#### Modality on Graphs

The (modal) logical background of our concept is worded by Johan van Benthem:

“Our basic idea is simply this: we describe properties of directed graphs consisting of points (‘possible worlds’ if you like grandeur) with directed links encoded in an ‘accessibility relation’ between points. A universal modality □ Ø is true at a point in a graph if Ø is true at all points reachable by a directed arrow.”.

(...)

“We interpret formulas in models M = (W, R, V), that may be viewed as directed graphs (W, R) with annotations for proposition letters, given by the valuation V sending each proposition letter p to the set of points V(p) where p is true. When evaluating complex formulas, one can take either the existential or the universal modality as a primitive (both have their comfort zones in logical research):

M, s ⊨ ◊ Ø iff for some t with Rst, M, t ⊨ Ø
M, s ⊨ □ Ø iff for all t with Rst, M, t ⊨ Ø ” (van Benthem, IEP)

Using the above truth definition, the formula ◊□◊p is true at 1,4 but it is false at 2,3.

Further I will no longer use the operator ◊ for possibility since for the proposed application, namely visualisation of laws of nature by structured directed analytic graphs, only necessity is needed. But first we must solve the problem causation has when applying the material implication of formal logic as axiomatised by Frege and Hilbert. A causal implication implies a logical implication, but not the other way around.

(p □→ q) →(p → q)

There are no rules in formal logic to determine if a particular causation is relevant for a given effect neither is there a rule that demands a real world relation between cause and effect. Therefore I will rely on the “Why-Because-Analysis” (WBA) as proposed be the researchers of incidence at the Bielefeld university building on counterfactuals initiated by David Lewis (Ladkin, 2018, p. 2-5). The formalism in WBA enables relative-completeness, objectivity, falsifiability and reproducibility of the analysis results to be as far as possible assured. The result of the WBA is a Why-Because-Graph (WBG). Mathematically, a WBG is a directed, acyclic graph. The nodes of the WBG represent causal factors of an incident. The directed edges represent cause-effect relationships between the factors. Tim Schürmann defines de Counterfactual Test as follows:

“To check the correctness of a cause-and-effect relationship, the Counterfactual Test (CT), based on work of David Lewis and David Hume, is used. "If the (potential) causal factor had not occurred, could the effect have occurred?" If this test is answered with "no", then the potential causal factor is a "necessary causal factor" (abbr. NCF). Use of the CT ensures that all nodes in the WBG are correctly linked.” (Schürman, WBA)

This Counterfactual Test conforms to the demands defined by Alexander Stephanov as “condition sine qua non” in 1985 (Stephanov, 1985). He needed this to define the use of objects in C++ whose libraries were written by him. I will also introduce definitions of a given time and a given location, since I will not only visualise incidents, but also statuses and processes at a given time and place, that even can be repeated at another given time and place but can still be attributed to the same real world.

Structured directed analytic graphs are used in this concept as mathematics without numbers. This doesn't mean it can't be used with numbers. In various research domains, weighted directed graphs are used as an analysis method of datasets in search of causal relationships, 'causal discovery' (Runge,2019). However, the causal relationships depicted in this concept are based on proven scientific knowledge. The graphs themselves are not a proof but a tool to present proofs based on logical and mathematical proof. The conditions formulated will be of course also mathematical expressions but they will be insulated in ‘Prolog’ terms. These must be seen as an exercise in formalisation, but they can also be used to build a formal database of rules and expressions.

#### Declarations for structured directed analytic graphs

Since the edges of the graph fix the order of the cause and effect, there is no need to use numbers. Also the process is fastened between the cause and effect, the order of processes does not need numbers either. So the elementary form of the proposed structured directed analytic graph is:

start_definition /= end_definition. Tautologies are banned

start_definition /= end_definition /= chained_definition. Tautologies are banned. This chain can of course also contain more than two shackles.

A real world example: Plant Transpiration

A (mutually) exclusive disjunction of processes, XOR has the form:

end_definition /= xor_definition. This difference is the reason for mutual exclusion

As in formal logic more than two processes can be all mutually exclusive.

A Real world example: Rain or Snow

A conjunction of processes AND has the form:

As in formal logic more than two processes can be conjunctive causes leading to only one effect.

A concatenation of processes between start_status and end_status is allowed but the these processes ought to be fully defined the way it will be defined below. This concatenation can have the form:

process(definition) /= concat_process(definition). Tautologies are banned

Compared with the incident analysis of WBA I use the same graph structure thought WBA uses different graphical containers, boxes for the incident and sub incidents, ellipses and octagons for the processes connecting to one another but finally ending in a box. WBA only uses a single line description in natural language in each node. Until now I used only boxes, but these will be expanded to tables having more lines and so containing the structured definitions of the nodes. This way the concept op WBA is altered or, let us say, amended.

