## Logic Syllabus • Discussion 2

JM:
Is [the “just one true” operator] the same or different to xor?  I have read that xor is true when an odd number of variables are true which would make it different.  But I also read somewhere that xor was true when only one is true.

Here’s my syllabus entry on Exclusive Disjunction (xor), also known as Logical Inequality, Symmetric Difference, and a few other names.  It’s my best effort so far at straightening out the reigning confusions and also at highlighting the links between the various notations and visualizations we find in practice.

Exclusive disjunction, also known as logical inequality or symmetric difference, is an operation on two logical values, typically the values of two propositions, which produces a value of true just in case exactly one of its operands is true.

To say exactly one operand is true is to say the other is false, which is to say the two operands are different, that is, unequal.

Expressed algebraically, $x_1 + x_2 = 1 ~ (\text{mod}~ 2).$

Viewed in that light, it is tempting to think a natural extension of xor to many variables $x_1, \ldots, x_m$ will take the form $x_1 + \ldots + x_m = 1 ~ (\text{mod}~ 2).$  And saying the bit sum of several boolean values is 1 is just another way of saying an odd number of the values are 1.

Sums of that order form a perfectly good family of boolean functions, ones we’ll revisit in a different light, but their kinship to the family of logical disjunctions is a bit more strained than uniquely natural.

## Logic Syllabus • Discussion 1

JM:
In a previous post you mentioned the minimal negation operator.  Is there also the converse of this, i.e. an operator which is true when exactly one of its arguments is true?  Or is this just xor?

Yes, the “just one true” operator is a very handy tool.  We discussed it earlier under the headings of “genus and species relations” or “radio button logic”.  Viewed in the form of a venn diagram it describes a partition of the universe of discourse into mutually exclusive and exhaustive regions.  Reading $\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_m \texttt{)}$ to mean just one of $x_1, \ldots, x_m$ is false, the form $\texttt{((} x_1 \texttt{),} \ldots \texttt{,(} x_m \texttt{))}$ means just one of $x_1, \ldots, x_m$ is true.

For two logical variables, though, the cases “condense” or “degenerate” and saying “just one true” is the same thing as saying “just one false”.

$\texttt{((} x_1 \texttt{),(} x_2 \texttt{))} = \texttt{(} x_1 \texttt{,} x_2 \texttt{)} = x_1 + x_2 = \textsc{xor} (x_1, x_2).$

## Inquiry Into Inquiry • On Initiative 5

JS:
That’s not how it works.  The model lacks agency.  It is a machine whose gears are cranked by the user’s prompt.  It can ask questions, but only when prompted to.  It is not doing anything at all when it isn’t being prompted.

Sure, I understand that.  The hedge “as it were” is used advisedly for the sake of the argument.  (I wrote my own language learner back in the 80s.)

Speaking less metaphorically, the program and its database are always in their respective states and the program has the capacity to act on the database even when not engaged with external prompts.

Is there any reason why the program’s “housekeeping” functions should not include one to measure its current state of “uncertainty” (entropy of a distribution) with regard to potential questions — or any reason why it should “hurt to ask”?

As it were …

### Resources

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## Mathematical Demonstration and the Doctrine of Individuals • 2

### Selection from C.S. Peirce’s “Logic Of Relatives” (1870)

In reference to the doctrine of individuals, two distinctions should be borne in mind.  The logical atom, or term not capable of logical division, must be one of which every predicate may be universally affirmed or denied.  For, let $\mathrm{A}$ be such a term.  Then, if it is neither true that all $\mathrm{A}$ is $\mathrm{X}$ nor that no $\mathrm{A}$ is $\mathrm{X},$ it must be true that some $\mathrm{A}$ is $\mathrm{X}$ and some $\mathrm{A}$ is not $\mathrm{X};$  and therefore $\mathrm{A}$ may be divided into $\mathrm{A}$ that is $\mathrm{X}$ and $\mathrm{A}$ that is not $\mathrm{X},$ which is contrary to its nature as a logical atom.

Such a term can be realized neither in thought nor in sense.

Not in sense, because our organs of sense are special — the eye, for example, not immediately informing us of taste, so that an image on the retina is indeterminate in respect to sweetness and non-sweetness.  When I see a thing, I do not see that it is not sweet, nor do I see that it is sweet;  and therefore what I see is capable of logical division into the sweet and the not sweet.  It is customary to assume that visual images are absolutely determinate in respect to color, but even this may be doubted.  I know no facts which prove that there is never the least vagueness in the immediate sensation.

