark & Megill (2010), Wahlberg (2017), Dumsday (2018), Loke (2018: ch.5), and Byerly (2019); Flynn & Gel (forthcoming) derive it from the distinction between a contingent entity and its existence. WL-PSR appears in Kremer (1997), Gale & Pruss (1999: 463), Pruss (2010), Rasmussen (2009), and Weaver (2016).

¹PI does not entail metaphysical nihilism: if the empty world ). Yet few philosophers still endorse the PSR. According to Bourget & Chalmers (2023), 57.26% of metaphysicians reject it, while only 32.66% affirm it. Objections to its logical coherence are raised by Van Inwagen (1983: 202), McDaniel (2019) and Briceño (2023: §5). These extend even to weaker forms: Oppy (2000: §2) argues that Pruss’s WL-PSR entails S-PSR if “has an explanation” is a dissective operator — a point Pruss concedes.

Such challenges do not refute the PSR conclusively, but they expose a pressing need for a cosmological argument that dispenses with it. In what follows, I aim to show that the existence of a necessary being can be demonstrated even under the radical assumption that contingent entities may exist—or fail to exist—without any reason, cause, ground, or explanation whatsoever.

2. Translation of QKB into Two-Sorted FOL

The reasoning is framed within the QKB modal system. Since handling certain metalanguage statements has proven cumbersome, the formulas are translated into a two-sorted FOL as follows:

Table 1. 

Category	Variables/Symbols	Description
Individuals	$\mathit{a},\mathit{b},\mathit{c},...$	Entities
Worlds	$\mathit{w},\mathit{v},\mathit{u},\mathit{t},...$	Possible worlds
Predicates	$\mathit{I}\mathit{n}(\mathit{a},\mathit{w})$,$\mathit{A}(\mathit{w},\mathit{v})$	Membership ($\mathit{I}\mathit{n}$) and accessibility ($\mathit{A}$)
Quantifiers	$\forall \mathit{a},\forall \mathit{w},\exists \mathit{a},\exists \mathit{w}$	Necessity and possibility for each sort
B axiom	$\forall w\forall v\left(A\right(w,v)\to A(v,w\left)\right)$	Symmetry of accessibility

Table 1. 

Category	Variables/Symbols	Description
Individuals	$\mathit{a},\mathit{b},\mathit{c},...$	Entities
Worlds	$\mathit{w},\mathit{v},\mathit{u},\mathit{t},...$	Possible worlds
Predicates	$\mathit{I}\mathit{n}(\mathit{a},\mathit{w})$,$\mathit{A}(\mathit{w},\mathit{v})$	Membership ($\mathit{I}\mathit{n}$) and accessibility ($\mathit{A}$)
Quantifiers	$\forall \mathit{a},\forall \mathit{w},\exists \mathit{a},\exists \mathit{w}$	Necessity and possibility for each sort
B axiom	$\forall w\forall v\left(A\right(w,v)\to A(v,w\left)\right)$	Symmetry of accessibility

Most derivations that follow are straightforward applications of standard inference rules, so only proof sketches are given.

3. Inaccessibility from the Empty World

In cosmological arguments from contingency, the PSR supplies contingent entities with a necessary condition: if every fact has an explanation, then the totality of contingent entities has one, and their existence thereby entails something beyond them. To preserve this structure without invoking the PSR, we need an alternative principle that still provides contingent entities with a necessary condition.

Suppose instead that a contingent entity’s possibility has no necessary condition. Then it would require no ontological configuration — nothing else would need to exist. A world containing such an entity would thus be accessible from the empty world. Therefore, we can provide contingent entities with a necessary condition through the following principle, stated explicitly by Armstrong (1989: 64):

Principle of Inaccessibility (PI). No non-empty world is accessible from an empty world. (¹ PI does not entail metaphysical nihilism: if the empty world is impossible, it is vacuously true. For discussion, see Baldwin (1996); Efird & Stoneham (2006); Coggins (2010); Thompson (2010: ch. 2); Goldschmidt (2012); Hansen (2012); and De Clerque (2023).)

Formally:

$$\forall w\forall v(\forall a\neg In(a,w)\wedge \exists bIn(b,v\left)\right)\to \neg A(w,v)$$

(PI)

Consider a model $M$ such that: 

Individual domain: $\left\{\mathit{a}\right\}$

World domain: $\{\mathit{w},\mathit{v}\}$

Valuations: $\mathit{M}\models \mathit{I}\mathit{n}(\mathit{a},\mathit{v});\neg \mathit{A}(\mathit{w},\mathit{v})$

PI holds in $\mathit{M}$, but WL-PSR* fails, since there is no $b$ such that $R(b,a)$. Hence PI does not entail any version of the PSR as strong as WL-PSR* or more.

