An Argument Against Free Will

In a YouTube Short, Alex O’Connor argues that there is no free will. In this post, I formalise his argument against free will.

Free Will Requires an Internal Cause

If there is an individual who has free will, then there is an individual, an effect and a cause such that the effect is entirely determined by the cause, and that the cause is entirely internal to the individual. Let \(F\) be a one-place predicate for an individual who has free will, \(D\) be a two-place predicate for an effect that is entirely determined by a cause (i.e., an effect that is not random) and \(I\) be a two-place predicate for a cause that is entirely internal to an individual else. In addition, let \(x\) be a variable for an individual, \(y\) be a variable for an effect and \(z\) be a variable for a cause. Then

\[ \exists xF\left(x\right)\rightarrow\exists x\exists y\exists z\left[D\left(y,z\right)\land I\left(z,x\right)\right] \tag{1}\]

In order to demonstrate Equation 1, let us start with the proposition that for all individuals, if the individual has free will, then there is an effect and a cause such that the effect is entirely determined by the cause (i.e., the effect not random), and that the cause is entirely internal to the individual.

\[ \forall x\left\{F\left(x\right)\rightarrow\exists y\exists z\left[D\left(y,z\right)\land I\left(z,x\right)\right]\right\} \tag{2}\]

Indeed, for all individuals, if the individual has free will, then there is an effect and a cause such that the effect is entirely determined by the cause (i.e., the effect is not random); if there is no effect and no cause such that the effect is entirely determined by the cause, then the individual does not have free will. In addition, if the individual has free will, then there is a cause that is entirely internal to the individual; if there is no cause that is entirely internal to the individual, then the individual does not have free will. Therefore, for all individuals, if the individual has free will, then there is an effect and a cause such that the effect is entirely determined by the cause (i.e., the effect not random), and that the cause is entirely internal to the individual.

Having demonstrated Equation 2, let us proceed to demonstrate Equation 1. Suppose that there is an individual such that the individual has free will. Further, suppose that the individual is Adam. From Equation 2, there is an effect and a cause such that the effect is entirely determined by the cause, and that the cause is entirely internal to Adam. In other words, there is an individual, an effect and a cause such that the effect is entirely determined by the cause, and that the cause is entirely internal to the individual. Therefore, if there is an individual who has free will, then there is an individual, an effect and a cause such that the effect is entirely determined by the cause, and that the cause is entirely internal to the individual. Which was to be demonstrated.

There is no Internal Cause

There is no individual, no effect and no cause such that the effect is entirely determined by the cause, and that the cause is entirely internal to the individual.

\[ \lnot\exists x\exists y\exists z\left[D\left(y,z\right)\land I\left(z,x\right)\right] \tag{3}\]

In order to demonstrate Equation 3, let us start with the proposition that for all individuals and effects, either there is a cause such that the effect is entirely determined by the cause, or there is no cause such that the effect is entirely determined by the cause.

\[ \forall x\forall y\left[\exists z D\left(y,z\right)\vee\lnot\exists z D\left(y,z\right)\right] \tag{4}\]

This is equivalent to the proposition that for all individuals, effects and causes, either the effect is entirely determined by the cause, or the effect is not entirely determined by the cause.

\[ \forall x\forall y\forall z\left[D\left(y,z\right)\vee\lnot D\left(y,z\right)\right] \tag{5}\]

For all individuals, effects and [proximal] causes, if the effect is entirely determined by the [proximal] cause, then there is a [distal] cause such that the cause is entirely determined by the [distal] cause. Let y be a variable for a [proximal] cause w be a variable for a [distal] cause. Then

\[ \forall x\forall y\forall z\left\{D\left(y,z\right)\rightarrow\exists w\left[D\left(z,w\right)\right]\right\} \tag{6}\]

For all individuals, effects and [proximal] causes, if the effect is entirely determined by the [proximal] cause, then either there is a [distal] cause such that the [proximal] cause is entirely determined by the [distal] cause, or there is no [distal] cause such that the [proximal] cause is entirely determined by the [distal] cause. If there is no [distal] cause such that the [proximal] cause is entirely determined by the [distal] cause, then the [proximal] cause is both random and not random. Which is absurd. Therefore, for all individuals, effects and [proximal] causes, if the effect is entirely determined by the [proximal] cause, then there is a [distal] cause such that the cause is entirely determined by the [distal] cause.

However, O’Conner goes one step farther and argues without proof that for all individuals, effects and [proximal] causes, if the effect is entirely determined by the [proximal] cause, then there is a [distal] cause such that the [proximal] cause is entirely determined by the [distal] cause, and that the [distal] cause is not entirely internal to the individual.

\[ \forall x\forall y\forall z\left\{D\left(y,z\right)\rightarrow\exists w\left[D\left(z,w\right)\land\lnot I\left(w,x\right)\right]\right\} \tag{7}\]

In particular, he argues that “if it is [the case that whatever it is that’s bringing about actions is not random], it ultimately terminates in something outside of yourself or something random”. Of course, for the abovementioned reason, nothing that brings about actions that is not random ultimately terminates in something random. In other words, “if it is [the case that whatever it is that’s bringing about actions is not random], it ultimately terminates in something outside of yourself [and not] something random”.

Let us also introduce the rather uncontroversial proposition that for all individuals, effects and [proximal] causes, if the [proximal] cause is entirely inside the individual, then there is no [distal] cause such that the [proximal] cause is entirely determined by the [distal] cause, and that the [distal] cause is not entirely internal to the individual.

\[ \forall x\forall y\forall z\left\{I\left(z,x\right)\rightarrow\lnot\exists w\left[D\left(z,w\right)\land\lnot I\left(w,x\right)\right]\right\} \tag{8}\]

After all, if there is a [distal] cause such that the [proximal] cause is entirely determined by the [distal] cause, and that the [distal] cause is not entirely internal to the individual, then the [proximal] cause is both entirely internal to the individual and not entirely internal to the individual. Which is absurd. Therefore, for all individuals, effects and [proximal] causes, if the [proximal] cause is entirely inside the individual, then there is no [distal] cause such that the [proximal] cause is entirely determined by the [distal] cause, and that the [distal] cause is not entirely internal to the individual.

From Equation 7 and Equation 8, we conclude that for all individuals, effects and causes, if the effect is entirely determined by the cause, then the cause is not entirely internal to the individual.

\[ \forall x\forall y\forall z\left[D\left(y,z\right)\rightarrow\lnot I\left(z,x\right)\right] \tag{9}\]

From Equation 5 and Equation 9, we conclude that for all individuals, effects and causes, either the effect is not entirely determined by the cause, or the cause is not entirely internal to the individual.

\[ \forall x\forall y\forall z\left[\lnot D\left(y,z\right)\vee\lnot I\left(z,x\right)\right] \tag{10}\]

This is equivalent to Equation 3. Which was to be demonstrated.

There is no Free Will

From Equation 1 and Equation 3, we conclude that there is no individual such that the individual has free will.

\[ \lnot\exists xF\left(x\right) \tag{11}\]

Which was to be demonstrated.