In thought, an absolutely determinate term cannot be realized, because, not being given by sense, such a concept would have to be formed by synthesis, and there would be no end to the synthesis because there is no limit to the number of possible predicates.

A logical atom, then, like a point in space, would involve for its precise determination an endless process.  We can only say, in a general way, that a term, however determinate, may be made more determinate still, but not that it can be made absolutely determinate.  Such a term as “the second Philip of Macedon” is still capable of logical division — into Philip drunk and Philip sober, for example;  but we call it individual because that which is denoted by it is in only one place at one time.  It is a term not absolutely indivisible, but indivisible as long as we neglect differences of time and the differences which accompany them.  Such differences we habitually disregard in the logical division of substances.  In the division of relations, etc., we do not, of course, disregard these differences, but we disregard some others.  There is nothing to prevent almost any sort of difference from being conventionally neglected in some discourse, and if $I$ be a term which in consequence of such neglect becomes indivisible in that discourse, we have in that discourse,

$[I] = 1.$

This distinction between the absolutely indivisible and that which is one in number from a particular point of view is shadowed forth in the two words individual (τὸ ἄτομον) and singular (τὸ καθ᾿ ἕκαστον);  but as those who have used the word individual have not been aware that absolute individuality is merely ideal, it has come to be used in a more general sense.  (CP 3.93)

### Note

Peirce explains his use of the square bracket notation at CP 3.65.

I propose to denote the number of a logical term by enclosing the term in square brackets, thus, $[t].$

The number of an absolute term, as in the case of $I,$ is defined as the number of individuals it denotes.

### References

• Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9, 317–378, 26 January 1870.  Reprinted, Collected Papers 3.45–149, Chronological Edition 2, 359–429.  Online (1) (2) (3).
• Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.

## Mathematical Demonstration and the Doctrine of Individuals • 1

### Selection from C.S. Peirce’s “Logic Of Relatives” (1870)

Demonstration of the sort called mathematical is founded on suppositions of particular cases.  The geometrician draws a figure;  the algebraist assumes a letter to signify a single quantity fulfilling the required conditions.  But while the mathematician supposes an individual case, his hypothesis is yet perfectly general, because he considers no characters of the individual case but those which must belong to every such case.

The advantage of his procedure lies in the fact that the logical laws of individual terms are simpler than those which relate to general terms, because individuals are either identical or mutually exclusive, and cannot intersect or be subordinated to one another as classes can.

Mathematical demonstration is not, therefore, more restricted to matters of intuition than any other kind of reasoning.  Indeed, logical algebra conclusively proves that mathematics extends over the whole realm of formal logic;  and any theory of cognition which cannot be adjusted to this fact must be abandoned.  We may reap all the advantages which the mathematician is supposed to derive from intuition by simply making general suppositions of individual cases.  (CP 3.92)

### References

• Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9, 317–378, 26 January 1870.  Reprinted, Collected Papers 3.45–149, Chronological Edition 2, 359–429.  Online (1) (2) (3).
• Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.

## Inquiry Into Inquiry • On Initiative 4

I think a lot of people who’ve been working all along on AI, intelligent systems, and computational extensions of human capacities in general are a little distressed to see the field cornered and re‑branded in the short‑sighted, market‑driven way we currently see.

The more fundamental problem I see here is the failure to grasp the nature of the task at hand, and this I attribute not to a program but to its developers.

Journalism, Research, and Scholarship are not matters of generating probable responses to prompts or other stimuli.  What matters is producing evidentiary and logical supports for statements.  That is the task requirement the developers of recent LLM‑Bots are failing to grasp.

There is nothing new about that failure.  There is a long history of attempts to account for intelligence and indeed the workings of scientific inquiry based on the principles of associationism, behaviorism, connectionism, and theories of that order.  But the relationship of empirical evidence, logical inference, and scientific information is more complex and intricate than is dreamt of in those reductive philosophies.

### Resources

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## Systems of Interpretation • 3

The “triskelion” figure in the previous post shows the bare essentials of an elementary sign relation or individual triple $(o, s, i).$  There’s a less skeletal figure Susan Awbrey and I used in an earlier paper, where our aim was to articulate the commonalities Peirce’s concept of a sign relation shares with its archetype in Aristotle.

$\text{Figure 1. The Sign Relation in Aristotle}$

Here is the corresponding passage from “On Interpretation”.

