# Looking for the Unified Modeling Language of philosophy and ideas

September 13, 2013 7:28 PM Subscribe

I'm looking for a way of diagramming the component parts of ideas and arguments, and their relationships to each other, formally and visually. Does such a thing exist?

There are hints of what I'm looking for in UML, flowcharts, Venn diagrams, and the notation of Boolean algebra and set theory, but has anyone developed something specifically for diagramming deductive argument?

It seems to me that diagramming an argument in this way would (a) force the argument toward higher standards of clarity and rigor, (b) facilitate the communication and debate of the idea, and (c) make it easier to identify (and illustrate) where fallacies exist.

Primitives would probably include individual objects, categories of objects, the relationships between categories, the properties of those objects and categories, the ways in which the objects can act on each other, the effects which follow those actions, etc.

By combining a few primitives, you could express simple ideas such as:

—"object X is a member of category Y" (e.g., "this object is a member of the category known as 'apples'")

—"object X has property Y" (e.g., "this apple is red")

—"all members of category X have property Y" (e.g., "all apples are red")

—"categories X and Y are mutually exclusive" (e.g., "no member of the category 'apples' is also a member of the category 'peaches', and vice-versa")

—"event X [necessarily|sometimes|never] follows as a consequence of event Y" (e.g. "if an apple is dropped, it will always fall to the ground")

—"if X [has property Y|belongs to category Z], then X [can never have property W|must have property V|must belong to category U|etc]"

—various opposites/negations of the above

—probably lots of other things I'm not thinking of

If the primitives (objects, categories, properties, etc.) are atoms, then these statements are molecules. And by assembling molecules, you could diagram the whole argument.

So: is there such a thing, or something like it? Does this even make sense? :)

There are hints of what I'm looking for in UML, flowcharts, Venn diagrams, and the notation of Boolean algebra and set theory, but has anyone developed something specifically for diagramming deductive argument?

It seems to me that diagramming an argument in this way would (a) force the argument toward higher standards of clarity and rigor, (b) facilitate the communication and debate of the idea, and (c) make it easier to identify (and illustrate) where fallacies exist.

Primitives would probably include individual objects, categories of objects, the relationships between categories, the properties of those objects and categories, the ways in which the objects can act on each other, the effects which follow those actions, etc.

By combining a few primitives, you could express simple ideas such as:

—"object X is a member of category Y" (e.g., "this object is a member of the category known as 'apples'")

—"object X has property Y" (e.g., "this apple is red")

—"all members of category X have property Y" (e.g., "all apples are red")

—"categories X and Y are mutually exclusive" (e.g., "no member of the category 'apples' is also a member of the category 'peaches', and vice-versa")

—"event X [necessarily|sometimes|never] follows as a consequence of event Y" (e.g. "if an apple is dropped, it will always fall to the ground")

—"if X [has property Y|belongs to category Z], then X [can never have property W|must have property V|must belong to category U|etc]"

—various opposites/negations of the above

—probably lots of other things I'm not thinking of

If the primitives (objects, categories, properties, etc.) are atoms, then these statements are molecules. And by assembling molecules, you could diagram the whole argument.

So: is there such a thing, or something like it? Does this even make sense? :)

http://www.iep.utm.edu/prop-log/

Sorry link addition is not working on my screen.

posted by sb3 at 7:39 PM on September 13, 2013

Sorry link addition is not working on my screen.

posted by sb3 at 7:39 PM on September 13, 2013

Have you taken logic?

First blush, what you're looking for is logic. A couple of starting places to see if this sounds like what you're after -

classical first-order logic

(more meta) model theory

Second possibility, are you interested in whether there is a way to break down every fact about the world into this kind of logical atoms-and-molecules kind of thing? Might look at -

logic and ontology (ontology is the study of what there is in the world)

correspondence theories of truth

other theories about what truth is

posted by LobsterMitten at 7:39 PM on September 13, 2013

First blush, what you're looking for is logic. A couple of starting places to see if this sounds like what you're after -

classical first-order logic

(more meta) model theory

Second possibility, are you interested in whether there is a way to break down every fact about the world into this kind of logical atoms-and-molecules kind of thing? Might look at -

logic and ontology (ontology is the study of what there is in the world)

correspondence theories of truth

other theories about what truth is

posted by LobsterMitten at 7:39 PM on September 13, 2013

You mention Venn diagrams ... They are often used to illustrate and test categorical syllogisms for validity in introductory logic classes.

