THE BLANK PAGE DENOTES TRUTH

My search for identity, the subject of my enquiry, has so far exhibited little logic.

Logic. noun

1. the science that investigates the principles governing correct or reliable inference.

2. a particular method of reasoning or argumentation: We were unable to follow his logic.

3. the system or principles of reasoning applicable to any branch of knowledge or study.

4. reason or sound judgment, as in utterances or actions: There wasn't much logic in her move.

5. convincing forcefulness; inexorable truth or persuasiveness: the irresistible logic of the facts.

6. Computers. logic circuit.

Peirce

I should have started with Charles Sanders Peirce. Peirce argued that logic is formal semiotic, the formal study of signs in the broadest sense, not only signs that are artificial, linguistic, or symbolic, but also signs that are semblances or are indexical such as reactions. Peirce held that "all this universe is perfused with signs, if it is not composed exclusively of signs", along with their representational and inferential relations. He argued that, since all thought takes time, all thought is in signs and sign processes ("semiosis") such as the inquiry process. He divided logic into: (1) speculative grammar, or stechiology, on how signs can be meaningful and, in relation to that, what kinds of signs there are, how they combine, and how some embody or incorporate others; (2) logical critic, or logic proper, on the modes of inference; and (3) speculative or universal rhetoric, or methodeutic, the philosophical theory of inquiry, including pragmatism.

Peirce's most important work in pure mathematics was in logical and foundational areas. He also worked on linear algebra, matrices, various geometries, topology and Listing numbers, Bell numbers, graphs, the four-colour problem, and the nature of continuity. He worked on applied mathematics in economics, engineering, and map projections (such as the Peirce quincuncial projection), and was especially active in probability and statistics. Peirce made a number of striking discoveries in formal logic and foundational mathematics, nearly all of which came to be appreciated only long after he died. He died destitute in Milford, Pennsylvania, on the 19^th April 1914.

Professor Robert Burch, Professor of Philosophy at Texas A&M University commented:

Burch

Currently, considerable interest is being taken in Peirce's ideas by researchers wholly outside the arena of academic philosophy. The interest comes from industry, business, technology, intelligence organizations, and the military; and it has resulted in the existence of a substantial number of agencies, institutes, businesses, and laboratories in which on going research into and development of Peircean concepts are being vigorously undertaken. (2010)

Robert Burch works primarily on the logical theories of C. S. Peirce and Josiah Royce. His current special interest is Inductive Logic, Abductive Logic, and the computer instrumentation of these kinds of logic. His published work includes the first proof of Peirce's Reduction Thesis and his computer programs are widely used in connection with problems in homeland security.

One of Peirce’s innovations was the development of the existential graph, which is a type of diagrammatic or visual notation for logical expressions. What I particularly like are the syntax and semantics of the Alpha graph.

Alpha Graphs

Peirce proposed three systems of existential graphs:

•    alpha, isomorphic to sentential logic and the two-element Boolean algebra;

•    beta, isomorphic to first-order logic with identity, with all formulas closed;

•    gamma, (nearly) isomorphic to normal modal logic.

Alpha nests in beta and gamma. Beta does not nest in gamma, quantified modal logic being more than even Peirce could envisage.

Herewith the Alpha:

The syntax is:

The blank page;

•    Single letters or phrases written anywhere on the page;

•    Any graph may be enclosed by a simple closed curve called a cut or sep. A cut can be empty. Cuts can nest and concatenate at will, but must never intersect.

Any well-formed part of a graph is a subgraph.

The semantics are:

•    The blank page denotes Truth;

•    Letters, phrases, subgraphs, and entire graphs may be True or False;

•    To enclose a subgraph with a cut is equivalent to logical negation or Boolean complementation. Hence an empty cut denotes False;

•    All subgraphs within a given cut are tacitly conjoined.

Hence the alpha graphs are a minimalist notation for sentential logic, grounded in the expressive adequacy of And and Not. The alpha graphs constitute a radical simplification of the two element Boolean algebra and the truth functors..

The depth of an object is the number of cuts that enclose it.

Rules of inference:

•    Insertion - Any subgraph may be inserted into an odd numbered depth.

•    Erasure - Any subgraph in an even numbered depth may be erased.

Rules of equivalence:

•    Double cut - A pair of cuts with nothing between them may be drawn around any subgraph. Likewise two nested cuts with nothing between them may be erased. This rule is equivalent to Boolean involution.

•    Iteration/Deiteration – To understand this rule, it is best to view a graph as a tree structure having nodes and ancestors. Any subgraph P in node n may be copied into any node depending on n. Likewise, any subgraph P in node n may be erased if there exists a copy of P in some node ancestral to n (i.e., some node on which n depends). For an equivalent rule in an algebraic context, see C2 in Laws of form.

A proof manipulates a graph by a series of steps, with each step justified by one of the above rules. If a graph can be reduced by steps to the blank page or an empty cut, it is what is now called a tautology (or the complement thereof). Graphs that cannot be simplified beyond a certain point are analogues of the satisfiable formulas of first order logic.

Beta

Peirce notated predicates using intuitive English phrases; the standard notation of contemporary logic, capital Latin letters, may also be employed. A dot asserts the existence of some individual in the domain of discourse. Multiple instances of the same object are linked by a line, called the "line of identity". There are no literal variables or quantifiers in the sense of first-order logic. A line of identity connecting two or more predicates can be read as asserting that the predicates share a common variable. The presence of lines of identity requires modifying the alpha rules of Equivalence.

The beta graphs can be read as a system in which all formula are to be taken as closed, because all variables are implicitly quantified. If the "shallowest" part of a line of identity has even (odd) depth, the associated variable is tacitly existentially (universally) quantified.

Gamma

Add to the syntax of alpha a second kind of simple closed curve, written using a dashed rather than a solid line. Peirce proposed rules for this second style of cut, which can be read as the primitive unary operator of modal logic.

And then there is this map - the Peirce quincuncial projection is a conformal map projection (except for four points where its conformality fails) that presents the sphere as a square. It was developed by Charles Sanders Peirce in 1879.

Peirce quincuncial projection of the world. The red equator is a square whose corners are the only four points on the map which fail to be conformal.