# Descartes’s First Proof of God’s Existence in First-Order Logic

2020-06-07, updated 2020-06-07 next - previous

In the language of first-order logic, I provide in this post a proof in natural deduction which translates the argument that Descartes gave as evidence of the existence of God.

## 1 Introduction

I am going to focus only on the logical structure of Descartes’s argument descartes2008a, to prove that this argument is, from a deductive point of view, intuitionistically valid. Let us recall these important definitions:

An argument is deductively valid if and only if its conclusion is true whenever its premises are all true. bonevac2003

An argument is sound if and only if it is valid and all its premises are true. bonevac2003

Therefore, the validity and the soundness of Descartes’s argument are different issues, and there are sometimes confused, for example, by Williams when he wrote:

The trouble with Descartes’s system is not that it is circular; nor that there is an illegitimate relation between the proofs of God and the clear and distinct perceptions; nor that there is a special problem about the proofs of God when they are not intuited. I have argued that in these respects, it is structurally sound. The trouble is that the proofs of God are invalid and do not convince even when they are supposedly being intuited.williams2005

Pace Williams, I am going to prove in first-order logic that Descartes’s argument is valid, that is deductively valid. To do this, I need the following formal lexicon:

• c = “the cogito”,
• d = “the idea of God”,
• $$x \geqslant y$$ = “the degree of reality of x is at least equal to that of y”,
• Sxy = “the formal reality x is the total and efficient cause of the objective reality y” (the meaning of “formal reality” and “objective reality” are explained below),
• Rcy = “y is a thought of the cogito”,
• Last, to the usual rules of natural deduction for first-order logic, I add this couple of rules derived from the definition of $$\geqslant$$, where defm. and defs. are the respective abbreviations for the definiens and the definiendum that are at the bottom of each rule:
\begin{array}[tb]{ccc} \dfrac{x \geqslant y}{(x > y) \lor (x = y)}\scriptsize{\geqslant defs.}& & \dfrac{ (x > y) \lor (x = y)}{x \geqslant y}\scriptsize{\geqslant defm.} \\ \end{array}

The core of the first proof of the existence of God in the Third Meditation is contained in the following text:

And so there remains only the idea of God, in which I must consider whether there is anything that could not derive from myself. By the name ‘God’ I understand an inﬁnite, independent, supremely intelligent, supremely powerful substance, by which I myself and whatever else exists (if anything else does exist) was created. But certainly, all these properties are such that, the more carefully I consider them, the less it seems possible that they can be derived from me alone. And so I must conclude that it necessarily follows from all that has been said up to now that God exists. For indeed, even if the idea of substance is in me as a result of the very fact that I am a substance, the idea of an inﬁnite substance would not therefore be in me, since I am ﬁnite, unless it derived from some substance that is really inﬁnite.descartes2008a

Descartes’s argument is the deduction of a conjunction of two statements, deduction which can therefore be divided into two subproofs. To use Descartes’s vocabulary, the last sentence in the previous quotation means that the objective reality of the idea of substance can be caused by the formal reality of the cogito because the cogito is a substance, but being a finite substance, the cogito cannot be the cause of his idea of God, because the objective reality of this idea is an infinite substance, and only a formal reality of an infinite substance can be the cause of this representation. On this point, Williams’s explanations are the clearest I have ever read:

He calls the reality that anything possesses intrinsically, its formal reality; and he calls the reality that an idea possesses in virtue of its object, its objective reality. Thus all ideas have the same degree of formal reality, but different degrees of objective reality – because their objects have, or would have, different degrees of (formal) reality. Lastly, in expressing the principle that the cause of anything must contain at least as much reality as the effect, he says that the reality of the effect must exist in the cause either formally or eminently: formally, if there is just as much reality in the cause as in the effect, and eminently, if there is more reality in the cause than in the effect, the cause being of some higher type than the effect (this will be so with works of art, the mind of the artist being of a higher type of reality than any of his products). Putting all these terms together, Descartes’s principle about the causation of ideas comes out like this: the cause of any idea must contain either formally or eminently as much reality as the idea possesses both formally and objectively.williams2005

In its simplest expression, the conclusion of Descartes’s argument says that the cogito is not the cause of the idea of ​​God and that there is something which is the cause of this idea and which is as perfect as this idea of ​​God, conjunction that can be translated by

\begin{equation} \label{one} \lnot Scd \land \exists x(Sxd \land (x = d)) \end{equation}

There will be therefore a first subproof to deduce the formula on the left of conjunction \eqref{one} that is

\begin{equation} \label{two} \lnot Scd \end{equation}

and the second subproof will be the deduction of the formula on the right of conjunction \eqref{one}.

