#boole — Public Fediverse posts

Alicia #Boole au pays des #polytopes :

Au départ, il y a les cinq « solides platoniciens » vénérés en géométrie depuis l'Antiquité : le cube, le tétraèdre, l’octaèdre, le dodécaèdre et l’icosaèdre. Mais pourquoi s’arrêter aux 3 dimensions de l’espace ordinaire ? Alicia Boole Stott a consacré sa vie à chercher des solides réguliers en dimension 4… et elle a trouvé !

Une inspiration pour la future itération des #polyharmonies ;-)

https://www.arte.tv/fr/videos/107398-006-A/voyages-au-pays-des-maths/

Alicia #Boole au pays des #polytopes :

Au départ, il y a les cinq « solides platoniciens » vénérés en géométrie depuis l'Antiquité : le cube, le tétraèdre, l’octaèdre, le dodécaèdre et l’icosaèdre. Mais pourquoi s’arrêter aux 3 dimensions de l’espace ordinaire ? Alicia Boole Stott a consacré sa vie à chercher des solides réguliers en dimension 4… et elle a trouvé !

Une inspiration pour la future itération des #polyharmonies ;-)

https://www.arte.tv/fr/videos/107398-006-A/voyages-au-pays-des-maths/

In 1847, George Boole developed Boolean algebra, on which all digital computers are based. #Poetry #Science #History #Logic #Boole (https://sharpgiving.com/thebookofscience/items/p1847b.html)

In 1847, George Boole developed Boolean algebra, on which all digital computers are based. #Poetry #Science #History #Logic #Boole (https://sharpgiving.com/thebookofscience/items/p1847b.html)

Overall thoughts: this is such a shockingly original work that no wonder it caught on slowly, and we can certainly forgive all errors and infelicities of presentation. #WSJevons (mentioned above) led one response by acknowledging #Boole 's insights but trying to fold them into the old paradigm. But with developments such as emphasing inference over equality, abandoning partially defined connectives, and new quantifiers to allow binary+ relations, the formal approach to #logic was unstoppable.

Overall thoughts: this is such a shockingly original work that no wonder it caught on slowly, and we can certainly forgive all errors and infelicities of presentation. #WSJevons (mentioned above) led one response by acknowledging #Boole 's insights but trying to fold them into the old paradigm. But with developments such as emphasing inference over equality, abandoning partially defined connectives, and new quantifiers to allow binary+ relations, the formal approach to #logic was unstoppable.

#Boole 's closing chapter, with untranslated quotes in French, Italian, and ancient Greek, was not an easy read, but he compares and contrasts the abstraction from physical observation to mathematics with his abstraction from thought to mathematical #logic , noting that correctness is a criteria that can be meaningfully applied to thought processes but not to physical ones; and observes that just as physical science cannot be entirely reduced to maths, so it is with the science of the intellect.

#Boole 's closing chapter, with untranslated quotes in French, Italian, and ancient Greek, was not an easy read, but he compares and contrasts the abstraction from physical observation to mathematics with his abstraction from thought to mathematical #logic , noting that correctness is a criteria that can be meaningfully applied to thought processes but not to physical ones; and observes that just as physical science cannot be entirely reduced to maths, so it is with the science of the intellect.

#FinishedReading #Boole 's Laws of Thought, whose final chapters partly shift attention from #logic to #probabilityTheory and #philosophyOfScience . I don't have a strong sense of his historical standing in either discipline, although for probability there is https://en.m.wikipedia.org/wiki/Boole%27s_inequality . The attraction of probability is clear, with its range of values from 0 to 1, use of '1 -' for negation, multiplication for conjunction (of independent events) etc. There are rhymes here with his logic at least

#FinishedReading #Boole 's Laws of Thought, whose final chapters partly shift attention from #logic to #probabilityTheory and #philosophyOfScience . I don't have a strong sense of his historical standing in either discipline, although for probability there is https://en.m.wikipedia.org/wiki/Boole%27s_inequality . The attraction of probability is clear, with its range of values from 0 to 1, use of '1 -' for negation, multiplication for conjunction (of independent events) etc. There are rhymes here with his logic at least

Oof. #Boole #logic

Oof. #Boole #logic

The section where #Boole compares his #logic with Aristotelian syllogisms, the dominant approach in the West for more than two millennia, would have been key for readers of the time. I imagine the attached quote would have been unbelievably spicy, but virtually nobody would disagree with it now. With the help of some tidying up and extensions from Peirce, Lewis, Schroeder, Frege etc, Boole's vision won comprehensively, and helped to build our modern world.

