A Fregean Diagramatic Notation for Classical Logic

One of the main philosophical peeves of mine are philosophers confounding symbols and what they represent, and in particular mistaking features of logical notation with actual logical facts. For example, is double negation an actual logical rule or is it actually a notational convention of certain logical notations? It seems to be the later, since we can have expressively equivalent logical notations where double negation is not even expressible, for example, if we consider diagrammatic systems where negation is represented by a reversible transformation of diagrammatic elements (Monroy forthcoming). For similar reasons, one might consider the  commutativity (of disjunction or conjunction) not as an actual logical rule but a notational convention of certain logical notations, since we can have expressively equivalent logical notations where commutativity is not even expressible, for example, if we consider expressions that are not sequences but sets of symbols. . 

It is very difficult to represent unary operators non-symbolically – impossible if you also want them to be iterable. Thus, most if not all so-called diagrammatic methods for logic usually still contain a symbol for negation (for example Carroll's trees, Cook Wilson’s hanging plant diagrams, Dave Beisecker and Amirouche Moktefi’s diagramatic methods, etc.)

Variables are ubiquitous, not only in logic and mathematics, but in all sorts of formal representations. Despite their ubiquity, they are not essential to all formal representations, since diagrammatic systems might prescind of them. 

Variables  play two essential roles in formal systems of representations, both diagrammatical and sequential: On the one hand, they serve to mark argument places; and on the other, they serve to mark intra- and trans-configurational argumentative coordination. Argumentative coordination is the notion developed by Fine (2007) to explain what variables do in formal languages. The idea, in short, is that two argument places are coordinated if the value assigned to one argument place restricts or determines the value assigned to the other. In most formal languages, for example, variables serve to indicate which argument places must be assigned the same value and which ones not. Two argument places marked by the same variable (that is, occupied by different tokens of the same variable type) indicate that they must be Assigned the same value.  In order to extend his account to other systems of representation I extent Panza and Longa’s distinction (2021) between inter- and trans-configurational analysis. Applying this distinction to Fine’s point, the basic idea is that, within a single configuration of  formulas or a single diagram, variables serve to indicate that two arguments are to be coordinated. In most cases, this means that they must be assigned the same value in interpretation. In multi-configurational contexts, in contrast, they allow us to identify syntactic items across modalities and configurations, so that we can see when an element in a formula, for example, corresponds to an element in a diagram, allow us to refer to an element of a diagram in a text, or to talk about the element of a formula in speech, etc. (as Netz 2000 has emphasized) This trans-configurational role of variables is essential to the multi-modal nature of most if not all mathematical argumentation.

Coordination, of course, is fundamentally a relation and as such can be notated any of the many ways we have developed to visually represent relations. (Barceló 2022) Variables are the traditional way of notating coordination, but we can also use external icons like  connecting lines.

So, we could easily have a diagrammatic system of representation for (classical) propositional logic without variables. This is what I will develop now:

It is very easy to represent iterable operators and relations of n-arity of more than one with external icons like connecting lines or parentheses; but if we have more than one of them (of the same n-arity) in a single system, most systems opt for symbolic representation, even if the same different can be marked through internal distinctions – width of line, square vs circles, etc. or external ones – like location, direction or inclination. This is the situation we face if we try to represent argument correlation iconically instead of symbolically, and in the following proposal this is just what I will do: I will use a simple external device for representing binary relations – a half circle connecting the relata – and distinguish between the two binary relations by location and direction using an imaginary baseline, so that half circles above this baseline represent the only binary logical operator in the system, while half circles under this same baseline represent the binary relation of coordination between arguments. In order for the edges of the semicircles to match, the curves above the baseline must be open at the bottom, while the curves underneath the baseline must be open at the top.

Divide diagrams in two parts: the top one for the operational part and the bottom one for the coordinative part – the top for operations and the bottom for their arguments, so to speak. Use curves and strikes on the top part for atomic (strike) and molecular (connecting curves) formulas. Use curves at the bottom for coordination – connect tow edges of the operational part of the diagram when they correspond to the same proposition. Use marks for the unary operator (negation) at edges or crests. If we want our system to be able to express double negation, we can use Frege’s technique of allowing for extra strokes (In other words, use Frege’s system for the operational part and add connecting curves for the coordinative part). If the system is single-conclusion, there is no need for a symbol separating premises and conclusions: it is enough to adopt the convention that the rightmost (operationally complete) diagram represents the conclusion (the inverse convention would do as well).