## 2 First Subproof

Three premises are needed to deduce \eqref{two}. The first of them is the definition of the idea of God:

The idea of God is “the idea of a supremely perfect and inﬁnite being.” descartes2008a

Descartes’s point is that the representative content of the idea of God is such that it is impossible for the cogito to get the intellection of a perfection that would be greater. Therefore, the formalization of Anselm’s definition of God can be used again to translate Descartes’s claim:

\begin{equation} \label{three} \forall y \lnot(y > d) \end{equation}

the idea of God is the idea of the being that is such that nothing greater can be thought of.

Contrarily to the idea of God, I can always conceive something more perfect than I am. Therefore, expressed at the first personn, our second premise would say “I have certainly not all the perfections that I conceive in the idea of God” and, in our lexicon, this second premise says “it is false that the degree of perfection of the cogito is the same as the degree of perfection of the idea of God”:

\begin{equation} \label{four} \lnot(c = d) \end{equation}

Some readers will find maybe useful the following explanation on a part of the first subproof. Note that from \eqref{three} and \eqref{four} it is possible to deduce $$\lnot(c \geqslant d)$$: $\forall y \lnot(y > d),\lnot(c = d) \vdash \lnot(c \geqslant d)$

\begin{prooftree} \AxiomC{$\overset{1}{(c > d) \lor (c = d)}$} \AxiomC{$\forall y \lnot(y > d)$} \RightLabel{$\scriptsize{\forall E}$} \UnaryInfC{$\lnot(c > d)$} \AxiomC{$\overset{2}{c > d}$} \RightLabel{$\scriptsize{\lnot E}$} \BinaryInfC{$\bot$} \AxiomC{$\lnot(c = d)$} \AxiomC{$\overset{2}{c = d}$} \RightLabel{$\scriptsize{\lnot E}$} \BinaryInfC{$\bot$} \RightLabel{$\scriptsize{\lor E, 2}$} \TrinaryInfC{$\bot$} \RightLabel{$\scriptsize{\lnot I, 1}$} \UnaryInfC{$\lnot((c > d) \lor (c =d))$} \RightLabel{$\scriptsize{\geqslant defm.}$} \UnaryInfC{$\lnot(c \geqslant d)$} \end{prooftree}

$$\textcolor{white}{text}$$

and, once erased all the discharged assumptions, the previous deduction is reducible to the following deduction that is going to be a part of the first subproof:

\begin{prooftree} \AxiomC{$\forall y\lnot(y > d)$} \RightLabel{$\scriptsize{\forall E}$} \UnaryInfC{$\lnot(c > d)$} \AxiomC{$\lnot(c = d)$} \RightLabel{$\scriptsize{DMi}$} \BinaryInfC{$\lnot(c \geqslant d)$} \end{prooftree}

Note that the sequent $\lnot(c > d), \lnot(c = d) \vdash \lnot((c > d) \lor (c = d))$ has just been proved above; it is an instance of De Morgan’s laws that is intuitionistically derivable (hence the label DMi). It is also well known that this other instance of De Morgan Law: $\lnot((c > d) \land (c = d)) \vdash \lnot(c > d) \lor \lnot (c = d)$ is not intuitionistically valid, but valid only in classical logic; such a deduction does not exist in Descartes’s argument.

Finally, a specific application of the causality principle is necessary for Descartes’ argument, as clearly shown by what he wrote:

Now, it is manifest by the natural light that there must be at least as much reality in the total and efficient cause as in its effect. For, I ask, from where could the effect derive its reality, if not from the cause? And how could the cause give it reality, if it did not also possess it? Hence it follows, both that nothing can come from nothing, and that what is more perfect (that is, what contains more reality within itself) cannot derive from what is less perfect. And this is not only plainly true of those effects whose reality is actual or formal, but also of ideas, in which only the objective reality is considered. descartes2008a

I call “principle of applied causality” this application of causality principle from formal realities to objective realities that represent them: “A formal reality x is the cause of an objective reality y only if the degree of perfection of x is at least equal to that of y,” which I translate by:

\begin{equation} \label{five} \forall x \forall y(Sxy \to x \geqslant y) \end{equation}

