## Theme One • A Program Of Inquiry 20

Back in the day when I was making The Big Bucks (time-adjusted dollars) consulting on research statistics in bioscience-medical-nursing-public-health settings, I noticed a certain analogy between propositional calculus research (PCR0) and polymerase chain reactions (PCR1).  I was going to say something about it on a previous thread where these topics collided but then I lost track of the links I needed, so I will go dig them up now.

## Sign Relations, Triadic Relations, Relation Theory • Discussion 2

Dear Edwina,

Analytic frameworks, our various theories of categories, sets, sorts, and types, have their uses but they tend to become à priori, autonomous, top-down, and top-heavy unless they are supported by a robust population of concrete examples arising in practical experience, one of the things the maxim of pragmatism advises us to remember.  That is why Peirce’s tackling of information and inquiry is even-handed with respect to their extensional and intensional sides.  And it’s why we need to pay attention when anomalies accumulate and the population of presenting cases rebels against the dictates of Procrustean predicates.  Times like that tell us we may need to reconceive our customary conceptual frameworks.

As it happens, I’ve been thinking a lot lately about a particular class of sign sequences, namely, proofs in propositional calculus regarded as cases of sign process, or semiosis.  Naturally I’ve been thinking of delving more deeply into Robert Marty’s work on paths through the lattice of sign classes but so far I’m still in the early stages of that venture.

For what they’re worth, here are my blog posts so far on Proof As Semiosis.

## Sign Relations, Triadic Relations, Relation Theory • Discussion 1

ET:
I particularly like your comment that “signhood is a role in a triadic relation, a role that a thing bears or plays in a given context of relationships — it is not an absolute, non-relative property of a thing-in-itself, one that it possesses independently of all relationships to other things”.
I myself emphasize that this context of the role is made up of relationships (plural) — which gives the triad its capacity for complexity.  Therefore, as we see in Robert Marty’s lattice, a thing is never a thing-in-itself but is an action, a process, composed of complex relations.

Dear Edwina,

Things grow complex rather quickly once we start to think about all the roles a sign may play on all the stages where it struts and frets its parts.  There is no unique setting, no one scene, but concentric and overlapping contexts of relationship all have their bearing on the sign’s significance.

One strategy we have for dealing with these complexities and avoiding being overwhelmed by them is to build up a stock of well-studied examples, graded in complexity from the very simplest to the increasingly complex.  The wide world may always present us with situations more complex than any in our inventory of familiar cases but the better our stock of ready examples the more aspects of novel situations we can capture and the greater our odds of coping with them.

## Sign Relations, Triadic Relations, Relation Theory • 1

To understand how signs work in Peirce’s theory of triadic sign relations, or “semiotics”, we have to understand, in order of increasing generality, sign relations, triadic relations, and relations in general, each as conceived in Peirce’s logic of relative terms and the corresponding mathematics of relations.

Toward that understanding, here are the current versions of articles I long ago contributed to Wikipedia and Wikiversity and continue to develop at a number of other places.

## Theme One • A Program Of Inquiry 19

It’s the usual thing to say scientific inquiry involves a combination of deductive and inductive reasoning.  A slightly different, 3-phase model, going back to Aristotle and revived by Charles S. Peirce, analyzes the process producing knowledge into abductive, deductive, and inductive stages.  Abductive inference is used to generate a hypothesis, deduction is used to derive its logical consequences, and inductive reasoning is how we test the hypothesis against experimental observations.

Here’s a few thoughts toward the design of software platforms for integrating these three components of inquiry.  (Also research and teaching.)

## Animated Logical Graphs • 44

RS:  DNA and proteins might be good places to look for logical graphs in nature since our tech for mapping those structures has become fairly proficient lately.  Do you think we could train some kind of neural net to find the patterns?  Might that lead to a real breakthrough in computational microbiology?

Dear Richard,

Models of neural nets are extremely various.  I don’t especially cotton to the ones based on threshold computation, as I think they’re bound to remain rather dumb.   I view all those blinking neurons as something like a night view of the earth’s cities from space.  What we see is only a measure of the raw power consumption occurring in the cities, buildings, and homes, not anything like the actual processes going on inside those sites.

### Reference

• Leibniz, Gottfried W. (1679–1686?), “Addenda to the Specimen of the Universal Calculus”, pp. 40–46 in G.H.R. Parkinson (ed., trans., 1966), Leibniz : Logical Papers, Oxford University Press, London, UK.

## Animated Logical Graphs • 43

RS:  I wonder if we might find such graphs in the physical microstructures of brains, cells, proteins, etc.

