Animated Logical Graphs • 47

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (30) (45) (46)

A logical concept represented by a boolean variable has its extension, the cases it covers in a designated universe of discourse, and its comprehension (or intension), the properties it implies in a designated hierarchy of predicates.  The formulas and graphs tabulated in previous posts are well-adapted to articulate the syntactic and intensional aspects of propositional logic.  But their very tailoring to those tasks tends to slight the extensional and therefore empirical applications of logic.  Venn diagrams, despite their unwieldiness as the number of logical dimensions increases, are indispensable in providing the visual intuition with a solid grounding in the extensions of logical concepts.  All that makes it worthwhile to reset our table of boolean functions on two variables to include the corresponding venn diagrams.

Venn Diagrams and Logical Graphs on Two Variables
\text{Boolean Function} \text{Entitative Graph} \text{Existential Graph}
f₀(x,y) Cactus Root
 
Cactus Stem
 
f_{0} \text{false} \text{false}
f₁(x,y) Cactus (xy)
 
Cactus (x)(y)
 
f_{1} \lnot (x \lor y) \lnot x \land \lnot y
f₂(x,y) Cactus (x(y))
 
Cactus (x)y
 
f_{2} \lnot x \land y \lnot x \land y
f₃(x,y) Cactus (x)
 
Cactus (x)
 
f_{3} \lnot x \lnot x
f₄(x,y) Cactus ((x)y)
 
Cactus x(y)
 
f_{4} x \land \lnot y x \land \lnot y
f₅(x,y) Cactus (y)
 
Cactus (y)
 
f_{5} \lnot y \lnot y
f₆(x,y) Cactus ((x,y))
 
Cactus (x,y)
 
f_{6} x \ne y x \ne y
f₇(x,y) Cactus (x)(y)
 
Cactus (xy)
 
f_{7} \lnot x \lor \lnot y \lnot (x \land y)
f₈(x,y) Cactus ((x)(y))
 
Cactus xy
 
f_{8} x \land y x \land y
f₉(x,y) Cactus (x,y)
 
Cactus ((x,y))
 
f_{9} x = y x = y
f₁₀(x,y) Cactus y
 
Cactus y
 
f_{10} y y
f₁₁(x,y) Cactus (x)y
 
Cactus (x(y))
 
f_{11} x \Rightarrow y x \Rightarrow y
f₁₂(x,y) Cactus x
 
Cactus x
 
f_{12} x x
f₁₃(x,y) Cactus x(y)
 
Cactus ((x)y)
 
f_{13} x \Leftarrow y x \Leftarrow y
f₁₄(x,y) Cactus xy
 
Cactus ((x)(y))
 
f_{14} x \lor y x \lor y
f₁₅(x,y) Cactus Stem
 
Cactus Root
 
f_{15} \text{true} \text{true}

Resources

cc: Cybernetics Communications (1) (2)FB | Logical Graphs • Ontolog Forum (1) (2)
cc: Peirce List (1) (2) (3) (4) (5) (6) (7) (8) • Structural Modeling (1) (2) • Systems (1) (2)

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Animated Logical Graphs • 46

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (30) (45)

Another way of looking at Peirce duality is given by the following Table, which shows how logical graphs denote boolean functions under entitative and existential interpretations.  Column 1 shows the logical graphs for the sixteen boolean functions on two variables.  Column 2 shows the boolean functions denoted under the entitative interpretation and Column 3 shows the boolean functions denoted under the existential interpretation.

\text{Logical Graphs} \stackrel{_\bullet}{} \text{Entitative and Existential Interpretations}

Logical Graphs • Entitative and Existential Interpretations

Resources

cc: Cybernetics Communications (1) (2)FB | Logical Graphs • Ontolog Forum (1) (2)
cc: Peirce List (1) (2) (3) (4) (5) (6) (7) (8) • Structural Modeling (1) (2) • Systems (1) (2)

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , | 3 Comments

Animated Logical Graphs • 45

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (30)

There’s a nice interplay between geometric and logical dualities in C.S. Peirce’s graphical systems of logic, rooted in his discovery of the amphecks \textsc{Nand} and \textsc{Nnor} and flowering in his logical graphs for propositional and predicate calculus.  Peirce’s logical graphs bear the dual interpretations he dubbed entitative and existential graphs.

