Inquiry Into Inquiry

Re: Logical Graphs, Truth Tables, Venn Diagrams • 6
Re: Amphecks

On the subject of Peirce’s ampheck operators, see the earlier discussion of their duality under entitative and existential interpretations.

• Animated Logical Graphs • 74

The ampheck operators are dual with respect to entitative and existential interpretations:

Under the existential interpretation:

Under the entitative interpretation:

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Re: Laws of Form • John Mingers • Lyle Anderson

Dear John, Lyle,

See:  Ampheck

Peirce discovered this about 1880 but did not publish it, leaving it to be named after Sheffer at a much later date.  In one discussion Peirce used simple concatenation for the abstract operation which can be interpreted in two ways:  “Both Not” (joint denial, Nnor) or “Not Both” (alternate denial, Nand).  In the passage linked above Peirce used a symbol for Nnor whose nearest facsimiles in HTML are (⋏) and “⥿” (⥿), adding an overbar for Nand.  Peirce used 2 × 2 matrices to represent the truth tables of all 16 boolean operators then converted the matrices into cursive symbols for the operators.  Warren S. McCulloch mentioned Peirce’s discovery and his matrices, referring to Nand and Nnor collectively as “amphecks” on account of their abstract duality.

Regards,

Jon

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Re: Laws of Form • Lyle Anderson
Re: Anderson, Lyle A. III (1981), “Systematic Analysis of Algorithms”,
Open Access Master’s Theses, Paper 1167, (1) (2).

Thanks, Lyle, your Chapter 4, “Dealing With Conditional Statements”, provides a detailed treatment of algorithmic branching constructs in general purpose programming languages but as you noted in saying, “we are already way outside the realm of truth tables with only and it tangos with a much-higher-maintenance date than the one John Mingers brought to the dance.

I think we are making this problem harder than it needs to be.  Let’s go back to the original question and try to view it with fresh eyes.  All we have to decide is which candidate among the three-variable boolean functions provides a reasonable mathematical proxy for what we mean when we say,   Experience with informal-to-formal translation tells us there may be no functional form capturing every nuance of a natural language idiom but there is usually one serving all practical purposes in empirical and mathematical contexts.

Resources

• Cactus Rules • NKS Forum • Inquiry List • Ontology List

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All we are saying is give Peirce a chance

Re: Laws of Form • John Mingers • Lyle Anderson

Dear John, Lyle,

I’ve seen too many ways of interpreting and implementing If‑Then‑Else clauses to know what any one person or processor means by them until they give me the truth table they have in mind, so if you write out the truth table you like for them I’ll be able to work with that and say something more definite about it.

More importantly, once we get the full power of Peirce’s logical graphs, Spencer Brown’s calculus of indications, and the extensions to cactus graphs and differential logic in gear we’ll find there are better, clearer, more efficient ways of handling Boolean Expansions and Case Analysis and more generally applying propositional logic to real problems.

Here’s the NKS Forum link again:

• NKS Forum • Cactus Rules

The anchor post of that series used to have a file attached with the full set of cactus graphs for propositions on three variables … but it looks like the file was not preserved.  There’s a couple of links to other copies below.

• Cactus Rules • Inquiry List • Ontology List

Regards,

Jon

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Re: Laws of Form • John Mingers • Lyle Anderson

Dear John, Lyle,

There is nothing simple about the interpretation of If-Then-Else constructions in ordinary language as they combine the equivocation between formal and material implication at the outset with the vacillation between exclusive and inclusive disjunction at the final Or-Else.

Nor is there anything straightforward about the implementation of If-Then-Else clauses in half-functional half-procedural programming languages like Pascal.  In settings like that they do not render as pure boolean expressions but as boolean tests determining a choice between procedural branches.  Multiply that by the diversity of evaluation strategies for boolean expressions, (complete | partial), (eager | greedy | lazy), etc., and the possibilities are legion.  That is all well and good, those are just the choices that are out there, and we can work with anyone’s understanding of If-Then-Else as a boolean function so long as they give us their intended truth table so we don’t have to guess what they have in mind.

I’ll touch on If-Then-Else again when we turn to what I regard as the proper handling of Case Analysis in the systems of logical graphs evolving from the work of C.S. Peirce and Spencer Brown.

As it happens, I did once write out all 256 boolean functions on three variables in cactus syntax several years ago — pursuant to discussions in Stephen Wolfram’s New Kind of Science (NKS) Forum regarding Elementary Cellular Automaton Rules (ECARs), which are in effect just that set of boolean functions.  I’ll have to dig up a passel of ancient links from the WayBack Machine, but see the following archive page for a hint of how it went.

• Cactus Rules

To be continued …

Jon

References

• “Number of Boolean Functions Distinct under Complementation/Permutation”, A000370, N.J.A. Sloane (ed.), The On-Line Encyclopedia of Integer Sequences.
• “The If..then..else statement”, in Michaël Van Canneyt (May 2021), Free Pascal Reference Guide.

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Re: Laws of Form • John Mingers

JM:

Most of the recent discussion is about two-variable logic forms where there is a logical relation between two logical variables.  I want to bring up the subject of three-variable logic which I think is very rich but not much discussed.

In two-variable logic, as we know, there are 16 possible relations.  With three variables, there are 8 rows in the truth table and so 2⁸ = 256 possibilities.  Many of these are the same at 2-variable, eg. AND(a,b,c) or OR(a,b,c) but some are different, eg. IF a THEN b ELSE c.  This latter one is really at the heart of all computer programming.

I haven’t seen much written about this although William Bricken has done some (see for example “Symmetry in Boolean Functions with Examples for Two and Three Variables”).  Here he shows that when you take into account reflections and rotations there are actually 14 distinct forms within the 256.

Dear John,

One of the biggest advantages of the systems of graphical forms derived from C.S. Peirce’s logical graphs and Spencer Brown’s calculus of indications is precisely the conceptual and computational efficiencies they afford us in dealing with propositional forms and boolean functions of many variables.  This has been one of my main motivations in pursuing their development for the last half century and I think we have hopes of enjoying those benefits once we’ve had our dose of minimum logical requirements and cross the threshold of first principles.

That said, I still have work to do on the logical graphs for two-variable boolean functions since I’ve been using those as logical man-in-the-moon marigolds to study the effects of the duality.  That duality is associated with a transformation group of order two which partitions the set of sixteen functions into ten orbits.  The groups William Bricken considers have much higher orders at each number of variables and thus partition their spaces of functions into many fewer orbits in each case.  See the first reference below.

Have to break here …

Jon

References

• “Number of Boolean Functions Distinct under Complementation/Permutation”, A000370, N.J.A. Sloane (ed.), The On-Line Encyclopedia of Integer Sequences.
• “The If..then..else statement”, in Michaël Van Canneyt (May 2021), Free Pascal Reference Guide.

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cc: FB | Logical Graphs • Laws of Form