What is a visual inference?

 A relation or property R between characters/tokens is meaning carrying (in a representational system) if there is a (possibly derived) semantic rule/generalization such that for all x, y if R(x, y) then I(R)(I(x), (I(y)). [Analogously, A relation or property R* between characters/tokens is derivability relevant if there is a (possibly derived) derivation rule/generalization which appeals to R. A relation or property R is syntactic if it is either meaning carrying or derivability relevant] We also say that I(R)(I(x), (I(y)) is observed (in the derived sense) from a representation if one actually observes (in the strict sense) that R(x, y) [that R(x, y) is called a ‘statement' by Stapleton et al. and a ‘representing fact’ by Shin] and derives I(R)(I(x), (I(y)) from R(x, y) through an adequate semantic inference. 

The simple schema of representations for information transmission requires codification and decoding, i.e., the transmitter starts with some information, applies the semantic rules of the system to produce a representation that contains that information and then the receiver performs an inverse process applying the same semantic rules to recover the information contained in the representation. If we want to model this process as a sort of (perceptual) inference, its form would be something like A|=A. Therefore, it is of no use, but as a starting point, to understand visual inference. In normal cases of visual inference, the inferrer starts with some information, applies the semantic rules of the system to produce a representation that contains that information (so, so far there is no difference between what happens in simple information transmission) and then, usually but not necessarily the same agent performs a independent process of information extraction that, hopefully, will deliver information that is somehow contained in the representation, but is different from the information that was explicitly codified in the construction of the representation.

More formally, this means that some meaning carrying relations are exploited in the construction of the representation – they correspond to what is given in the problem, i.e., to the premises of the visual inference so to speak – while others are exploited for inferring new information. For example, in Euler diagrams with existential import, using a closed curve to represent a non-empty set makes the fact that the set thus represented is not empty observable (in the derived sense, because that there is a closed curve in the diagram is observable in the strict sense), but given. A visual inference is an inference such that premises are represented in the diagram through meaning carrying relations, and the conclusion is observed. The inference is truly visual if the conclusion was not given. Notice that, in strict sense, since not all meaning-carrying relations need be observable, it is possible for A to be not visually inferable from itself for at least some A. In normal diagrams, i.e., those where the meaning-carrying relations exploited in representing what is given are observable, A is usually visually inferable from itself, but not truly visually inferable. Notice also that derivations are not visual inferences.

A system of representations is observational devoid, in my sense, which is very close to Stapleton et al.’s, if nothing is observable unless it was given – and truly observational devoid if every observable fact involves the observation of the meaning carrying relation used to represent it as given. A system of representations is observational complete if every fact in its target domain is observable.