Here’s a Table of Boolean Functions on Two Variables, using an extension of Peirce’s graphs from trees to cacti, illustrating the duality so far as it affects propositional calculus.

\text{Boolean Functions on Two Variables}

Boolean Functions on Two Variables

Resources

cc: Cybernetics Communications (1) (2)FB | Logical Graphs • Ontolog Forum (1) (2)
cc: Peirce List (1) (2) (3) (4) (5) (6) (7) (8) • Structural Modeling (1) (2) • Systems (1) (2)

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , | 3 Comments

Problems In Philosophy • 12

Re: R.J. Lipton and K.W. ReganThe Night Of The Ethical Algorithm
Re: K.W. ReganThe Election Night Time Warp
Re: Ontolog ForumJohn Sowa

JFS:
C.S. Peirce made a very clear and sharp distinction between formal or mathematical logic and logic as semiotic.
\cdots
Short summary:  When Peirce uses the word ‘logic’ by itself, it’s important to check the context to see whether he’s talking about formal logic or logic as semiotic.

Dear John,

The first post of this series was prompted by a post 4 years ago on the Gödel’s Lost Letter and P=NP blog which jumped from the frying pan of problems in programming to the fire of problems in philosophy.  Then last week two more posts, linked above, made the leap to two of the most flagrant problems in politics, namely, (1) the passage from effective and efficient algorithms to ethical algorithms and (2) the perils of navigating turbulent seas in a ship of state guided by elective representation, where the people pick their pilots from among themselves to represent their collective will and whatever wits they can muster.

Bearing all that in mind, I would like to keep exploring the ancient issues of aesthetics, ethics, and logic from our contemporary algorithmic perspective.  There the descriptive and normative orientations to knowledge parallel the systems-theoretic dimensions of information and control.  And there we find normative sciences appearing under the banner of “design sciences”.  In that frame the art of crafting a ship of state becomes a question of optimal design for a human society.

When it comes to logic, then, a generic conception will do for now, leaving Peirce’s definition of logic as formal semiotic and fine points of the difference between mathematical logic and mathematics of logic to another day.

Resources

cc: CyberneticsOntolog ForumPeirce ListStructural ModelingSystems Science

Posted in Aesthetics, Algorithms, Animata, Automata, Beauty, C.S. Peirce, Ethics, Inquiry, Justice, Logic, Model Theory, Normative Science, Peirce, Philosophy, Pragmatism, Problem Solving, Proof Theory, Summum Bonum, Truth, Virtue | Tagged , , , , , , , , , , , , , , , , , , , | 1 Comment

Problems In Philosophy • 11

Re: Problems In Philosophy 9Richard Saunders

RS:
BTW I’m not sure I really see a distinction between descriptive and normative (prescriptive?) science except in the set of aims, goals, etc. that are entertained.  It might be useful to try to characterize some distinctions in the goals of each.

Re: Problems In Philosophy 10Richard Saunders

RS:
Jon, the philosophy of science is all about the aims of science \textsc{and} good ways of achieving them.  I’m still not seeing a clear distinction, traditions notwithstanding, between descriptive and normative science.  I do see the recursive entanglement though, and I’m still wondering if we can find common axioms that underlie both.

\textsc{Saturday, November 7}

Dear Richard,

Sue and I will be downing some bubbly and sleeping it off till the dawn’s early light, but Sue was into this Policy-Theory Reunion stuff well before I clued into it, so here’s one of her earlier papers you might find of interest in the interim.

  • Scott, David K., and Awbrey, Susan M. (1993), “Transforming Scholarship”, Change : The Magazine of Higher Learning, 25(4), 38–43.  Online (1) (2) (3).

\textsc{Monday, November 9}

I am still trying to unscramble my brains after the week’s events but I’m surprised to see so much difficulty over the difference between descriptive sciences, the special sciences as Peirce called them, and normative sciences like aesthetics, ethics, and logic.  I deferred to common idiom and conventional wisdom regarding the irreducibility of “Ought” to “Is” but roughly the same dimension and tension is recognized under a legion of names — policy vs. theory, procedural vs. declarative, deontic vs. ontic, and many others.

