blobs and arrows as basis for graph theory
relators and relationships
Abstractly relators are things that are related, while relationships are how those are related. These things go together in what might be throught of as a structure, or form. Cliology and memetics focuses on a couple of aspects. Memes can be seen as an information structure where components of a meme are assembled to form a larger replicating whole. Memes may be held by individuals, but are transmitted through a population. The pattern of transmission is largely goverend my the social structures – who is communicating wth whom. This social network can easily be seen in terms of relators (the individuals) and relationships (links between individuals).
Graph theory is a branch of mathematics used to describe and depict relators and relationships. Usually the relators are shown as some kind of blob shape (circle etc.) while relationships are indicated by lines between those blobs. Graph theory has its own lexicon, but the terms adopted here are nodes (blobs) and edges (lines). The social network among a population would consist of a set of nodes, one for each individual and shown as a blob, while the social links would be a set of edges shown as lines. Graph theory is used extensivly in modelling social networks and its value has been inherited by cliology.
Relationships can often be directional, such as who transmits a particular meme to whom. In such cases, the edges are shown as arrows, producing a graph known as a flow diagram.
Graph theory is a major theme in computer science, and is widely implemented as software. Hence, it is also useful in the tech-enablement of cliology. The aim, herein, is not mathamatical purity, but rather, practical implementation.
conventional logic symbols (explained by others)
Blobs and arrows are endemic to whiteboards, in both business and academia!
An arrow, again in an abstract sense, often depicts some relationship: flow, causality, inference, indication and so on. Mathematicians have given names to properties of relationships such as reflexive, symmetric and transitive. Most arrows indicate relationships that are neither reflexive nor symmetric but transitive (though there are exceptions) meaning that they indicate a one-way relationship. in the proposition “John is taller than Liz” the is taller than is a relationship that could be depicted by an arrow (as could is shorter than). We would need to know what the arrows mean for any diagram to make sense. As a diagramatic note John and Liz are the set of nodes; blobwise, these could be written with or without a circle drawn around them (ie a transparent blob). A reflexive relationship would be a node with an arrow looping back on itself, while a reflexive relationship would either have a pair of arrows in opposing directions, or a double ended arrow.
Some important arrow-like relationships in propositional logic concern implication, which in computing terms, usually makes for an ‘if… then…’ statement.
The most basic and useful statement is that of material implication aka the rule of inference in the form, ‘x -> y’ . This says that ‘x implies y’ which means that if x is true, then y must also be true. From this we also know that if y is false then x must also be false; and if x is fasle that y could be either true or false. Examples can easily be found on line. For the present purpose,the ‘->’ arrow is important. A whiteboard depiction in a computing lecture might be written as ‘x’ and ‘y’ in transparent blobs with a one way arrow between them. As a proposition, ‘x -> y’ might actually be a false proposition, usually if seen in a larger statement. It could be seen as more like a test which of which the result isn’t necessarily true. Another symbol, a meta-logical one, is ‘=>’ says that the statement is always true. Again, the issue here is not to discuss logic but rather its depiction as blobs and arrows. Furthermore, there are reflexive, or two way versions, such as ‘<->’ and ‘<=>’ which are part of graph theory and amenable to blob and arrow diagrams.
Causality can be modelled similarly. Causality can be thought of as being similar to implication in its reasoning. By way of a idealised example statement, if action a causes some resulting state b, then where a is preformed then b will result. This can also be denoted ‘a -> b’, and blobs and arrows ensue.
Such blobs and arrows diagrams are capable of showing complex networks of things like causality. The dispersal of information in a societal network of individuals has causal aspects therefore the flow diagram would be one portraying the relationships of material implication. Material implication arrows are therefore essential to cliological modelling.
propriatary conventions
Material implication is purely a logical thing. Interpreting an arrow as a causal would be one of the ways that a diagram can be employed. There are numerous applications and interpretations, so to be clear, it is worth declaring the type of relationship explicitly. A simple way suggested here is to simply state the nature of the relationship near to the arrow. The relationship of ’causes’ is the reverse of ‘because’; they are both variations on the form of ‘therefore’: in this convention be expressed anticendent ->causes consequent.
Arrows can be drawn in any direction and so it is the position of the arrow head that determines the direction of the logic. and sentences are read ccw from the arrow head. For western language directions, the relational symbol could be placed under the arrow (or in subscript after it).
So, a proposition like ‘sunshine causes sunburn’ (whether true or not) would, in this convention, be expressed as: sunshine ->causes sunburn
the example would be read top-down, left to right, counter clockwise from the arrow head yielding the original. to reverse, making a ‘because’ statement then ‘because’ would be after the arrow head ccw, therefore above the arrow. Again, a counter-clockwise reading would make sense.
the ccw convention preserves the meaning of the relationship
annotation using this convention allows for meaningful graphs and understanding of complex relationships.
it contributes towards cliology’s aspiration of producing a practical visual calculus for interveining in cultural systems of practice.
Tech enabled allows for data visualisation, static or dynamic, of important information such as the flow of memes and influence within a social network topology.