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Differential Propositional Calculus • 10

Special Classes of Propositions (cont.)

Let’s pause at this point and get a better sense of how our special classes of propositions are structured and how they relate to propositions in general.  We can do this by recruiting our visual imaginations and drawing up a sufficient budget of venn diagrams for each family of propositions.  The case for 3 variables is exemplary enough for a start.

Linear Propositions

The linear propositions, may be written as sums:

One thing to keep in mind about these sums is that the values in are added “modulo 2”, that is, in such a way that

In a universe of discourse based on three boolean variables, the linear propositions take the shapes shown in Figure 8.

At the top is the venn diagram for the linear proposition of rank 3, which may be expressed by any one of the following three forms.

Next are the venn diagrams for the three linear propositions of rank 2, which may be expressed by the following three forms, respectively.

Next are the three linear propositions of rank 1, which are none other than the three basic propositions,

At the bottom is the linear proposition of rank 0, the everywhere false proposition or the constant function, which may be expressed by the form or by a simple

Resources

• Logic Syllabus
• Survey of Differential Logic

cc: Academia.edu • Cybernetics • Structural Modeling • Systems Science
cc: Conceptual Graphs • Laws of Form • Mathstodon • Research Gate

#Amphecks #Animata #BooleanAlgebra #BooleanFunctions #CSPeirce #CactusGraphs #CategoryTheory #Change #Cybernetics #DifferentialAnalyticTuringAutomata #DifferentialCalculus #DifferentialLogic #DiscreteDynamics #EquationalInference #FunctionalLogic #GraphTheory #Hologrammautomaton #IndicatorFunctions #InquiryDrivenSystems #Leibniz #Logic #LogicalGraphs #Mathematics #MinimalNegationOperators #PropositionalCalculus #Time #Topology #Visualization

Peirce's 1885 “Algebra of Logic” • Discussion 2
• https://inquiryintoinquiry.com/2024/04/03/peirces-1885-algebra-of-logic-discussion-2/

Re: FB | Daniel Everett

One thing I've been trying to understand for a very long time is the changes in Peirce's writing about math and logic from 1865 to 1885. If there's anything I've learned from reading Peirce in the often dim light of intellectual history it is to be wary of progressivist assumptions — but unlike many of his other fans I apply that caution also within the body of his own work. Long story short, from 1865 to 1885 I see progress on several fronts but also bits of backsliding from his more prescient early insights. So it's a puzzle … and it will take more study to ravel out the reasons why.

Resources for reconciling Peirce's two accounts —
1. The 1870 account of logical involution
2. The 1885 account of universal quantification

Peirce's 1870 “Logic of Relatives” • Selection 12 • The Sign of Involution
• https://inquiryintoinquiry.com/2014/06/09/peirces-1870-logic-of-relatives-selection-12/
Comments —
(1) https://inquiryintoinquiry.com/2014/06/10/peirces-1870-logic-of-relatives-comment-12-1/
(2) https://inquiryintoinquiry.com/2014/06/11/peirces-1870-logic-of-relatives-comment-12-2/
(3) https://inquiryintoinquiry.com/2014/06/12/peirces-1870-logic-of-relatives-comment-12-3/
(4) https://inquiryintoinquiry.com/2014/06/14/peirces-1870-logic-of-relatives-comment-12-4/
(5) https://inquiryintoinquiry.com/2014/06/15/peirces-1870-logic-of-relatives-comment-12-5/

Peirce's 1885 “Algebra of Logic” • Selections
(1) https://inquiryintoinquiry.com/2024/03/24/peirces-1885-algebra-of-logic-selection-1/
(2) https://inquiryintoinquiry.com/2024/03/26/peirces-1885-algebra-of-logic-selection-2/
(3) https://inquiryintoinquiry.com/2024/03/30/peirces-1885-algebra-of-logic-selection-3/
(4) https://inquiryintoinquiry.com/2024/04/01/peirces-1885-algebra-of-logic-selection-4/

Peirce, C.S. (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, American Journal of Mathematics 7, 180–202.
• https://www.jstor.org/stable/2369451

#Peirce #Logic #AlgebraOfLogic #LogicOfRelatives #RelationTheory #CategoryTheory
#Semiotics #PredicateCalculus #Quantification #LogicalInvolution #ComputerScience

Peirce's 1885 “Algebra of Logic” • Discussion 1
• https://inquiryintoinquiry.com/2024/04/02/peirces-1885-algebra-of-logic-discussion-1/

Re: FB | Daniel Everett

DE:
❝One of the most important papers in the history of logic. “On the Algebra of Logic” was the first to introduce the term “quantifier”.

❝Peirce, C.S. (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, American Journal of Mathematics 7, 180–202.
• https://www.jstor.org/stable/2369451 ❞

As far as quantification by any other word goes, Peirce had already introduced a more advanced and “functional” concept of quantification in his 1870 “Logic of Relatives”. The subsequent passage to Fregean styles of first order logic would turn out to be a retrograde movement toward syntacticism (a species of nominalism), as seen in the general run of what fol‑lowed in the fol‑lowing years.

See ☞ Peirce's 1870 “Logic of Relatives”
• https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-overview/

Especially ☞ “The Sign of Involution”
• https://inquiryintoinquiry.com/2014/06/09/peirces-1870-logic-of-relatives-selection-12/

The connection between logical involution and universal quantification which Peirce put to use in his 1870 Logic of Relatives will turn up again a century later with the application of category theory to computer science and both of those in turn to logic. Just one more time Peirce was that far ahead of it.