All status definitions have the same table form. For convenience I introduce the container status(type) that stands for all the different forms of statuses in different contexts. So they are defined als follows:

status(type): {

<thead> = prolog(status_expression) & <tbody> = prolog(condition) &

prolog(condition)…}

Prolog(expression) is specified as follows:

prolog(status_expression): {

element_name, element_quality, element_specification, T|P }

Then we define also T:

T : time specification.

Then we define P:

P : place specification.

Definitions of time specification and place specifications:

time specification: {time range | time description | time moment}

place specification: {specific place| other defined place}

At last the conditions are defined:

prolog(condition) : {result of measurement | result of observation}

For processes I will use process(type) for convenience.

process(type): {

<thead> = prolog(process_description) & <tbody>= prolog(process_condition) &

prolog(process_condition)…}

But for processes the prolog(process_condition) is optional unless it is not defined by a preceding and following status(type), where the difference, in other words the working of the process should normally be well defined in the definition and conditions of start_status and end_status. If there are specific process conditions for a process to initiate and continuate, these conditions must also be given by definitions and conditions.

For the use and application on a digital platform Scalable Vector Graphics (SVG) an XML-based vector image format for defining two-dimensional graphics has been chosen. Although SVG was already defined for the first time by WC3 in 2008 and the second version dates from 2011 (Dahlström et al., 2011) it is less known by the public. For instance you can not upload SVG on social media or blogs. But these SVG is very powerfull, it can also hold and handle hyperlinks to relate to textual proof or context or a mix of textual contexts and other graphical elements.

Using tables a general form looks like this:

#### Empiric verification and Resources

A sdag(type) must contain al least one xlink:href reference to an external document describing the proof of the causal relation. The xlink:href must apply probative information. These can be provided externally in textual format or a combination of textual and graphic format. This can be coded as an HTML link with the "rel" attribute:rel="external" or rel="search". The source of this textual material must be certified from an encyclopedia (e.g. Wikipedia), from a scientific institute (e.g. the ESA climate agency) or created by a teacher. A measurement system analysis is de most strict form of certification.

What can be used as evidence is determined by the international community of scientists. Where and when the evidence is valid is also determined by the international community. But these scientists are only competent in the domain in which they have authority.

It is therefore not sufficient that a mathematical logical proof is consistent and well formulated. This evidence may be deficient for several reasons: (1) incompleteness; (2) not relevant (3) based on incorrect observations or unrepeatable experiments, and so on. It is therefore up to the international community of scientists to decide whether a proof is valid and sufficient.

It becomes complicated when different domains are involved in a problem definition, such as climate change. However, this problem is solved by international consultation and international meetings, where a panel then drafts an interdisciplinary synthesis report, the way IPCC did in 2023.

#### References

Dahlström, Erik et al, (2011) Scalable Vector Graphics (SVG) 1.1 (Second Edition), W3C Recommendation 16 August 2011, <https://www.w3.org/TR/2011/REC-SVG11-20110816/>

Ladkin, Bernard (2018), Causal Analysis of Incidents with Why-Because Analysis using the SERAS® , Software Toolkit, CAUSALIS Ingenieurgesellschaft mbH, 2008, revised 2018-02-14, <https://rvs-bi.de/research/WBA/WBA-NewIntro20180214.pdf>

Lewis, David, (2004). “Causation as Influence” (expanded version), in Collins, Hall, and Paul 2004, 75–106, <https://www.andrewmbailey.com/dkl/Causation_As_Influence_long.pdf>

Menzies, Peter, (2019), Counterfactual Theories of Causation, 29 October 2019, Stanford Encyclopedia of Pholosophy, <https://plato.stanford.edu/entries/causation-counterfactual/>

Reed, Stephen K. (2012). Cognition : theories and applications. Wadsworth, Cengage Learning, 12 April 2012, ISBN 978-1-133-49228-3. OCLC 1040947645, <https://www.worldcat.org/nl/title/1040947645>

Runge, Jakob, et al. , (2019), Detecting and quantifying causal associations in large non-linear time series datasets. Sci. Adv.5,eaau4996 (2019). DOI: <https://www.science.org/doi/10.1126/sciadv.aau4996>

Schürmann, Tim, (WBA) 'Counterfactual Test’, Workgroup RVS, Faculty of Technology, Bielefeld University) <https://rvs-bi.de/research/WBA/IntroWBA-ENG.pdf>

Stepanov, Alexander (1985), Towards a Theory of Causal Implication, Department of Electrical Engineering and Computer Science, Polytechnic University of New York, 1985, <http://stepanovpapers.com/TOWARDS%20A%20THEORY%20OF%20CAUSAL%20IMPLICATION.pdf >

van Benthem, Johan, (IEP), Modal Logic: A Contemporary View, University of Amsterdam, Stanford University, and Tsinghua University, The Netherlands, U. S. A., and China, <https://iep.utm.edu/modal-lo/>

Weatherson, Brian, "David Lewis", (2021)The Stanford Encyclopedia of Philosophy (Winter 2021 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/win2021/entries/david-lewis/>.