Rotational Logic

FP Ramsey makes the following observation in the 1927 “Symposium: Facts and Propositions”:

We might, for instance, express negation not by inserting a word “not,” but by writing what we negate upside down. Such a symbolism is only inconvenient because we are not trained to perceive complicated symmetry about a horizontal axis, and if we adopted it we should be rid of the redundant “not-not,” for the result of negating the sentence “p” twice would simply be the sentence “p” itself. (p161-2) $\newcommand{\RR}[2]{\mathrel{\style{display: inline-block; transform: rotate(#1deg)}{#2}}}$ $\newcommand{\dotcup}{\mathbin{\unicode{x228D}}}$ $\newcommand{\dotg}{\mathbin{\unicode{x22D7}}}$

Ramsey’s observation is a side comment in a larger argument about the nature of relations between propositions, but it suggests a different approach to logical notation. Rather than writing $\lnot p$, we can write $\RR{180}{p}$.¹ Perhaps surprisingly, this rotated negation extends to more complex formulae, capturing the De-Morgan laws. $$\RR{180}{(p \lor q)} \leftrightarrow \lnot (p \lor q) \leftrightarrow (\lnot p \land \lnot q)$$ Using the standard notation for disjunction ($\lor$) and conjunction ($\land$), we can easily read off the meaning of a rotationally negated formula simply by treating all logical connectives as they appear, and rotated propositions as being negated.

Unfortunately, this happy situation does not remain for other connectives. If $\supset$ is the material conditional, it is natural to interpret $\RR{180}{\supset}$ as the material conditional in the other direction, but this does not correspond to negation. $$\RR{180}{(p\supset q)} \leftrightarrow \lnot(p \supset q) \leftrightarrow (p \land \lnot q) \not\leftrightarrow (\lnot q \supset\lnot p)$$ Interpreting $\RR{180}{\supset}$ simply as the negation of $\supset$ feels inelegant. Instead, we should hope that the rotated notation intuitively conveys its meaning.

This blog post is an attempt to elaborate on what such a rotational notation for logic would look like. During the course of exposition, we will discover a new branch of logic (rotational logic) which deals with invariance properties over permutations of logical connectives.

Rotational Logic

Of Ramsey’s observation, we will keep only the idea of rotating formulas. In fact, we will make a great number of changes. Where for Ramsey, rotation gives negation, for us it will merely result in a change of logical connectives, not necessarily negation. Where Ramsey rotated propositional letters, we will keep them upright, and rotate only the connective. Where Ramsey’s formulas were rotated overall ($(p \lor q)$ becomes $\RR{180}{(p \lor q)}$), ours will be rotated in place ($(p \RR{180}{\land}q)$ becomes $(p \land q)$). Finally, where Ramsey employs only 180 degree rotations, we will also utilise 90 degree rotations.

Formally, the language of rotational logic is defined as follows: $$\varphi:: = p \mid (\varphi\land\varphi) \mid (\varphi\supset\varphi) \mid {\varphi}^\circ \mid {\varphi }^\cdot$$ The notation ${\varphi}^\circ$ is understood to mean a single 90-degree rotation of the connectives in a formula. I will abbreviate this by writing: $$\begin{aligned} {p}^\circ &= p \\ {(\varphi\sqcup\psi)}^\circ &= ({\varphi}^\circ\RR{-90}{\sqcup}{\psi}^\circ), \end{aligned}$$ where $\sqcup$ is any binary operation (in any rotation). The notation ${\varphi}^\cdot$ indicates a change in the first truth value in a truth table for each connective. I will abbreviate this by writing a $\cdot$ inside each connective.

The language is fundamentally propositional, so we give semantic definitions in terms of truth tables. The definition of each symbol depends on the rotation of that symbol. We begin with the semantics for the point symbol. $$\begin{array}{c c | c | c | c | c} \varphi & \psi & (\varphi\land\psi) & (\varphi\RR{-90}{\land}\psi) & (\varphi\RR{180}{\land} \psi) & (\varphi\RR{90}{\land}\psi)\\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}$$ As can be seen, the $\land$ symbol matches closely with the standard conjunction and disjunction symbols. The $\RR{-90}{\land}$ symbol can be read as “the truth value on the left”, and $\RR{90}{\land}$ can be read as “the truth value on the right”. In this way, the symbols and their rotations align with how we expect them to behave.

We have a similar correspondence with $\supset$, which behaves identically to the material conditional. $$\begin{array}{c c | c | c | c | c} \varphi & \psi & (\varphi\supset\psi) & (\varphi\RR{-90}{\supset}\psi) & (\varphi\RR{180}{\supset} \psi) & (\varphi\RR{90}{\supset}\psi)\\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & 1 & 1 \end{array}$$ As can be seen, $\RR{180}{\supset}$ behaves like material conditional in the other direction. $\RR{-90}{\supset}$ is the biconditional and $\RR{90}{\supset}$ is binary tautology.