Unlike the PSR, PI postulates no explanatory relation $R(b,a)$. It merely implies that if something is possible but not actual, something else is actual:

$$\forall a\forall w\forall v\left(\right(\neg In(a,w)\wedge In(a,v)\wedge A(w,v))\to \exists bIn(b,w\left)\right))$$

(1)

The derivation of (1) from PI is trivial. Assume $\neg In(a,w)\wedge In(a,v)\wedge A(w,v)$. If $\forall b\neg In(b,w)$, then by PI we would have $\neg \mathit{A}(\mathit{w},\mathit{v})$, contradicting the assumption. Hence, $\exists bIn(b,w)$. (1) follows by universal generalisation.

Thus, if an entity can come into existence, it can only do so if something else exists. Or, as Billy Preston puts it, “nothin’ from nothin’ leaves nothin’.”

4. Philosophical Grounds for PI

Armstrong takes PI as a consequence of combinatorialism — the view that possible worlds are recombinations of actual entities and properties. Among the three major theories of possible worlds, combinatorialism enjoys a clear advantage: unlike concretism and abstractionism, it avoids ontological inflation while preserving a realist account of possibility and necessity, immanently realised in actual entities. But in the empty world there are no entities or properties, and thus no recombinations. If nothing can be recombined, no world can be constructed from it; hence, no world is accessible from the empty one. Under this interpretation, combinatorialism entails PI.

A second line of support derives from the nature of time. On the two dominant views, time is either an entity (substantivalism) or emerges from relations among entities (relationism). As Shoemaker (1969) observed, time seems irreducible to change, yet change presupposes time. The emergence of something from nothing would constitute a change. But in the empty world there are neither entities nor relations from which time could arise. Without time, nothing can change; hence again, no non-empty world can be accessible from the empty one.

Finally, PI articulates a principle as old as metaphysics itself: ex nihilo nihil fit. No philosophical principle is beyond dispute, but every argument must begin from what is most intuitive and widely accepted. Given its proximity to the PSR, this principle has likely enjoyed similar historical acceptance —often tacit, yet near universal. Thinkers who diverge on nearly everything else can still converge on PI as a credible starting point.

5. Circularity and Infinite Regress

We now show that the existence of contingent entities requires a non-contingent one.

We define a contingent entity as one that can both enter and exit existence. Assume, for reductio, that only contingent entities exist:

$$\forall a\exists w\exists v(\neg In(a,w)\wedge In(a,v)\wedge A(w,v\left)\right)$$

(2)

Take any world $v$ in which $a$ is actual. From (2), there exists a world $w$ in which $a$ is not actual but, given (1), another entity $b$ is. Likewise, there exists a world $u$ in which $b$ is not actual, but some $c$ is. Iterating this reasoning, for every contingent we can define the following dependency relation:

$$F(b,a)\leftrightarrow \exists w\exists v(\neg In(a,w)\wedge In(b,w)\wedge In(a,v)\wedge A(w,v\left)\right)$$

(F)

which reads: “$b$ exists in a world $w$ where $a$ does not exist, and $w$ accesses a world $v$ where $a$ exists”.

Let $G$ be the directed graph whose vertices represent entities and whose edges represent the relation $F$. Each vertex $e_n$ is such that there exists another vertex $e_{n+1}$ with a directed edge $(e_{n+1},e_n)$. Since every contingent entity is in a $F$-relation with another, every $e_n$ lies on a trail $(...,(e_{n+3},e_{n+2}),(e_{n+1},e_n\left)\right)$.

Either some vertex $e_m$ with $m<n$ coincides with $e_n$—yielding a circularity—or no vertex ever repeats — yielding an infinite regress. If both infinite regresses and circularities are impossible, it follows that there must exist at least one non-contingent entity. But why should they be impossible?

6. Well-Foundness

Consider the following pedagogical example: first there is an already existing entity, and only then can a contingent entity begin to exist. In an infinite regress or circularity, no entity is already existing — each requires some prior existence elsewhere. Hence, none can ever begin to exist.

This captures a familiar intuition. Aquinas (2017: Iª q. 2 a. 3 co.) writes: “if everything is possible not to be, then at one time there could have been nothing in existence [and] if this were true, even now there would be nothing in existence”. Similarly, Leibniz (1989: 85) states that “every being derives its reality only from the reality of those beings of which it is composed, so that it will not have any reality at all if each being of which it is composed is itself a being by aggregation”. Schaffer (2010: 62) expresses the same idea: if everything were grounded in something else, “being would be infinitely deferred, never achieved”.