A pragmatic semiotician’s ears will naturally perk up at reading the word irreducibility above and lead to wondering whether the irreducibility of normative to descriptive has anything to do with the irreducibility of triadic relations to dyadic relations.

To my way of thinking, yes, it does.

Resources

cc: CyberneticsOntolog ForumPeirce ListStructural ModelingSystems Science

Posted in Aesthetics, Algorithms, Animata, Automata, Beauty, C.S. Peirce, Ethics, Inquiry, Justice, Logic, Model Theory, Normative Science, Peirce, Philosophy, Pragmatism, Problem Solving, Proof Theory, Summum Bonum, Truth, Virtue | Tagged , , , , , , , , , , , , , , , , , , , | 1 Comment

Problems In Philosophy • 10

Re: Ontolog ForumDavid Whitten

DW:
Why does classical tradition or any tradition consider logic to be a normative science?

Dear David,

A science is called that because it deals in knowledge (Latin scientia).  Knowing what is the case in a given domain of experience may be distinguished from knowing what ought to be in a given set of circumstances, and people who think in threes, like Kant and Peirce and me, add knowing what may be hoped to the mix.

In the quest to understand how science works a praxis/pragmatist like myself gives the process, inquiry, equal billing with the product, knowledge.  People have gotten used to seeing sciences as bodies of ostensible knowledge (BOOKs) and taking their analysis as a matter of assigning them distinctive catalogue numbers and sorting them to the indicated library shelves.  That is all well and good but it leaves an all too static impression of science if we settle for that.

Here are capsule summaries on the Sciences of Is and the Sciences of Ought from the Wikiversity articles on Descriptive Science and Normative Science.

Descriptive Science
A descriptive science, or a special science, is a form of inquiry, typically involving a community of inquiry and its accumulated body of provisional knowledge, which seeks to discover what is true about a recognized domain of phenomena.
Normative Science
A normative science is a form of inquiry, typically involving a community of inquiry and its accumulated body of provisional knowledge, which seeks to discover good ways of achieving recognized aims, ends, goals, objectives, or purposes.
The three normative sciences, according to traditional conceptions in philosophy, are aesthetics, ethics, and logic.

Resources

cc: CyberneticsOntolog ForumPeirce ListStructural ModelingSystems Science

Posted in Aesthetics, Algorithms, Animata, Automata, Beauty, C.S. Peirce, Ethics, Inquiry, Justice, Logic, Model Theory, Normative Science, Peirce, Philosophy, Pragmatism, Problem Solving, Proof Theory, Summum Bonum, Truth, Virtue | Tagged , , , , , , , , , , , , , , , , , , , | 2 Comments

Problems In Philosophy • 9

Re: FB | Ecology Of Systems ThinkingRichard Saunders

RS:
Hume’s is/ought dichotomy:  are these as Gould said “non-overlapping magisteria” or are they concentric domains?  Is a science of aesthetics at the core?  If memory serves it seems like that was what Wittgenstein suggested at the end of Tractatus.  In The Moral Landscape, Harris narrows the aesthetic focus to a distinction between the minimum and maximum suffering of all sentient beings.  Maximum suffering is bad or ugly and minimum suffering is good or beautiful.  The relationship of conduct to result is the subject of consequentialism, isn’t it?  Isn’t that also the subject of science?

I know a lot of people see a cut and dried dichotomy here and conventional wit says you can’t derive Ought from Is.  My tracings of the boundaries though tend to find them recursively entangled.

RS:
Recursively entangled is a nice phrase, like the the chicken and the egg.  But I’m still wondering about the catch-22.  On what general axiom is aesthetics/ethics/logic based?  Harris suggests it’s minimizing net suffering.  (That doesn’t imply the elimination of suffering, because some suffering has a net positive result.)

I got no absolutes here.  I have my personal aesthetic, but a personal aesthetic is the moral equivalent of a religion, and folks are pretty free about that.

I’ll have more to say about my personal aesthetic … all in good time.

