Re: Animated Logical Graphs • (57) • (58) • (59) • (60) • (61) • (62) • (63) • (64) • (65) • (66) • (69) • (70)

Our investigation has brought us to the point of seeing both a transformation group and a triadic sign relation in the duality between entitative and existential interpretations of logical graphs.

Given the level of the above abstractions it helps to anchor them in concrete structural experience. In that spirit we’ve been pursuing the case of a group action and a sign relation where is the set of boolean functions on two variables and is a set of logical graphs denoting those functions. We drew up a Table combining the aspects of both structures and sorted it according to the orbits induces on and consequently on

- Logic Syllabus
- C.S. Peirce • Logic and Signs
- C.S. Peirce • Logic as Semiotic
- Sign Relation • Triadic Relation
- Survey of Animated Logical Graphs
- Survey of Semiotics, Semiosis, Sign Relations

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Re: Peirce List (1) (2) • John Sowa (1) (2)

Re: Peirce List • Jon Alan Schmidt

- JAS:
- Thanks for providing a longer excerpt of that passage than I did, including Peirce’s statement about “the lower animals.” I see now that I was wrong when I said on Tuesday, “Peirce makes no claim in the relevant texts about non-human animals at all.” What I should have said is that he makes no claim in the relevant texts about whether non-human animals can “learn to recognize negations.” He merely expresses doubt that they “have any clear and steady conception of falsehood,” and adds that “without a knowledge of falsehood no development of discursive reason can take place.”
- In other words, it is not reasoning per se that distinguishes humans from other animals, since no notion of falsity is required for that — only a capacity for drawing inferences, which in “non-relative formal logic” corresponds to the relation of implication such that one proposition necessarily (i.e., deductively) follows from another. Instead, what distinguishes humans from other animals is discursive reasoning, which does require “a knowledge of falsehood” and the more sophisticated “logic of relatives.”
- In accordance with Peirce’s own words, it is important to keep in mind that these points all have to do with logic, not psychology or linguistics. His thesis is that we acquire the notion of falsity and associate it with the formal relation of negation through “the avenue of experience and logical reflexion,” when reality confronts us with surprising observations that call for explanation, thus compelling us to initiate the process of inquiry by which we eventually revise our previous beliefs that grounded our incorrect expectations.

Dear Jon Alan,

Thanks for your comments, which I look forward to studying further. Earlier I mentioned CP 3.488–490 as one of “the very doors I first walked through into the wonderland of logic à la Peirce”. That is because, just the other side of that door, at CP 3.491, Peirce introduces a triadic relative term signifying with reference to three elements in the universe of discourse that is neither nor

Now, a relation among three elements of an arbitrary universe of discourse is more general than a relation among three logical values but they are kin enough to connect the passage with Peirce’s marking of the amphecks Nand and Nnor and their sole sufficiency among boolean operations.

What flashed me back this time, though, was John Sowa raising the topic of reflection on visual diagrams, those being forms of expression and calling to mind what Peirce wrote about “logical reflexion” being “the observation of thoughts in their expressions”.

I’m now going to hold off further commentary on this passage and stand back to take in a broader view of its context, Peirce’s 1897 Logic of Relatives, as I’m seeing many issues I did not appreciate, much less understand in times past. This looks like it will take me no little time …

Regards,

Jon

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Re: Laws of Form • Lyle Anderson

- LA:
- Here Peirce is confusing Truth and Falsity with Good and Evil. The Creator of the Universe (CotU) created Truth and Falsity when He drew the First Distinction. The Universe started with Bet (2) not Aleph (1). What was the First Commandment? Do NOT eat of the Tree of the Knowledge of Good and Evil. Good and Evil are many levels of abstraction away from True and False.