See ☞ Lambek and Scott (1986), Introduction to Higher Order Categorical Logic, Cambridge University Press.
• https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems#Lambek

#Peirce #Logic #AlgebraOfLogic #LogicOfRelatives #RelationTheory #CategoryTheory
#Semiotics #PredicateCalculus #Quantification #LogicalInvolution #ComputerScience

Peirce's 1870 “Logic of Relatives” • Selection 3.2
• https://inquiryintoinquiry.com/2014/01/30/peirces-1870-logic-of-relatives-selection-3/

❝§3. Application of the Algebraic Signs to Logic❞

❝The Signs of Inclusion, Equality, Etc.❞

❝But not only do the significations of \(=\) and \(<\) here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations. Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes.

❝So, to write \(5 < 7\) is to say that \(5\) is part of \(7,\) just as to write \(\mathrm{f} < \mathrm{m}\) is to say that Frenchmen are part of men. Indeed, if \(\mathrm{f} < \mathrm{m},\) then the number of Frenchmen is less than the number of men, and if \(\mathrm{v} = \mathrm{p},\) then the number of Vice-Presidents is equal to the number of Presidents of the Senate; so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities.❞

(Peirce, CP 3.66)

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory

Peirce's 1870 “Logic of Relatives” • Selection 2.1
• https://inquiryintoinquiry.com/2014/01/29/peirces-1870-logic-of-relatives-selection-2/

❝§3. Application of the Algebraic Signs to Logic❞

❝Numbers Corresponding to Letters❞

❝I propose to use the term “universe” to denote that class of individuals about which alone the whole discourse is understood to run. The universe, therefore, in this sense, as in Mr. De Morgan's, is different on different occasions. In this sense, moreover, discourse may run upon something which is not a subjective part of the universe; for instance, upon the qualities or collections of the individuals it contains.

❝I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual. Thus in a universe of perfect men \((\mathrm{men}),\) the number of “tooth of” would be 32. The number of a relative with two correlates would be the average number of things so related to a pair of individuals; and so on for relatives of higher numbers of correlates. I propose to denote the number of a logical term by enclosing the term in square brackets, thus, \([t].\)❞

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory

Peirce's 1870 “Logic of Relatives” • Selection 1.2
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-selection-1/

❝The conjugative term involves the conception of third, the relative that of second or other, the absolute term simply considers an object. No fourth class of terms exists involving the conception of fourth, because when that of third is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship. Whether this reason for the fact that there is no fourth class of terms fundamentally different from the third is satisfactory of not, the fact itself is made perfectly evident by the study of the logic of relatives.❞

One thing that strikes me about the above passage is a pattern of argument I can recognize as invoking a closure principle. This is a figure of reasoning Peirce uses in three other places: his discussion of continuous predicates, his definition of a sign relation, and his formulation of the pragmatic maxim itself.

One might also call attention to the following two statements:

❝Now logical terms are of three grand classes.❞

❝No fourth class of terms exists involving the conception of fourth, because when that of third is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship.❞

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory

Peirce's 1870 “Logic of Relatives” • Selection 1.1
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-selection-1/

We pick up Peirce's text at the following point.

❝§3. Application of the Algebraic Signs to Logic❞

❝Use of the Letters❞

❝The letters of the alphabet will denote logical signs.

❝Now logical terms are of three grand classes.

❝The first embraces those whose logical form involves only the conception of quality, and which therefore represent a thing simply as “a ──”. These discriminate objects in the most rudimentary way, which does not involve any consciousness of discrimination. They regard an object as it is in itself as such (quale); for example, as horse, tree, or man. These are absolute terms.

❝The second class embraces terms whose logical form involves the conception of relation, and which require the addition of another term to complete the denotation. These discriminate objects with a distinct consciousness of discrimination. They regard an object as over against another, that is as relative; as father of, lover of, or servant of. These are simple relative terms.

❝The third class embraces terms whose logical form involves the conception of bringing things into relation, and which require the addition of more than one term to complete the denotation. They discriminate not only with consciousness of discrimination, but with consciousness of its origin. They regard an object as medium or third between two others, that is as conjugative; as giver of ── to ──, or buyer of ── for ── from ──. These may be termed conjugative terms.❞

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory

Peirce's 1870 “Logic of Relatives” • Preliminaries 5
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/

Individual terms are taken to denote individual entities falling under a general term. Peirce uses upper case Roman letters for individual terms, for example, the individual horses \(\mathrm{H}, \mathrm{H}^{\prime}, \mathrm{H}^{\prime\prime}\) falling under the general term \(\mathrm{h}\) for horse.

The path to understanding Peirce's system and its wider implications for logic can be smoothed by paraphrasing his notations in a variety of contemporary mathematical formalisms, while preserving the semantics as much as possible. Remaining faithful to Peirce's orthography while adding parallel sets of stylistic conventions will, however, demand close attention to typography-in-context.

Current style sheets for mathematical texts specify italics for mathematical variables, with upper case letters for sets and lower case letters for individuals. So we need to keep an eye out for the difference between the individual \(\mathrm{X}\) of the genus \(\mathrm{x}\) and the element \(x\) of the set \(X\) as we pass between the two styles of text.

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory

Peirce's 1870 “Logic of Relatives” • Preliminaries 4
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/

Conjugative Terms (Higher Adic Relatives)
• https://inquiryintoinquiry.files.wordpress.com/2021/11/peirces-1870-lor-e280a2-conjugative-terms-higher-adic-relatives-2.0.png

The Table displays the single-letter abbreviations and their verbal equivalents for the “conjugative terms” (or “higher adic relative terms”) used in Peirce's examples of logical formulas. Peirce used a distinctive typeface for the abbreviations of higher adic relative terms, rendered here as LaTeX “mathfrak”, Fraktur, or Gothic.

#Peirce #Logic #LogicOfRelatives #RelationTheory #LOR1870
#Boole #LogicalCalculus #MathematicalLogic #LogicalGraphs
#PropositionalCalculus #PredicateCalculus #CategoryTheory