Formula \eqref{five} provides the third and last premise necessary to get the first step of Descartes’s argument that concludes that the cogito is not the cause of the idea of God. Indeed, formula \eqref{two} is deducible from the conjunction of \eqref{three}, \eqref{four} and \eqref{five}:

\begin{equation} \label{six} \forall y \lnot(y > d), \lnot(c = d), \forall x \forall y(Sxy \to x \geqslant y) \vdash \lnot Scd \end{equation}
\begin{prooftree} \AxiomC{$\forall y\lnot(y > d)$} \RightLabel{$\scriptsize{\forall E}$} \UnaryInfC{$\lnot(c > d)$} \AxiomC{$\lnot(c = d)$} \RightLabel{$\scriptsize{DMi}$} \BinaryInfC{$\lnot(c \geqslant d)$} \AxiomC{$\forall x\forall y(Sxy \to x \geqslant y)$} \RightLabel{$\scriptsize{\forall E}$} \UnaryInfC{$\forall y(Scy \to c \geqslant y)$} \RightLabel{$\scriptsize{\forall E}$} \UnaryInfC{$Scd \to c \geqslant d$} \AxiomC{$\overset{1}{Scd}$} \RightLabel{$\scriptsize{\to E}$} \BinaryInfC{$c \geqslant d$} \RightLabel{$\scriptsize{\lnot E}$} \BinaryInfC{$\bot$} \RightLabel{$\scriptsize{\lnot I,1}$} \UnaryInfC{$\lnot Scd$} \end{prooftree}

$$\textcolor{white}{text}$$

## 3 Second Subproof

The second step of Descartes’s argument must prove that there is a cause of the idea of God and that this formal reality of this cause has necessarily the same perfection as the objective reality of this idea, in other words, that this cause is God. In the language of first-order logic with our lexicon, the second subproof of Descartes’s argument must be the deduction of the following formula which is on the right of conjunction \eqref{one}:

\begin{equation} \label{seven} \exists x(Sxd \land (x = d)) \end{equation}

The deduction of $$\exists x(Sxd)$$ is not difficult and is based on two premises only; “Every idea that the cogito thinks has a cause” for the first, and “The idea of God is thought by the cogito” for the second, respectively:

\begin{equation} \label{eight} \forall y(Rcy \to \exists x Sxy) \end{equation}

and

\begin{equation} \label{nine} Rcd \end{equation}

It is now quite easy to deduce $$\exists x(Sxd)$$ from \eqref{eight} and \eqref{nine}:

\begin{prooftree} \AxiomC{$\forall y(Rcy \to \exists xSxy)$} \RightLabel{$\scriptsize{\forall E}$} \UnaryInfC{$Rcd \to \exists x Sxd$} \AxiomC{$Rcd$} \RightLabel{$\scriptsize{\to E}$} \BinaryInfC{$\exists x Sxd$} \end{prooftree}

From this result, it is now possible to deduce \eqref{seven} without additional premise, but it needs more work. Here is the second part of the proof that shows that sequent \eqref{ten} is valid:

\begin{equation} \label{ten} \forall y(Rcy \to \exists x Sxy), Rcd, \forall x \forall y(Sxy \to x \geqslant y), \forall y \lnot(y > d) \vdash \exists x(Sxd \land (x = d)) \end{equation}
\begin{prooftree} \AxiomC{$\forall y(Rcy \to \exists xSxy)$} \RightLabel{$\scriptsize{\forall E}$} \UnaryInfC{$Rcd \to \exists x Sxd$} \AxiomC{$Rcd$} \RightLabel{$\scriptsize{\to E}$} \BinaryInfC{$\exists x Sxd$} \AxiomC{$\overset{2}{Sbd}$} \AxiomC{$\forall x \forall y(Sxy \to x \geqslant y)$} \RightLabel{$\scriptsize{\forall E}$} \UnaryInfC{$\forall y(Sby \to b \geqslant y)$} \RightLabel{$\scriptsize{\forall E}$} \UnaryInfC{$Sbd \to b \geqslant d$} \AxiomC{$\overset{2}{Sbd}$} \RightLabel{$\scriptsize{\to E}$} \BinaryInfC{$b \geqslant d$} \RightLabel{$\scriptsize{\geqslant defs.}$} \UnaryInfC{$((b > d)\lor(b = d))$} \AxiomC{$\forall y\lnot(y > d)$} \RightLabel{$\scriptsize{\forall E}$} \UnaryInfC{$\lnot(b > d)$} \AxiomC{$\overset{3}{b > d}$} \RightLabel{$\scriptsize{\lnot E}$} \BinaryInfC{$\bot$} \RightLabel{$\scriptsize{\bot E}$} \UnaryInfC{$b = d$} \AxiomC{$\overset{3}{b = d}$} \RightLabel{$\scriptsize{\lor E, 3}$} \TrinaryInfC{$b = d$} \RightLabel{$\scriptsize{\land I}$} \BinaryInfC{$Sbd \land (b = d)$} \RightLabel{$\scriptsize{\exists I}$} \UnaryInfC{$\exists x(Sxd \land (x = d))$} \RightLabel{$\scriptsize{\exists E, 2}$} \BinaryInfC{$\exists x(Sxd \land (x = d))$} \end{prooftree}