Dear Lyle,

Thanks for making that observation. I had seen what Peirce did there, too, and it’s one reason I saw him as making an allegorical or parabolic use of *felix culpa* at that point, since I know he knows his Scripture too well to imagine it anything but intentional. Read in that spirit Peirce is inviting us to contemplate a particular form of distinction — a threshold between an immersive state of being and a reflective stage of critical thought.

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Re: FB | Medieval Logic • Kollbjorn Oldtheyn

Dear Kollbjorn,

I used to think I knew what Peirce was talking about in this passage but it looks like it may be time to make a new examination of that.

**Guess 1.**Strictly speaking he’s talking about his earlier system of “entitative graphs” which are logically dual to existential graphs as far as propositional calculus goes. That may not affect his point, except he did not extend the entitative graphs to cover the logic of relative terms, so he may be talking about the limitations of absolute versus relative terms.**Guess 2.**He may be alluding to the complex way he treated negation in his 1870 Logic of Relatives, which is very tricky but worth revisiting.**Guess 3.**He may be talking about the threshold between first intentional and second intentional relatives, which may or may not be the same thing as first order versus second order logic.

At any rate, I’ll be looking further into it …

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Re: FB | Medieval Logic • Kollbjorn Oldtheyn

Dear Kollbjorn,

The way I understand Peirce’s parable, he is asking, “How do we arrive at a condition far enough removed from our immersion in a current experience to question it, to reflect on it, and thereby conceive the possibility of something other?” Until we do that we do not have a concept of “not”.

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Re: Paradisaical Logic and the After Math • Comment (1) (2)

I am getting a feeling I occasionally get when discussing Peirce’s work in a group setting — as though we had a grand feast set before us while the dining philosophers wrangle over the *amuse-bouche*. With that in mind I’d like to return to the inciting text, the one so exciting me all those years ago, and see what’s truly substantial and tantalizing in it.

488. The general method of graphical representation of propositions has now been given in all its essential elements, except, of course, that we have not, as yet, studied any truths concerning special relatives; for to do so would seem, at first, to be “extralogical”.

Logic in this stage of its development may be called *paradisaical logic*, because it represents the state of Man’s cognition before the Fall. For although, with this apparatus, it is easy to write propositions necessarily true, it is absolutely impossible to write any which is necessarily false, or, in any way which that stage of logic affords, to find out that anything is false. The mind has not as yet eaten of the fruit of the Tree of Knowledge of Truth and Falsity.

Probably it will not be doubted that every child in its mental development necessarily passes through a stage in which he has some ideas, but yet has never recognised that an idea may be erroneous; and a stage that every child necessarily passes through must have been formerly passed through by the race in its adult development. It may be doubted whether many of the lower animals have any clear and steady conception of falsehood; for their instincts work so unerringly that there is little to force it upon their attention. Yet plainly without a knowledge of falsehood no development of discursive reason can take place.

489. This paradisaical logic appears in the study of non-relative formal logic. But *there* no possible avenue appears by which the knowledge of falsehood could be brought into this Garden of Eden except by the arbitrary and inexplicable introduction of the Serpent in the guise of a proposition necessarily false. The logic of relatives affords such an avenue, and *that*, the very avenue by which in actual development, this stage of logic supervenes. It is the avenue of experience and logical reflexion.

490. By *logical* reflexion, I mean the observation of thoughts in their expressions. Aquinas remarked that this sort of reflexion is requisite to furnish us with those ideas which, from lack of contrast, ordinary external experience fails to bring into prominence. He called such ideas *second intentions*. Is is by means of *relatives of second intention* that the general method of logical representation is to find completion.

- Charles S. Peirce, “The Logic of Relatives”,
*The Monist*, vol. 7, 161–217, (1897). Reprinted, CP 3.456–552.

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Dear Mauro,

My access to the internet is limited today — maybe I can make a start toward addressing your comments by linking to an article on “sole sufficient operators” in boolean algebra and propositional calculus.

There’s more information about Peirce’s “amphecks”, tantamount to what we now call Nand and Nnor, in the following article.