Yet justifying this intuition has proven difficult. Although metaphysical foundationalism is often treated as the standard view, no argument against the possibility of actual downwardly non-terminating dependency chains commands general assent. Foundationalism thus appears axiomatic.

Bohn (2018: 170) rightly objects that the pedagogical example treats certain relations as “a diachronic, dynamic physical relation”, whereas they are instead “synchronic, static mathematical relation[s]”.

Indeed, $F$ is an atemporal relation. (1) states that a contingent entity is possible only if it is in an $F$-relation with some actual entity. Yet, given $F(b,a)$, there is an intuitive ontological—not temporal—precedence: while $a$ and $b$ may be synchronous, (1) shows that $b$’s existence in $w$ is the necessary condition for $a$’s mere possibility in $w$. Furthermore, there is a sense in which $a$ passes from a more fundamental state (possibility) to a less fundamental one (existence).

The pedagogical example assumes that the temporal precedence relation is well-founded. The same results follows even in an atemporal context via the following principle:

Well-Foundness (WF). The relation $F$ is well-founded.

That is, there exists and entity $a$ which is not in an $F$-relation with any $b$:

$$\exists a\forall b\neg F(b,a)$$

(WF)

WF is weaker than Schaffer's. Schaffer predicates well-foundness for an inheritance of being: the existence of one entity requires another’s. WF instead expresses, at most, an inheritance of possibility.

WF directly contradicts ($2$). By De Morgan’s law, WF predicates the existence of an $a$ such that, for every world, $In(a,w)\vee \neg In(b,w)\vee \neg In(a,v)\vee \neg A(w,v)$. The truth of any disjunct contradicts a conjunct of (2), except $\neg In(b,w)$. Assume that $\neg In(b,w)$ is true while all other disjuncts are false. Then $\neg In(a,w)\wedge In(a,v)\wedge A(w,v)$ holds, which, by (1), entails $In(b,w)$, contradicting our assumption. Therefore, at least one other disjunct must be true. WF and (2) are thus incompatible given PI.

Accordingly, we can rewrite WF as:

$$\exists a\forall w\forall v\left(In\right(a,w)\vee \neg In(a,v)\vee \neg A(w,v\left)\right)$$

(3)

7. Impossible Entities

At least one of (3)’s disjuncts must be true. Which entity satisfies it?

Since an entity either exist or not exist, we can distinguish two classes of worlds:

$$W_{+}:=\left\{w\right|In(a,w)\}$$

(W₊)

$$\begin{gathered} (W_{+} \\ {W}_{-}:=\left\{w\right|\neg In(a,w)\} \end{gathered}$$

(W₋)

Accordingly, three possibilities arises:

α

Necessary entity. If $W_{+}$ comprises all worlds, $a$ exists in every world. It satisfies the first disjuncts—or the first and the third jointly—since it exists in both $w$ and $v$, which may or may not be interaccessible.

β

Impossible entity. If $W_{-}$ comprises all worlds, $a$ exists in none. It satisfies the second disjuncts—or the second and the third jointly—since it exists in neither $w$ nor $v$, regardless of accessibility.

ɣ

Isolated entity. If both $W_{+}$ and $W_{-}$ are non-empty, the first two disjuncts are either both true or both false. If both are false, the third must be true. If both are true, the third must hold too, otherwise (2) would follow. Hence $W_{+}$ and $W_{-}$ are non-communicating: no world of one class accesses a world of the other. This satisfies the third disjunct — or all three jointly.

By the symmetry of accessibility, there can be no partially isolated entities — those that can begin but not cease to exist (only $W_{-}$ accesses $W_{+}$) or vice versa (only $W_{+}$ accesses $W_{-}$).

Formally, the three classes of non-contingent entities can be represented as follows:

$$\exists a\forall w\left(In\right(a,w\left)\right)$$

(α)

$$\exists a\forall w(\neg In(a,w\left)\right)$$

(β)

$$\exists a\forall w\forall v\left(In\right(a,w)\wedge \neg In(a,v)\wedge \neg A(w,v\left)\right)$$

(γ)

Of these, the impossible ones are the most problematic. They can be ruled out by the following principle:

Non-Meinongism (NM).          There are no nonexistent entities.

Formally:

$$\forall a\exists wIn(a,w)$$

(NM)

which is the negation of ($\beta$). NM is widely accepted: it forbids existentially quantification over entities that are not actual in any world. There is no Meinong’s jungle: to exist is to exist in some world.

Thus, only two possibilities remain: at least one entity is either necessary or isolated.

