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Artificial Intelligence

Inside out logic

If you live in Edinburgh, and Edinburgh is in Scotland, then you live in Scotland. This reasoning draws on the containment metaphor, and the transitivity of containment. It is easy to represent as a diagram, eg, as a Venn diagram. If something is in A then it is also in B.

The diagram also applies to “all As are B.” Draw a small circle or a dot in A and label it C. This is the basic representation of the logical syllogism. You can see at a glance (even without drawing it) that C must be a B. The diagram demonstrates the following

All As are B; C is an A; therefore C is a B.

If a cat is an animal, and I own a cat, then I own an animal. The classical example of the syllogism reads

All men are mortal
Socrates is a man
Therefore Socrates is mortal.

In minimally abstract terms

A implies B
A
therefore B.

C.S. Peirce worked with Venn diagrams (such as the one above) and even developed a means of extending their scope to include variables. But he also developed his own system of logical diagrams that look superficially like Venn diagrams.

Because I am so wedded to the idea that nested shapes indicate containment, I find it difficult to interpret enclosing shapes to be read in any other way. But in his existential graph system, Peirce’s enclosing shapes indicate exclusion.

Negation

Peirce proposed a method for representing logical propositions diagrammatically, and in such a way that you can transform that diagram (through a series of rules) to demonstrate a logical proof.

Instead of thinking of bounding lines (the circles in the Venn diagram) as including what is in them, think of the lines as demarcating exclusion, i.e. as “untruths,” or more precisely, propositions that are not the case in the domain of discussion.

Peirce thought of a line bounding a space is as a “cut.” A continuous (circular) cut line excludes a certain part of the set/world it is in. (I think of one of those crenelated circular pastry cutters, or groups of activists corralled by police in a demonstration.)

The grey rectangle in the diagram below is the domain of discourse. It is the page on which we construct the drawings. Peirce called this the “sheet of assertions.” (He did not use shading, but some commentators think this helps in reading the diagrams.)

Whatever is contained in the outer circle below is excluded from consideration. It is untrue, i.e. not the case. The diagram asserts simply that there is nothing that is A and not B. Peirce called this kind of diagram an “existential graph.”

Here’s how to read the diagram. If two or more entities are enclosed in the same shape then link them with “and.” When you see a boundary line say “not.” Start from the outermost boundary. In other words the circles are brackets prefaced by “not.”

The diagram above reads not(A and not B). This is Boolean terminology familiar to anyone who has searched a database using the standard query language (SQL).

Substitute Edinburgh for A and Scotland for B. Then the diagram reads: you can’t live in Edinburgh and not live in Scotland. It is equivalent to saying that living in Edinburgh implies you live in Scotland — A implies B.

From what I can gather, there is nothing particularly two-dimensional about such diagrams. The circles cannot overlap. What they represent could be laid out as a single string with nested brackets. The advantage of the 2D shapes is that there’s room to introduce other symbols, such as points and lines indicating qualities. (I’ll ignore that feature here.)

Peirce also devised a series of rules to operate on such diagrams to convert one set of propositions into another. The diagram below demonstrates the working of the syllogism: A implies B, A, therefore B. I have left out the transformations demonstrating how the final proposition it derived. But here is the starting diagram and what it looks like at the end of the demonstration (proof).

In Boolean calculus that’s

not(A and not B) and A
therefore B

As he worked in threes, Peirce described the kinds of graphs indicated above as alpha graphs, with further developments designated as beta and gamma graphs. (The examples he and his commentators use extend beyond the simple demonstration I have given here.)

The 5 rules Peirce devised for translating a diagram so as to demonstrate a proof include removing double cuts, i.e. not(not(A)) is simplified to A. There’s no need to iterate these rules here, and I think they are similar to basic Boolean calculus, but in graphical form.

Possibility

How does this diagrammatic method cope with abduction, i.e. the proposition that if someone lives in Scotland then there’s the possibility that they live in Edinburgh, but no logical certainty that they do?

A implies B
B
therefore perhaps A

As we would expect, the formulation

not(A and not B) and B
therefore A

does not hold according to the transformation rules.

Such speculative propositions that someone might live in Edinburgh if we know they live in Scotland are supported by Peirce’s gamma graphs and the idea of the “broken cut” (a dashed boundary) meaning the “possibility of,” requiring 2 extra rules to process. The start and end points of the demonstration (proof) would look like this.

Proving that something is a possibility is easy, and the method supports such a demonstration. But to ascertain how likely it is requires extra evidence, as well as some extra-logical processing. See post Calculating belief.

Bibliography

Note

About Richard Coyne

The cultural, social and spatial implications of computers and pervasive digital media spark my interest ... enjoy architecture, writing, designing, philosophy, coding and media mashups.

Discussion

6 thoughts on “Inside out logic

  1. Venn diagrams make for very iconic representations of their universes of discourse. That is one of the main sources of their intuitive utility and also the main source of their logical limitations — they begin to exceed our human capacity for visualization once we climb to 4 or 5 circles (Boolean variables) or so.

    Peirce’s logical graphs at the Alpha level (propositional calculus) are somewhat iconic but far less so than Venn diagrams. They are more properly regarded as symbolic representations, in a way that exceeds the logical capacities of icons. That is the source of their considerably greater power as a symbolic calculus.

    Here’s a primer on all that:

    Logical Graphs • Introduction

    Posted by Jon Awbrey | January 27, 2018, 12:50 pm

Trackbacks/Pingbacks

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