The section where #Boole compares his #logic with Aristotelian syllogisms, the dominant approach in the West for more than two millennia, would have been key for readers of the time. I imagine the attached quote would have been unbelievably spicy, but virtually nobody would disagree with it now. With the help of some tidying up and extensions from Peirce, Lewis, Schroeder, Frege etc, Boole's vision won comprehensively, and helped to build our modern world.

On page 170 we finally see the conditional, if-then. This is in a section on 'secondary propositions' which relate the truth of propositions. If y then x is not, perhaps surprisingly given #Boole 's mission to arithmetise #logic , encoded as the exponent xʸ, but instead via introduction of a new unknown v, as y=vx (v and x). Given that conjunction as multiplication led us to elimination via semantically dubious propositional division, I suppose I should feel lucky we avoided logical logarithms!

On page 170 we finally see the conditional, if-then. This is in a section on 'secondary propositions' which relate the truth of propositions. If y then x is not, perhaps surprisingly given #Boole 's mission to arithmetise #logic , encoded as the exponent xʸ, but instead via introduction of a new unknown v, as y=vx (v and x). Given that conjunction as multiplication led us to elimination via semantically dubious propositional division, I suppose I should feel lucky we avoided logical logarithms!

To modern eyes a big thing missing in #Boole 's #logic is any proof of soundness, let alone completeness; he has what we would call a semantics (propositions as subsets of all entities in the universe of discourse, connectives as set operations e.g. conjunction as intersection) and many formula manipulations, but no verification of the manipulations in general. The approach is justified in Chapter 1 by " the general truths of Logic... when presented to the mind... at once command assent".

To modern eyes a big thing missing in #Boole 's #logic is any proof of soundness, let alone completeness; he has what we would call a semantics (propositions as subsets of all entities in the universe of discourse, connectives as set operations e.g. conjunction as intersection) and many formula manipulations, but no verification of the manipulations in general. The approach is justified in Chapter 1 by " the general truths of Logic... when presented to the mind... at once command assent".

A more twisted, but important, example: say x=yz, e.g. slithy = lithe and slimy. What can we say about y (lithe)? We rearrange as y=x/z, but division has no logical interpretation, so we develop the right hand side to xz + (1/0)x(1-z) + (0/0)(1-x)(1-z), with the (1-x)z case disappearing under coefficient 0. Conclusion: the lithe things are all the slithy slimy things, plus no, some, or all of the non-slithy non-slimy things. Independently, nothing is both slithy and not-slimy. #Boole #logic

A more twisted, but important, example: say x=yz, e.g. slithy = lithe and slimy. What can we say about y (lithe)? We rearrange as y=x/z, but division has no logical interpretation, so we develop the right hand side to xz + (1/0)x(1-z) + (0/0)(1-x)(1-z), with the (1-x)z case disappearing under coefficient 0. Conclusion: the lithe things are all the slithy slimy things, plus no, some, or all of the non-slithy non-slimy things. Independently, nothing is both slithy and not-slimy. #Boole #logic

What if the coefficients, e.g. f(x,y) for x,y ranging over 0 and 1, are not themselves 0 or 1? #Boole 's specific answer for case 0/0 is that the truth value is indeterminate; for any other value, including 1/0, we get the independent conclusion that the part of the developed proposition that the coefficient is applied to is 0. For example, x+y develops to 2xy + x + y, which is x + y with independent conclusion that the conjunction xy is 0 (false) - in keeping with + as disjoint union #logic

What if the coefficients, e.g. f(x,y) for x,y ranging over 0 and 1, are not themselves 0 or 1? #Boole 's specific answer for case 0/0 is that the truth value is indeterminate; for any other value, including 1/0, we get the independent conclusion that the part of the developed proposition that the coefficient is applied to is 0. For example, x+y develops to 2xy + x + y, which is x + y with independent conclusion that the conjunction xy is 0 (false) - in keeping with + as disjoint union #logic

#Boole sees #logic as arithmetic specialised to 0 and 1, as any function can be 'developed' e.g. by sending f(x,y) to f(1,1)xy + f(1,0)x(1-y) + f(0,1)(1-x)y + f(0,0)(1-x)(1-y). This looks like truth tables, considering all combinations of true and false, but this development only needs to happen at the end of a chain of reasoning, so intermediate terms do not have to be logically interpretable, by analogy with the use of i in finding real roots of polynomials. This raises soundness questions!