For both sets of truth tables, the behaviour of the unfamiliar operators $\RR{-90}{\supset}$, $\RR{90}{\supset}$, $\RR{-90}{\land}$ and $\RR{90}{\land}$ seems at first arbitrary. The justification for their definitions can be seen in how we deal with ${\varphi}^\cdot$. The reader will notice that in all truth functions thus far described, the output is $1$ when both $\varphi$ and $\psi$ have true value $1$. The $\cdot$ operator expands the system by changing the value of the first bit. In particular, we have the following truth tables: $$\begin{array}{c c | c | c | c | c} \varphi & \psi & (\varphi\RR{-90}{\dotg}\psi) & (\varphi\RR{-180}{\dotg}\psi) & (\varphi\RR{90}{\dotg} \psi) & (\varphi\RR{0}{\dotg}\psi)\\ 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}$$ $$\begin{array}{c c | c | c | c | c} \varphi & \psi & (\varphi\RR{-90}{\dotcup}\psi) & (\varphi\RR{180}{\dotcup}\psi) & (\varphi\RR{90}{\dotcup} \psi) & (\varphi\RR{}{\dotcup}\psi)\\ 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & 1 & 1 \end{array}$$

So where is the justification for our choice of semantics? Well, we now have a relationship between the pointed operators $\land$ and the curved operators $\supset$! What relationship, I hear you cry? Consider a truth table for $\supset$ and $\dotg$. $$\begin{array}{c c | c | c} \varphi & \psi & (\varphi \supset\psi) & (\varphi\dotg\psi)\\ 1 & 1 & 1 & 0\\ 1 & 0 & 0 & 1\\ 0 & 1 & 1 & 0\\ 0 & 0 & 1 & 0 \end{array}$$ The table for $\dotg$ is the table for the negation of $\supset$! In fact, any binary operation can be negated by switching between curve and point, and adding a dot. We leave it to the reader to check this correspondence.

Let us summarise what we have defined. We now have a language with a separate operator for each of the 16 possible binary boolean functions. Our language intuitively allows us to select an operation to apply, by combining rotations and dots. There is a natural correspondence between the curved and the pointed symbols. Further, as the reader can check, every pointed symbol takes $\varphi = \psi = 0$ to be false, and every curved to be true. The system is made even more elegant by the lack of unary operators, without losing expressiveness. To write $\neg p$, we can instead use $(p\RR{-90}{\dotcup}\varphi)$ for any formula $\varphi$ (again, the reader is encouraged to check this).

Defining Classical Logic

Call a formula rotationally invariant if it gives the same truth values on every rotation. Is every formula equivalent to a rotationally invariant formula? More specifically, are we able to define rotationally invariant versions of the classical operators?

Negation is easy. Take $$\neg\varphi := \varphi\dotcup\varphi.$$ It is easy to check this is equivalent to our classical $\neg \varphi$ for every rotation. Now can we obtain classical conjunction, $\&$? Indeed we can. Take $$\varphi\mathbin{\&}\psi := (((\psi\RR{180}{\land}\varphi) \RR{90}{\land} \psi ) \land(\varphi \RR{-90}{\land} (\psi\RR{180}{\land}\varphi) )).$$ Then we have $$\begin{aligned} (\varphi\mathbin{\&}\psi) &= (((\psi\RR{180}{\land}\varphi) \RR{-90}{\land} \psi ) \land(\varphi \RR{90}{\land} (\psi\RR{180}{\land}\varphi) )) \\&\approx (\psi \land\varphi) \\\\ {(\varphi\mathbin{\&}\psi)}^\circ &= (((\psi\RR{-90}{\land}\varphi) \land\psi ) \RR{90}{\land} (\varphi \RR{180}{\land} (\psi\RR{-90}{\land}\varphi) ))\\ &\approx (((\psi\RR{-90}{\land}\varphi) \land\psi ) \\&\approx (\varphi \land\psi) \\\\ {{(\varphi\mathbin{\&}\psi)}^\circ}^\circ &= (((\psi\land\varphi) \RR{90}{\land} \psi ) \RR{180}{\land} (\varphi \RR{-90}{\land} (\psi\land\varphi) )) \\ &\approx (((\psi\land\varphi)) \RR{180}{\land} ((\psi\land\varphi) )) \\&\approx (\varphi \land\psi) \\\\ {{{(\varphi\mathbin{\&}\psi)}^\circ}^\circ}^\circ &= (((\psi\RR{90}{\land}\varphi) \RR{180}{\land} \psi ) \RR{-90}{\land} (\varphi \land(\psi\RR{90}{\land}\varphi) )) \\ &\approx(\varphi \land(\psi\RR{90}{\land}\varphi) )) \\&\approx (\varphi \land\psi) \end{aligned}$$