Dear Richard,

You are reading my mind.  See the following post on the Standard Upper Ontology List, where I took a simple example of a propositional expression and proceeded by way of logical graphs to prove its equivalence to a syntactically simpler expression.

Reflecting on the form of the proof, I concluded with the following remark.

JA:  For some reason I always think of that as the way that our DNA would prove it.

There’s further discussion of that example at the following location.

### Reference

• Leibniz, Gottfried W. (1679–1686?), “Addenda to the Specimen of the Universal Calculus”, pp. 40–46 in G.H.R. Parkinson (ed., trans., 1966), Leibniz : Logical Papers, Oxford University Press, London, UK.

## Animated Logical Graphs • 42

Re: Richard J. LiptonLogical Complexity Of Proofs
Re: Animated Logical Graphs • (35) (36) (37) (38) (39) (40) (41)

Now that our propositional formula is cast in the form of a graph its evaluation proceeds as a sequence of graphical transformations where each graph in turn belongs to the same formal equivalence class as its predecessor and thus of the first.  The sequence terminates in a canonical graph making it manifest whether the initial formula is identically true by virtue of its form or not.

To be continued …

### Reference

• Leibniz, Gottfried W. (1679–1686?), “Addenda to the Specimen of the Universal Calculus”, pp. 40–46 in G.H.R. Parkinson (ed., trans., 1966), Leibniz : Logical Papers, Oxford University Press, London, UK.

## Pragmatic Semiotic Information • Discussion 20

A little bit of history recoded …

It may be worth noting the Information Revolution in our understanding of science began in the mid 1860s when C.S. Peirce laid down what he called the “Laws of Information” in his lectures on the “Logic of Science” at Harvard University and the Lowell Institute.  Peirce took up “the puzzle of the validity of scientific inference” and claimed it was “entirely removed by a consideration of the laws of information”.

Here’s a collection of excerpts and commentary I assembled on the subject.

## Animated Logical Graphs • 41

Re: Richard J. LiptonLogical Complexity Of Proofs
Re: Animated Logical Graphs • (35) (36) (37) (38) (39) (40)

Last time we looked at a formula of propositional logic Leibniz called a Praeclarum Theorema (PT).  We don’t concur it’s a theorem, of course, until there’s a proof it’s identically true and Leibniz gave an argument to demonstrate that.  Written out in one of our more current formalisms, PT takes the following form.

$((a \Rightarrow b) \land (d \Rightarrow c)) \Rightarrow ((a \land d) \Rightarrow (b \land c))$

Somewhat in the spirit of Reduced Instruction Set Computing, we reformulated PT in a propositional calculus using just two primitive operations, writing the logical negation of a proposition $p$ as $\texttt{(} p \texttt{)}$ and the logical conjunction of two propositions $p, q$ as $pq.$  That gave us a text string in teletype parentheses and proposition letters, formatted two ways below.

Our next transformation of the theorem’s expression exploits a standard correspondence in combinatorics and computer science between parenthesized symbol strings and trees with symbols attached to the nodes.

We can see the correspondence between text and tree in the case of PT by starting at the root of the tree and reading off the characters of the text string as we traverse the edges and nodes of the tree in the following manner.  The initial $``\texttt{(}"$ tells us to ascend the first edge, the next $``\texttt{(}"$ tells us to ascend the next edge on the left, where we find the letter $``a"$ from the string checks with the letter $``a"$ attached to the node of the tree where we are.  Another $``\texttt{(}"$ takes us up another edge, where we find the letter $``b"$ from the string checks with the letter $``b"$ on the current tree node.  Reading the first $``\texttt{)}"$ on the string entitles us to descend an edge and reading another $``\texttt{)}"$ gives us licence to descend another.  The way of things is most likely clear by this point — at any rate, I leave the exercise to the reader.

On the scene of the general correspondence between formulas and graphs the action may be summed up as follows.  The tree, called a parse tree or parse graph, is constructed in the process of checking whether the text string is syntactically well-formed, in other words, whether it satisfies the prescriptions of the associated formal grammar and is therefore a member in good standing of the prescribed formal language.  If the text string checks out, grammatically speaking, we call it a traversal string of the corresponding parse graph, because it can be reconstructed from the graph by a process like that illustrated above called traversing the graph.

To be continued …

### Reference

• Leibniz, Gottfried W. (1679–1686?), “Addenda to the Specimen of the Universal Calculus”, pp. 40–46 in G.H.R. Parkinson (ed., trans., 1966), Leibniz : Logical Papers, Oxford University Press, London, UK.