Let’s start with a simple Fregean diagrammatic language:

Inductive basis: A vertical stroke is a well formed operational diagram.

Inductive steps: 

For material implication: Given two well formed operational diagrams, joining their tops with a curved line above them is another well formed operational diagram.

For negation: Given a well formed operational diagram, adding a small black circle at its top  is another well formed operational diagram.

Closure clause: Nothing else is a well formed operational diagram.

Coordinative clause: Adding at least zero curves underneath a well formed operational diagram joining its edges results in a well formed diagram.

Notice that operational diagrams are just Begriffschrift formulas turned 90 degrees clockwise with curves instead of straight lines. What is original of this diagrammatic system is that instead of using propositional variables, we use curves also to represent coordination, i.e., when different argument places ought to be filled by the same (propositional) values.

A stylized variation:

Inductive bases: 

A vertical stroke is a well formed operational diagram. 

A half a circle with its edges down is a well formed operational diagram.

Terminological convention: Call a well formed operational diagram’s rightmost edge, its basis.

Inductive steps: 

For material implication: Given two well formed operational diagrams, joining their bases with a curved line above them is another well formed operational diagram.

For negation: Given a well formed operational diagram, adding a small black circle at  either its top or any of its edges is another well formed operational diagram.

Closure clause: Nothing else is a well formed operational diagram.

Coordinative clause: Adding at least zero curves underneath a well formed operational diagram joining its edges results in a semi-well formed diagram.

Stylizing clause: Deleting from a semi-well formed diagram all un-attached strokes results in a well-formed diagram.

Now, let us see how this works by translating a formula in our traditional notation of arrows, parentheses and letters into this diagrammatic notation:

Consider Łukasiewicz's Axiom of Material Implication:

((P → Q) →R) →((R →P) →(S →P))

The first and simplest step is to turn the arrows and parentheses into curves, one by one, starting from the basic ones:

For the curves that stand for the arrows that link molecular formulas, remember that the convention is to link the rightmost edges of the curves. This convention is clearly arbitrary. We could've chosen to link the rightmost edge, the top or any other point of the curve. There might be pragmatic ergonomic or aesthetic advantages in each one of these choices, but they all are equally apt for representing this sort of relation. (Frege famously thought that, for example, picking the topmost edge in his conceptual notation had the cognitive advantage of being more intuitive for mathematicians already used to making proofs by having premises above conclusions.)

Until finally reaching the main operator.

So far, this is mostly just Frege's Conceptual Notation but with curves instead of straight lines, and horizontal instead of vertical.

It is also interesting to notice that in this diagram, the main operator is very easy to identify in so far as it is the largest and topmost curve, and in general, shorter curves are more basic. However, the inverse is untrue. If we look at the two leftmost curves, the leftmost is subordinated to the second one, yet both

are the same size. This shortcoming of this notation could not be resolved by choosing the leftmost edge, but can be solved by choosing the middle of the curve or some other point in the curve. (Notice that Frege solved this problem by using straight lines instead of curves and demanding that the lines do not link edge to edge, as my notation does).

Now for the bottom part of the diagrams, we use also curves to link argument spaces that are coordinated, this is, the spaces marked by the same letters in the traditional notation, thus:

Notice that now, all the information contained in the original expression is now included in the curves that constitute the diagram.

Examples of well formed diagrams of this stylized freeman diagrammatic logical notation:

Axiom 1 of Implication from Mendelson

Modus Ponens

Transposition

Modus Tollens

Also, but perhaps not as aesthetically pleasing, we could have a diagrammatic system of representation for (classical) propositional logic where variables were used only for operations, but not for coordination among their arguments. Very likely, this might require TWO unary operators: one for assertion and another for negation. The resulting system would be very similar to my 2000 system and, therefore, I assume, to the diagrammatic systems of Cook and of Bolander.

Modus Ponens

Modus Tollens

Axiom 1 of Implication from Mendelson

Axiom 1 of Implication from Mendelson (Alternative)

Material Implication (Negative)

The problem here is that it is very likely that we will need an extra mechanism to distinguish premises and conclusion.