8. Isolated Entities

Isolated entities are metaphysically peculiar: they are globally contingent but locally necessary. They fail to exist in some worlds, yet they exist in all worlds where they are possible, and cannot begin or cease to exist.

In metaphysics, most arguments are framed within S5. Although it is coherent with the existence of multiple independent equivalence classes, most metaphysicians assume that accessibility forms a single equivalence class — that of the actual world. This entails universal accessibility, thereby excluding isolated entities altogether.

On combinatorialism, a world $v$ is accessible from $w$ iff $v$ is a recombination of the entities and properties of $w$. Since the domain of possible worlds is that of the recombinations of actual entities and properties, every world is accessible from the actual world. Thus, every world is accessible from every other.  Combinatorialism entails both PI and universal accessibility.

Yet the rejection of isolated entities requires far less than universal accessibility. It suffices to assume the following condition:

Interaccessibility (IA). At least one $W_{-}$-worlds access at least one $W_{+}$-worlds.

Formally:

$$\forall a(\exists w\exists v(In(a,w)\wedge \neg In(a,v))\to \exists u\exists t(In(a,u)\wedge \neg In(a,t)\wedge A(u,t)\left)\right)$$

(IA)

IA directly denies ($\gamma$). For if an entity satisfies $\forall w\forall v\left(In\right(a,w)\wedge \neg In(a,v)\wedge \neg A(w,v\left)\right)$, IA entails $\exists u\exists t\left(A\right(u,t\left)\right)$ from it.

Given WF, only one options remain: the existence of at least one necessary entity.

9. Evaluation of the Axioms

The existence of a necessary entity follows from four axioms:

(PI) $\forall w\forall v(\forall a\neg In(a,w)\wedge \exists bIn(b,v\left)\right)\to \neg A(w,v)$

(WF) $\exists a\forall w\forall v\left(In\right(a,w)\vee \neg in(a,v)\vee \neg A(w,v\left)\right)$

(NM) $\forall a\exists wIn(a,w)$

(IA) $\forall a(\exists w\exists v(In(a,w)\wedge \neg In(a,v))\to \exists u\exists t(In(a,u)\wedge \neg In(a,t)\wedge A(u,t)\left)\right)$

Consider the following models:

Table 2. 

	${\mathit{M}}_{\mathit{C}}$	${\mathit{M}}_{\mathit{P}\mathit{I}}$	${\mathit{M}}_{\mathit{W}\mathit{F}}$	${\mathit{M}}_{\mathit{N}\mathit{M}}$	${\mathit{M}}_{\mathit{I}\mathit{A}}$
Individual domain	$\{\mathit{a},\mathit{b}\}$	$\left\{\mathit{a}\right\}$	$\{\mathit{a},\mathit{b}\}$	$\{\mathit{a},\mathit{b},\mathit{c}\}$	$\{\mathit{a},\mathit{b}\}$
World domain	$\{\mathit{w},\mathit{v}\}$	$\{\mathit{w},\mathit{v}\}$	$\{\mathit{w},\mathit{v}\}$	$\{\mathit{w},\mathit{v}\}$	$\{\mathit{w},\mathit{v}\}$
Valuations	${\mathit{M}}_{\mathit{C}}\models$ $\mathit{I}\mathit{n}(\mathit{a},\mathit{w})\mathit{I}\mathit{n}(\mathit{b},\mathit{w})\mathit{I}\mathit{n}(\mathit{b},\mathit{v})\mathit{A}(\mathit{w},\mathit{v})$	${\mathit{M}}_{\mathit{P}\mathit{I}}\models$ $\mathit{I}\mathit{n}(\mathit{a},\mathit{v})$ $\mathit{A}(\mathit{w},\mathit{v})$	${\mathit{M}}_{\mathit{W}\mathit{F}}\models$ $\mathit{I}\mathit{n}(\mathit{a},\mathit{w})$ $\mathit{I}\mathit{n}(\mathit{b},\mathit{v})$ $\mathit{A}(\mathit{w},\mathit{v})$	${\mathit{M}}_{\mathit{N}\mathit{M}}\models$ $\mathit{I}\mathit{n}(\mathit{a},\mathit{w})$ $\mathit{I}\mathit{n}(\mathit{b},\mathit{w})$ $\mathit{I}\mathit{n}(\mathit{b},\mathit{v})$ $\mathit{A}(\mathit{w},\mathit{v})$	${\mathit{M}}_{\mathit{I}\mathit{A}}\models$ $\mathit{I}\mathit{n}(\mathit{a},\mathit{w})$ $\mathit{I}\mathit{n}(\mathit{b},\mathit{w})$ $\mathit{I}\mathit{n}(\mathit{b},\mathit{v})$
Axioms satisfied	All	All except PI	All except WF	All except NM	All except IA

Table 2. 