Words spoken are symbols or signs (symbola) of affections or impressions (pathemata) of the soul (psyche);  written words are the signs of words spoken.  As writing, so also is speech not the same for all races of men.  But the mental affections themselves, of which these words are primarily signs (semeia), are the same for the whole of mankind, as are also the objects (pragmata) of which those affections are representations or likenesses, images, copies (homoiomata).  (De Interp. i. 16a4).

### References

• Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.  ArchiveJournal.  Online (doc) (pdf).
• Awbrey, S.M., and Awbrey, J.L. (2001), “Conceptual Barriers to Creating Integrative Universities”, Organization : The Interdisciplinary Journal of Organization, Theory, and Society 8(2), Sage Publications, London, UK, 269–284.  AbstractOnline.
• Awbrey, S.M., and Awbrey, J.L. (September 1999), “Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century”, Second International Conference of the Journal ‘Organization’, Re‑Organizing Knowledge, Trans‑Forming Institutions : Knowing, Knowledge, and the University in the 21st Century, University of Massachusetts, Amherst, MA.  Online.

## Systems of Interpretation • 2

Let’s start as simply as possible.  The following Figure is typical of many I have used to illustrate sign relations from the time I first began studying Peirce’s theory of signs.

$\text{Figure 2. An Elementary Sign Relation}$

The above variant comes from a paper Susan Awbrey and I presented at a conference in 1999, a revised version of which was published in 2001.

As the drafter of that drawing I can speak with authority about the artist’s intentions in drawing it and also about the conventions of interpretation forming the matrix of its conception and delivery.

Just by way of refreshing my own memory, here is how we set it up —

Figure 2 represents an “elementary sign relation”.  It is a single transaction taking place among three entities, the object $o,$ the sign $s,$ and the interpretant sign $i,$ the association of which is typically represented by means of the ordered triple $(o, s, i).$

One of the interpretive conventions implied in that setup is hallowed by long tradition, going back to the earliest styles of presentation in mathematics.  In it one draws a figure intended as “representative” of many figures.  Regarded as a concrete drawing the figure is naturally imperfect, individual, peculiar, and special but it’s meant to be taken purely as a representative of its class — generic, ideal, and typical.  That is the main convention of interpretation which goes into giving diagrams and figures their significant power.

### References

• Awbrey, S.M., and Awbrey, J.L. (2001), “Conceptual Barriers to Creating Integrative Universities”, Organization : The Interdisciplinary Journal of Organization, Theory, and Society 8(2), Sage Publications, London, UK, 269–284.  AbstractOnline.
• Awbrey, S.M., and Awbrey, J.L. (September 1999), “Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century”, Second International Conference of the Journal ‘Organization’, Re‑Organizing Knowledge, Trans‑Forming Institutions : Knowing, Knowledge, and the University in the 21st Century, University of Massachusetts, Amherst, MA.  Online.

## Systems of Interpretation • 1

Questions have arisen about the different styles of diagrams and figures used to represent triadic sign relations in Peircean semiotics.  What do they mean?  Which style is best?  Among the most popular pictures some use geometric triangles while others use the three‑pronged graphs Peirce used in his logical graphs to represent triadic relations.

Diagrams and figures, like any signs, can serve to communicate the intended interpretants and thus to coordinate the conduct of interpreters toward the intended objects — but only in communities of interpretation where the conventions of interpretation are understood.  Conventions of interpretation are by comparison far more difficult to communicate.

That brings us to the first question we have to ask about the possibility of communication in this area, namely, what conventions of interpretation are needed to make sense of these diagrams, figures, and graphs?

## Inquiry Into Inquiry • On Initiative 3

The more fundamental problem I see here is the failure to grasp the nature of the task at hand, and this I attribute not to a program but to its developers.

Journalism, Research, and Scholarship are not matters of generating probable responses to prompts or other stimuli.  What matters is producing evidentiary and logical supports for statements.  That is the task requirement the developers of recent LLM‑Bots are failing to grasp.

There is nothing new about that failure.  There is a long history of attempts to account for intelligence and indeed the workings of scientific inquiry based on the principles of associationism, behaviorism, connectionism, and theories of that order.  But the relationship of empirical evidence, logical inference, and scientific information is more complex and intricate than is dreamt of in those reductive philosophies.

Note.  The above comment was originally posted on March 1st but appears to have been deleted accidentally.

### Resources

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