You may also be interested in the square of opposition (see also) and the logical hexagon.

posted by Monsieur Caution at 9:35 PM on September 13, 2013

You may also be interested in the square of opposition (see also) and the logical hexagon.

posted by Monsieur Caution at 9:35 PM on September 13, 2013

*has anyone developed something specifically for diagramming deductive argument?*

Isn't this what formal logic is for?

In particular, you may be interested in

*second-order logics*, which enable you say things like "Peaches and apples have some properties in common" as well as saying things like "Apples are red." (In slightly more technical terms, they let you quantify over properties and not just over entities.)

There's been a lot of work in linguistics on how to formalize natural-language descriptions of events, including things like the difference between past and present tense: your search keyword for this is

*event semantics*.

If you want to talk about cause-and-effect relationships, you might also want to look at

*modal logics*, which include notions of necessity and possibility — letting you say things like "If an object is dropped, it will necessarily fall" and "If an object is dropped, it's possible that it will break."

It may also be useful to allow for blanket generalizations that have some exceptions. For instance, you might want a rule that says "Birds fly," and then a more specific rule that says "...but penguins don't." There are logics that can handle this too, known as

*non-monotonic logics*.

(These properties — higher-order quantification, modality, non-monotonicity, etc. — can be mix-and-matched. So for instance, there are second-order modal logics, non-monotonic modal logics, etc.)

Long story short: just about anything you need, there's a logic that can do it. Start with good old-fashioned first-order logic, like everyone else suggests. But if that seems too limited to do what you want, keep in mind that a lot of more exotic logics are available, with different useful properties.

posted by Now there are two. There are two _______. at 10:10 PM on September 13, 2013 [4 favorites]

You might also be interested in

posted by Now there are two. There are two _______. at 10:12 PM on September 13, 2013

*formal ontologies*like SUMO. These are attempts to build up a big knowledge base of specific facts like the ones you describe ("apples are fruit" "peaches are fruit" "apples aren't peaches"), described in a well-defined and precise way.posted by Now there are two. There are two _______. at 10:12 PM on September 13, 2013

The goals you list as a, b & c are those behind much formal logic, as others have remarked. So, yeah, check out formal logic and the predicate calculus. Kalish and Montague's book is a widely used introdution. But you might also want to look as well into the Cognitive Grammar tradition, which uses visual diagrams to capture structures of meaning in a different and richly comprehensive way. (Browse around online in this book to get an idea.)

Since your interest seems to be towards the appraisal of real arguments, I'd recommend too Toulmin's The Uses of Argument to alert you to a tendency that formal logic's interest in formalization has of producing mathematical logics -- and a mathematical mind-set -- which can make them more or less irrelevant to the real business of reasoning.

posted by bertran at 12:22 AM on September 14, 2013 [2 favorites]

Since your interest seems to be towards the appraisal of real arguments, I'd recommend too Toulmin's The Uses of Argument to alert you to a tendency that formal logic's interest in formalization has of producing mathematical logics -- and a mathematical mind-set -- which can make them more or less irrelevant to the real business of reasoning.

posted by bertran at 12:22 AM on September 14, 2013 [2 favorites]

(Oh, good call on the Langacker/Cognitive Grammar recommendation.)

posted by Now there are two. There are two _______. at 7:11 AM on September 14, 2013

posted by Now there are two. There are two _______. at 7:11 AM on September 14, 2013

Oh, this just makes my Saturday morning awesome. Thanks!