	${\mathit{M}}_{\mathit{C}}$	${\mathit{M}}_{\mathit{P}\mathit{I}}$	${\mathit{M}}_{\mathit{W}\mathit{F}}$	${\mathit{M}}_{\mathit{N}\mathit{M}}$	${\mathit{M}}_{\mathit{I}\mathit{A}}$
Individual domain	$\{\mathit{a},\mathit{b}\}$	$\left\{\mathit{a}\right\}$	$\{\mathit{a},\mathit{b}\}$	$\{\mathit{a},\mathit{b},\mathit{c}\}$	$\{\mathit{a},\mathit{b}\}$
World domain	$\{\mathit{w},\mathit{v}\}$	$\{\mathit{w},\mathit{v}\}$	$\{\mathit{w},\mathit{v}\}$	$\{\mathit{w},\mathit{v}\}$	$\{\mathit{w},\mathit{v}\}$
Valuations	${\mathit{M}}_{\mathit{C}}\models$ $\mathit{I}\mathit{n}(\mathit{a},\mathit{w})\mathit{I}\mathit{n}(\mathit{b},\mathit{w})\mathit{I}\mathit{n}(\mathit{b},\mathit{v})\mathit{A}(\mathit{w},\mathit{v})$	${\mathit{M}}_{\mathit{P}\mathit{I}}\models$ $\mathit{I}\mathit{n}(\mathit{a},\mathit{v})$ $\mathit{A}(\mathit{w},\mathit{v})$	${\mathit{M}}_{\mathit{W}\mathit{F}}\models$ $\mathit{I}\mathit{n}(\mathit{a},\mathit{w})$ $\mathit{I}\mathit{n}(\mathit{b},\mathit{v})$ $\mathit{A}(\mathit{w},\mathit{v})$	${\mathit{M}}_{\mathit{N}\mathit{M}}\models$ $\mathit{I}\mathit{n}(\mathit{a},\mathit{w})$ $\mathit{I}\mathit{n}(\mathit{b},\mathit{w})$ $\mathit{I}\mathit{n}(\mathit{b},\mathit{v})$ $\mathit{A}(\mathit{w},\mathit{v})$	${\mathit{M}}_{\mathit{I}\mathit{A}}\models$ $\mathit{I}\mathit{n}(\mathit{a},\mathit{w})$ $\mathit{I}\mathit{n}(\mathit{b},\mathit{w})$ $\mathit{I}\mathit{n}(\mathit{b},\mathit{v})$
Axioms satisfied	All	All except PI	All except WF	All except NM	All except IA

Each axiom is therefore independent and jointly consistent.

10. No Second Stage Under Mysterianism

Following Rowe’s (1975:6) distinction, the foregoing constitutes the first stage of the cosmological argument. It is generally followed by a second stage, that aims to identify a necessary being with God.

As Theron (1987) notes, God possesses entitative and operational properties. Entitative properties follow from necessity and thus belong to every necessary entity. What distinguishes God are the operational properties such as thought and will. To identify a necessary entity with God, one must therefore show that it possesses a conscious mind.

Consider the following thesis:

Mysterianism        Beliefs about other minds cannot be justified.

Reasons for adopting mysterianism include: the belief that indirect evidence is insufficient for justification; that philosophical zombies are metaphysically possible; that the human cognitive architecture does not have access to such facts; that empirio-criticism or phenomenalism—according to which only sense data are knowable—is true.

Given mysterianism, the second stage becomes impossible. Alternatively, one might adopt:

Entitlement An unjustified belief can nevertheless be warranted.

Burge (2020) distinguishes justification (warrant with reasons) from entitlement (warrant without reasons). Following Wright (2004: §3), one is entitled to a belief when (i) they possess no evidence against it and (ii) acting as if it were true is essential to pursuing valued goals, regardless of its actual truth-value. The first condition is automatically satisfied under mysterianism.

This renders the second stage subjectively possible. One might, for instance, desire a personal relationship with an entity culminating in participation in its necessity. If such a conscious necessary entity exists, the dominant strategy is to attempt to relate to it. If it does not, the goal becomes unattainable, and all strategies become equal. Hence, one may be entitled to believe in a necessary conscious entity. Yet since entitlement depends on the subjective evaluation of goals, no objective second stage can be constructed.

Famously, after establishing the existence of a necessary entity, Aquinas concludes: "this all men speak of as God”. The mysterianist, by contrast, might say: "this some are entitled to believe is God."