(H9) Proof of Incompatibilism¹

Selmer Bringsjord
Philosophy of AI

Here, first, is the basic setup:

• Let `P' denote any true proposition whatsoever.
• Let `L' denote the conjunction of all the laws of nature.
• Let `P<sub>0</sub>' denote the conjunction of all facts about the universe at some point in the remote past, during the age of dinosaurs, say.
• Let the operator `N' be such that the result of prefixing it to some proposition p is to say that `p is true, and no one has, or ever had, any choice about whether p.'
• We employ two rules of inference that seem to be above reproach, viz.,

  $\alpha$

  $\Box p \vdash \mbox{N}p$

  $\beta$

  N( $p \rightarrow q$), N$p \vdash$ Nq

Now here is the proof:

1.

$\Box$((P$_0 \wedge$ L) $\rightarrow$ P) (definition of determinism)

2.

$\Box$(P $_0 \rightarrow$ (L $\rightarrow$ P)) (from 1)

3.

N(P $_0 \rightarrow$ (L $\rightarrow$ P)) (from 2 by $\alpha$)

4.

NP₀ (premise)

5.

N(L $\rightarrow$ P) (from 3 and 4 by $\beta$)

6.

NL (premise)

7.

NP (from 5 and 6 by $\beta$)

Since P can stand for any true proposition, let it stand for your decision to vote for the candidate of your choice. The proof shows that no one has, or ever had, any choice about whether you voted this way or not.