#Boole sees #logic as arithmetic specialised to 0 and 1, as any function can be 'developed' e.g. by sending f(x,y) to f(1,1)xy + f(1,0)x(1-y) + f(0,1)(1-x)y + f(0,0)(1-x)(1-y). This looks like truth tables, considering all combinations of true and false, but this development only needs to happen at the end of a chain of reasoning, so intermediate terms do not have to be logically interpretable, by analogy with the use of i in finding real roots of polynomials. This raises soundness questions!

The treatment of quantification by #Boole is interesting (quantifiers came along 20ish years later, with Frege). 'All x are (have property) y' is translated as x = vy (read concatenation as conjunction) where v is a distinguished 'indefinite' proposition, except that it must contain at least one y, understood to indicate that it is possible to find a subset of y that matches the collection x. 'Some x are y' is translated as vx = vy, where v must contain at least one each of x and y. #logic

The treatment of quantification by #Boole is interesting (quantifiers came along 20ish years later, with Frege). 'All x are (have property) y' is translated as x = vy (read concatenation as conjunction) where v is a distinguished 'indefinite' proposition, except that it must contain at least one y, understood to indicate that it is possible to find a subset of y that matches the collection x. 'Some x are y' is translated as vx = vy, where v must contain at least one each of x and y. #logic

The more I think about it, the more baffling I find #Boole 's insistence that x and x and x cannot be said to equal x (in his notation, x³ = x). One would think that two applications of the axiom x² = x (and replacing equals by equals) would get him there. It's like he confused the failure of a proof with the failure of the theorem. #logic

The more I think about it, the more baffling I find #Boole 's insistence that x and x and x cannot be said to equal x (in his notation, x³ = x). One would think that two applications of the axiom x² = x (and replacing equals by equals) would get him there. It's like he confused the failure of a proof with the failure of the theorem. #logic

This footnote is an example of the arithmetical approach to #logic making life terrible for #Boole ; given that x² (i.e. x and x) = x is an axiom, shouldn't x³ = x hold? Apparently not, as x³ - x = 0 'factorises' into gibberish terms like 1 + x (we can't add new things to the universe), or -1, which has no meaning at all (not to be confused with the negation of 1, which is 1 - 1 = 0). I must admit to my doubts about the well-definedness of this whole enterprise!

This footnote is an example of the arithmetical approach to #logic making life terrible for #Boole ; given that x² (i.e. x and x) = x is an axiom, shouldn't x³ = x hold? Apparently not, as x³ - x = 0 'factorises' into gibberish terms like 1 + x (we can't add new things to the universe), or -1, which has no meaning at all (not to be confused with the negation of 1, which is 1 - 1 = 0). I must admit to my doubts about the well-definedness of this whole enterprise!

#Boole develops #logic by close analogy with arithmetic, though he is at pains to say this is mere analogy and there is no a priori reason the rules should be the same. So while we usually think of logic as being about entailment, Boole virtually ignores it in the early going and makes equality primary; see the attached proof of the principle of contradiction (here 1 stands for the whole universe, and x - y, defined only if y is a subset of x, is set difference), with its arithmetical flavour.

#Boole develops #logic by close analogy with arithmetic, though he is at pains to say this is mere analogy and there is no a priori reason the rules should be the same. So while we usually think of logic as being about entailment, Boole virtually ignores it in the early going and makes equality primary; see the attached proof of the principle of contradiction (here 1 stands for the whole universe, and x - y, defined only if y is a subset of x, is set difference), with its arithmetical flavour.