Resources

cc: CyberneticsOntolog ForumPeirce ListStructural ModelingSystems Science

Posted in Aesthetics, Algorithms, Animata, Automata, Beauty, C.S. Peirce, Ethics, Inquiry, Justice, Logic, Model Theory, Normative Science, Peirce, Philosophy, Pragmatism, Problem Solving, Proof Theory, Summum Bonum, Truth, Virtue | Tagged , , , , , , , , , , , , , , , , , , , | 3 Comments

Problems In Philosophy • 8

Re: Ontolog ForumDavid Whitten

Dear David,

I’ll extend this post tomorrow, apocalypse permitting, but while I wait for the election returns I’ll post just a pair of links to the Wikiversity articles on Descriptive Science and Normative Science, forked over from the Wikipedia articles as I left them a decade and a half ago.  I have no idea what, if anything exists on Wikipedia itself these days but this much gives the basic ideas in a couple of nutshells.

Resource

cc: CyberneticsOntolog ForumPeirce ListStructural ModelingSystems Science

Posted in Aesthetics, Algorithms, Animata, Automata, Beauty, C.S. Peirce, Ethics, Inquiry, Justice, Logic, Model Theory, Normative Science, Peirce, Philosophy, Pragmatism, Problem Solving, Proof Theory, Summum Bonum, Truth, Virtue | Tagged , , , , , , , , , , , , , , , , , , , | Leave a comment

Problems In Philosophy • 7

Re: FB | Charles S. Peirce Society
(a) John Corcoran • Cosmic Justice Hypotheses
(b) John Corcoran • The Inseparability of Logic and Ethics

Peirce emphasized the intricate relationships among the Big Three Normative Sciences — Aesthetics, Ethics, Logic — a topic early and often discussed in the secondary literature and on the Peirce List.  One might also compare Theodore Parker’s well-known thesis:

I do not pretend to understand the moral universe;
the arc is a long one, my eye reaches but little ways;
I cannot calculate the curve and complete the figure by
the experience of sight;  I can divine it by conscience.
And from what I see I am sure it bends towards justice.

Theodore Parker

Questions about the interdependence of the principal normative sciences — Aesthetics, Ethics, Logic — just came up on another blog and prompted me to go looking for some of my earlier grapplings with the subject.  There’s an initial fragment of that harvest in the following post.

cc: CyberneticsOntolog ForumPeirce ListStructural ModelingSystems Science

Posted in Aesthetics, Algorithms, Animata, Automata, Beauty, C.S. Peirce, Ethics, Inquiry, Justice, Logic, Model Theory, Normative Science, Peirce, Philosophy, Pragmatism, Problem Solving, Proof Theory, Summum Bonum, Truth, Virtue | Tagged , , , , , , , , , , , , , , , , , , , | Leave a comment

Problems In Philosophy • 6

Another one of those recurring questions I’m constantly forgetting what I wrote about or where before just came up again on the Gödel’s Lost Letter blog.

Re: R.J. Lipton and K.W. ReganThe Night Of The Ethical Algorithm

Classical tradition views logic as a normative science, one whose object is truth.  This puts logic on a par with ethics, whose object is justice or morality in action, and aesthetics, whose object is beauty or the admirable for its own sake.

The pragmatic spin on this line of thinking treats logic, ethics, aesthetics as a concentric series of normative sciences, each a subdiscipline of the next.  Logic tells us how we ought to conduct our reasoning in order to achieve the goals of reasoning in general.  Thus logic is a special case of ethics.  Ethics tells us how we ought to conduct our activities in general in order to achieve the good appropriate to each enterprise.  What makes the difference between a normative science and a prescriptive dogma is whether this telling is based on actual inquiry into the relationship of conduct to result, or not.

Here’s a bit I wrote on this a long time ago in a galaxy not far away —

cc: CyberneticsOntolog ForumPeirce ListStructural ModelingSystems Science

Posted in Aesthetics, Algorithms, Animata, Automata, Beauty, C.S. Peirce, Ethics, Inquiry, Justice, Logic, Model Theory, Normative Science, Peirce, Philosophy, Pragmatism, Problem Solving, Proof Theory, Summum Bonum, Truth, Virtue | Tagged , , , , , , , , , , , , , , , , , , , | 2 Comments