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Re: Animated Logical Graphs • (57) • (58) • (59) • (60) • (61) • (62) • (63) • (64) • (65) • (66) • (69)

Our study of the duality between entitative and existential interpretations of logical graphs has brought to light its fully sign-relational character. We can see this in the sign relation linking an object domain with two sign domains, whose signs denote the objects in two distinct ways. We illustrated the general principle using an object domain consisting of the sixteen boolean functions on two variables and a pair of sign domains consisting of representative logical graphs for those functions, as shown in the following Table.

- Column 1 shows the object domain as the set of 16 boolean functions on 2 variables.
- Column 2 shows the sign domain as a representative set of logical graphs denoting the objects in according to the existential interpretation.
- Column 3 shows the interpretant domain as the same set of logical graphs denoting the objects in according to the entitative interpretation.

Additional aspects of the sign relation’s structure can be brought out by sorting the Table in accord with the orbits induced on the object domain by the action of the transformation group inherent in the dual interpretations. Performing that sort produces the following Table.

That’s enough bytes to chew on for one post — we’ll extract more information from the Tables next time.

- Logic Syllabus
- C.S. Peirce • Logic and Signs
- C.S. Peirce • Logic as Semiotic
- Sign Relation • Triadic Relation
- Survey of Animated Logical Graphs
- Survey of Semiotics, Semiosis, Sign Relations

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Re: Peirce List • (1) (2) (3) (4) (5) (6)

One of those recurring themes — you might call it “The Power of Negative Thinking” — arose this time on the Peirce List and it took me back to a post I wrote nine Aprils ago and that took me even further back to the very doors I first walked through into the wonderland of logic à la Peirce.

I fixed the links broken by the ravages of time and the impings of web developers and I added more links to the original context of discussion. A partial transcript follows.

Not too coincidentally with the mention of Peirce’s existential graphs, a tangent of discussion elsewhere brought to mind an old favorite passage from Peirce, where he is using his entitative graphs to expound the logic of relatives. Here is the observation I was led to make.

Negative operations (NOs), if not more important than positive operations (POs), are at least more powerful or generative, because the right NOs can generate all POs, but the reverse is not so.

Which brings us to Peirce’s *amphecks*, NAND and NNOR, either of which is a sole sufficient operator for all boolean operations.

In one of his developments of a graphical syntax for logic, that described in passing an application of the Neither-Nor operator, Peirce referred to the stage of reasoning before the encounter with falsehood as “*paradisaical logic*, because it represents the state of Man’s cognition before the Fall.”

Here’s a bit of what he wrote there —

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Re: Animated Logical Graphs • (57) • (58) • (59) • (60) • (61) • (62) • (63) • (64) • (65) • (66)

*“I know what you mean but I say it another way”* — it’s a thing I find myself saying often enough, if only under my breath, to rate an acronym for it ☞ IKWYMBISIAW ☜ and not too coincidentally it’s a rubric of relevance to many situations in semiotics where sundry manners of speaking and thinking converge, more or less, on the same patch of pragmata.

We encountered just such a situation in our exploration of the duality between entitative and existential interpretations of logical graphs. The two interpretations afford distinct but equally adequate ways of reasoning about a shared objective domain. To cut our teeth on a simple but substantial example of an object domain, we picked the space of boolean functions or propositional forms on two variables. This brought us to the following Table, highlighting the sign relation involved in switching between existential and entitative interpretations of logical graphs.

- Column 1 shows the object domain as the set of 16 boolean functions on 2 variables.
- Column 2 shows the sign domain as a representative set of logical graphs denoting the objects in according to the existential interpretation.
- Column 3 shows the interpretant domain as the same set of logical graphs denoting the objects in according to the entitative interpretation.

- C.S. Peirce • Logic and Signs
- C.S. Peirce • Logic as Semiotic
- Sign Relation • Triadic Relation
- Survey of Semiotics, Semiosis, Sign Relations

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