#Boole 's propositions do not range merely across 0 and 1, as often presented today, but across subsets of all objects in the universe (or some agreed upon universe of discourse). If this sounds like Boolean Algebra, you're half right; conjunction is indeed intersection, but disjunction (which he writes +) is *disjoint* union, so x+y is not meaningfully defined in general, as with x/y in arithmetic (as y might be 0). This strikes me as something which might cause trouble later. #logic

#Boole 's propositions do not range merely across 0 and 1, as often presented today, but across subsets of all objects in the universe (or some agreed upon universe of discourse). If this sounds like Boolean Algebra, you're half right; conjunction is indeed intersection, but disjunction (which he writes +) is *disjoint* union, so x+y is not meaningfully defined in general, as with x/y in arithmetic (as y might be 0). This strikes me as something which might cause trouble later. #logic

#AmReading this 1854 book by George #Boole , which summarises his thoughts (first published a few years earlier) on #logic , as well as probability. Boole built the world I live in as a logician (and to extent, the world we all live in in the age of computers) but this is the first time I've read him in the original, so I thought I might make a thread with a few notes in it as I read it over the next few weeks.

#AmReading this 1854 book by George #Boole , which summarises his thoughts (first published a few years earlier) on #logic , as well as probability. Boole built the world I live in as a logician (and to extent, the world we all live in in the age of computers) but this is the first time I've read him in the original, so I thought I might make a thread with a few notes in it as I read it over the next few weeks.

Après Uncle Bob et son Clean code, bouquin que je recommande d'ailleurs, j'ai eu envie de m'intéresser aux travaux de George Boole.
Bref, si on vous demande mon livre de chevet pour les jours qui viennent, le voici.
#Boole

On #Boole https://www.instagram.com/p/CtmdOW-s9TC

On #Boole https://www.instagram.com/p/CtmdOW-s9TC

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 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

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2023/03/28/survey-of-animated-logical-graphs-5/

This is a Survey of blog and wiki posts on Logical Graphs, encompassing several families of graph-theoretic structures originally developed by Charles S. Peirce as graphical formal languages or visual styles of syntax amenable to interpretation for logical applications.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#Boole #BooleanAlgebra #BooleanFunctions #ModelTheory #ProofTheory
#SpencerBrown #LawsOfForm #PropositionalCalculus #LogicAsSemiotics

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2023/03/28/survey-of-animated-logical-graphs-5/

This is a Survey of blog and wiki posts on Logical Graphs, encompassing several families of graph-theoretic structures originally developed by Charles S. Peirce as graphical formal languages or visual styles of syntax amenable to interpretation for logical applications.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#Boole #BooleanAlgebra #BooleanFunctions #ModelTheory #ProofTheory
#SpencerBrown #LawsOfForm #PropositionalCalculus #LogicAsSemiotics

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 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 1.2
• 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” • 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 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

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

Peirce's 1870 “Logic of Relatives” • Preliminaries 3
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/

Simple Relative Terms (Dyadic Relatives)
• https://inquiryintoinquiry.files.wordpress.com/2021/11/peirces-1870-lor-e280a2-simple-relative-terms-dyadic-relatives-2.0.png

The Table displays the single-letter abbreviations and their verbal equivalents for the “simple relative terms” (or “dyadic relative terms”) used in Peirce's examples of logical formulas. Peirce used a distinctive typeface for the abbreviations of dyadic relative terms, rendered here as LaTeX “mathit” or Italics.

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

Peirce's 1870 “Logic of Relatives” • Preliminaries 3
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/

Simple Relative Terms (Dyadic Relatives)
• https://inquiryintoinquiry.files.wordpress.com/2021/11/peirces-1870-lor-e280a2-simple-relative-terms-dyadic-relatives-2.0.png

The Table displays the single-letter abbreviations and their verbal equivalents for the “simple relative terms” (or “dyadic relative terms”) used in Peirce's examples of logical formulas. Peirce used a distinctive typeface for the abbreviations of dyadic relative terms, rendered here as LaTeX “mathit” or Italics.

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

Peirce's 1870 “Logic of Relatives” • Preliminaries 2
• https://inquiryintoinquiry.com/2014/01/27/peirces-1870-logic-of-relatives-preliminaries/

Absolute Terms (Monadic Relatives)
• https://inquiryintoinquiry.files.wordpress.com/2021/11/peirces-1870-lor-e280a2-absolute-terms-monadic-relatives-2.0.png

The Table displays the single-letter abbreviations and their verbal equivalents for the “absolute logical terms” (or “monadic relative terms”) used in Peirce's examples of logical formulas throughout the rest of the paper. Peirce used a distinctive typeface for the absolute term abbreviations, rendered here as LaTeX “mathrm” or Roman.

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