Thursday, April 12, 2018

Logical Representations

An odd claim in the SEP article on logical diagrams (by Shin, Lemon, and Mumma):

Once a purely intuitive notion, non-psychological claims about “efficacy” of diagrammatic systems can be examined in terms of standard formal properties of languages (Lemon et al. 1999). In particular, many diagrammatic systems are self-consistent, incorrect, and incomplete, and complexity of inference with the diagrams is NP-hard. By way of contrast, most sentential logics, while able to express inconsistencies, are complete and correct.

But the last sentence is certainly false. For any complete and correct sentential representations of logical relations you can make modifications that give you a sentential representation that is not complete or correct, and often multiple different such representations. The only thing that can be meant here is that most sentential logics, specifically proposed as complete and correct in a context of inquiry that vets sentential representations for completeness and correctness, are complete and correct. This is arguably not true, either, when you consider how many sentential systems, even among those that had originally been thought to be complete and correct by their proponents, have been proven not to be so, but it is at least not blatantly false. But the comparison seems to be to diagrammatic representations of logical relations generally, and thus the entire argument seems to be based on an equivocation. 'Sentential logics' are formal sentential representations systematically developed in order to have a complete, correct representation, but 'diagrammatic systems' are not usually developed toward this end at all, and only recently has any work on diagrammatic logical system been done on this point, outside of a few people like Euler, Venn, Carroll, and Peirce -- and even in those cases, only Peirce really goes all the way in trying to establish that the diagrammatic system has the formal properties that the article has in mind, rather than establishing an analogy that works practically as long as certain conditions are met or practical rules are followed.

What is true is that diagrammatic representation of logical relations, as we usually find it, is more analogous to natural-language sentential representation of logical relations than to artificial-language sentential representation. Only a handful of diagrammatic systems have been developed along artificial-language lines. One would not in general expect there to be a significant difference between the diagrammatic and the sentential at this level. In essence, what diagrams do is take advantage of the fact that the modal logic for mereotopological relations can be similar to the modal logic for inferential relations, and then they just add whatever assumptions are needed to close the gap; this is analogous to the case of sentential logics, which take advantage of the fact that the modal logic for grammatical relations can be similar to the modal logic for inferential relations. Because mereotopological relations for a spatial whole are not exactly like grammatical relations for a sentence, one would expect that there are kinds of logical representation the latter can do easily that the former would have difficulty doing, but also vice versa. It's always possible, of course, that the analogy between grammar and inference is much closer than the analogy between mereotopology and inference, in which case you'd have to do much more system-tweaking (adding special rules or assumptions or guides to interpretation) to diagrams than to sentences in order to represent logical inferences generally. I don't know of any argument that this is in fact true; one of the difficulties is that so much more work has been done on sentential representation of logic than on diagrammatic representation that there is a danger of conflating the sentential representation with the logical relation itself. But it's certainly possible that there is that sort of difference, even if the work really hasn't been done to establish it; the problem is that when we are talking about the systems we are usually only talking about the systems that have been so tweaked (on both sides), and you wouldn't expect much of a difference between diagrams and sentences there.

Incidentally, given the analogy, that diagrammatic systems make use of the similarity between mereotopology and general inference as sentential systems make use of the similarity between grammar and general inference, it raises the question of what other modal domains are similar enough to the modal logic of inferential consequence to allow them to be used in representing the latter. What other usable representations of logical relations are there besides sentential and diagrammatic? One obvious example is that of behavior or action, which we usually describe deontically and temporally, since both deontic and temporal relation have enough similarities to inferential relations that we often talk about the latter as if they were deontic or temporal relations. Just as we can have a diagrammatic logic or a sentential logic, we can have a system-of-processes logic; if you think about it, that's essentially what is going on in a computer -- computer programs are not actually lines of code (which are sentential representations) but processes in circuitry. But very little work has been done on logical systems outside of the sentential and diagrammatic, and even the diagrammatic hasn't been studied very closely until recently.

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