Disclaimer: I'm not a UML expert, either academically or in practice. That said, my first reaction as I started reading was that UML actually sounds like exactly what you need, until you started getting into the complex expressions. I don't think UML has a good way to visualize that. However, if visualization isn't really a primary goal, then the Object Constraint Language might be a reasonable approach. Hope that helps, and good luck!

Now, I have to decide whether to spend hours geeking out on the links above or actually go outside.

posted by finnegans at 8:14 AM on September 14, 2013

Disclaimer: I'm not a UML expert, either academically or in practice. That said, my first reaction as I started reading was that UML actually sounds like exactly what you need, until you started getting into the complex expressions. I don't think UML has a good way to visualize that. However, if visualization isn't really a primary goal, then the Object Constraint Language might be a reasonable approach. Hope that helps, and good luck!

Now, I have to decide whether to spend hours geeking out on the links above or actually go outside.

posted by finnegans at 8:14 AM on September 14, 2013

For a really rather alluring example of ways your quest can go horribly, horribly wrong, take a look at G. Spencer-Brown's Laws of Form:

And if you're in the mood to mess with Mr. In-between, give René Thom's Catastrophe theory a quick glance.

posted by jamjam at 10:18 AM on September 14, 2013 [2 favorites]

Ostensibly a work of formal mathematics and philosophy, LoF became something of a cult classic, praised in the Whole Earth Catalog. Those who agree point to LoF as embodying an enigmatic "mathematics of consciousness," its algebraic symbolism capturing an (perhaps even the) implicit root of cognition: the ability to distinguish. LoF argues that the pa (primary algebra) reveals striking connections among logic, Boolean algebra, and arithmetic, and the philosophy of language and mind.For something with at least superficial similarities to your desired object and which has turned out to have great power despite controversial beginnings, see Category theory:

...

LoF has influenced, among others, Heinz von Foerster, Louis Kauffman, Niklas Luhmann, Humberto Maturana, Francisco Varela and William Bricken. Some of these authors have modified the primary algebra in a variety of interesting ways.

LoF claimed that certain well-known mathematical conjectures of very long standing, such as the Four Color Theorem, Fermat's Last Theorem, and the Goldbach conjecture, are provable using extensions of the pa. Spencer-Brown eventually circulated a purported proof of the Four Color Theorem, but it met with skepticism.[2] (The Four Color Theorem and Fermat's Last Theorem were proved in 1976 and 1995, respectively, using methods owing nothing to LoF.)

Higher-dimensional categoriesI'll be trying to read Baez's article later this weekend myself.

Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can be situated into the context of higher-dimensional categories. Briefly, if we consider a morphism between two objects as a "process taking us from one object to another", then higher-dimensional categories allow us to profitably generalize this by considering "higher-dimensional processes".

For example, a (strict) 2-category is a category together with "morphisms between morphisms", i.e., processes which allow us to transform one morphism into another. We can then "compose" these "bimorphisms" both horizontally and vertically, and we require a 2-dimensional "exchange law" to hold, relating the two composition laws. In this context, the standard example is Cat, the 2-category of all (small) categories, and in this example, bimorphisms of morphisms are simply natural transformations of morphisms in the usual sense. Another basic example is to consider a 2-category with a single object; these are essentially monoidal categories. Bicategories are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative "up to" an isomorphism.

This process can be extended for all natural numbers n, and these are called n-categories. There is even a notion of ω-category corresponding to the ordinal number ω.

Higher-dimensional categories are part of the broader mathematical field of higher-dimensional algebra, a concept introduced by Ronald Brown. For a conversational introduction to these ideas, see John Baez, 'A Tale of n-categories' (1996).

And if you're in the mood to mess with Mr. In-between, give René Thom's Catastrophe theory a quick glance.

posted by jamjam at 10:18 AM on September 14, 2013 [2 favorites]

Frege's Begriffsschrift, a precursor to modern propositional calculus, took the geometric form of a sort of tree.

posted by phrontist at 11:20 AM on September 14, 2013 [2 favorites]

posted by phrontist at 11:20 AM on September 14, 2013 [2 favorites]

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posted by sb3 at 7:38 PM on September 13, 2013