$$\textcolor{white}{text}$$ It is now trivial to deduce \eqref{one} from the conjunction of the proofs of \eqref{six} and \eqref{ten}:

\begin{equation} \label{eleven} \forall y \lnot(y > d),\lnot(c = d), \forall x \forall y(Sxy \to x \geqslant y), \forall y(Rcy \to \exists x Sxy), Rcd \vdash \lnot Scd \land \exists x(Sxd \land (x = d)) \end{equation}

From the respective proofs of sequents \eqref{six} and \eqref{ten}, \eqref{eleven} is proved by application of the rule of conjunction introduction. Therefore \eqref{one} is an analytic consequence of the conjunction of assumptions \eqref{three}, \eqref{four}, \eqref{five}, \eqref{eight} and \eqref{nine}:

The fact that \eqref{eleven} is intuitionistically valid has been also checked thanks to some provers and theorem checkers; see this post.

$$\textcolor{white}{text}$$

## 4 Conclusion

Three points to conclude. First, this analysis of Descartes’s argument provides a logical result that is in agreement with Vuillemin’s point of view according to which Descartes’s philosophical system is intuitionistic vuillemin1981,vuillemin1996. Indeed, note that, by contrast with Anselm’s, Descartes’s does not need any specifically classical rule, nor existence predicate.1

Second, once this argument is faithfully formalized in natural deduction, becomes transparent a logical relation between two principles which, according to Gueroult, are basic in Descartes’s theory of truth: the principle of correspondence of true ideas with the objects represented by these ideas, and the principle of causality. Gueroult wrote:

Without the principle of causality, it would be impossible for me to attain the existence of God. Without the principle of the correspondence of the idea with what is ideated, it would be impossible to demonstrate that my idea of God is the image of his being and allows me to know him, since to prove that something is the effect of something else is not sufficient to establish that it resembles it. gueroult1984

The principle of the conformity of the idea with what is ideated only affirms a resemblance between two terms; the principle of causality affirms an equation -ad minimum- between two terms.

And he concluded:

The quantity of perfection contained in an idea cannot be evaluated by its resemblance to the thing, since this resemblance is unknown a priori, and must, on the contrary, be established.gueroult1984

The proof above shows that the resemblance that is the equality of perfection between the idea of God and God is logically deduced from the principle of causality and, moreover, that such a resemblance between an idea and “what is ideated” can be deduced only in the case of the idea of God.

Third, the result of this logical analysis now leads me to a single question: Descartes’s argument being deductively valid, is it sound, I mean, are these five premises all true? For the sake of brevity, I reserve an answer to this question for another publication.

# Bibliography

• [descartes2008a] Descartes, Meditations on First Philosophy: with Selections from the Objections and Replies, OUP Oxford (2008).
• [bonevac2003] Daniel Bonevac, Deduction: Introductory Symbolic Logic, Blackwell Pub. (2003).
• [williams2005] Williams, Descartes: The Project of Pure Enquiry, Routledge (2005).
• [vuillemin1981] Vuillemin, Trois philosophes intuitionnistes: Epicure, Descartes et Kant, Dialectica, 35, 21-41 (1981).
• [vuillemin1996] Jules Vuillemin, Necessity or Contingency: The Master Argument, CSLI Publications (1996).
• [gueroult1984] Martial Gueroult, Descartes’ Philosophy Interpreted According to the Order of Reasons. Vol. 1, The Soul and God, University of Minnesota Press (1984).

## Footnotes:

1

The fact that \eqref{eleven} is intuitionistically valid has been also checked with some provers; see this post.

Date: 2020-06-07 dim. 17:37

Made with Emacs 26.3 (Org mode 9.3.7) and Org export head (Many thanks to Ivan Tadeu Ferreira Antunes Filho!;)