31:T686,{ const PADDING = {top: 10, bottom: 10, left: 10, right: 10}; const TYPE = prefs.REPR; const id0 = DOM.uid(), id1 = DOM.uid(), id2 = DOM.uid(), id3 = DOM.uid(); yield md`

3=1: ${`${univ1ui_a.v3}`}

` formform.graph.createGraph(TYPE,univ1ui_a.v0,{parentId: id0.id, styleClass: prefs.STYLECLASS, padding: PADDING}); formform.graph.createGraph(TYPE,univ1ui_a.v1,{parentId: id1.id, styleClass: prefs.STYLECLASS, padding: PADDING}); formform.graph.createGraph(TYPE,univ1ui_a.v2,{parentId: id2.id, styleClass: prefs.STYLECLASS, padding: PADDING}); formform.graph.createGraph(TYPE,univ1ui_a.v3,{parentId: id3.id, styleClass: prefs.STYLECLASS, padding: PADDING}); }32:T65a,mdfn`Fortunately, it turns out that if you have two (different) terms that conjugate the same set of variables, a *XOR*-FORM relating them will always be reducible to a simple relation. If you take a closer look on the FORM above you can see why this is the case. If you need to be absolutely sure, here is a simple proof (you may skip it if you like): > Imagine *a* and *b* in the *XOR*-FORM above to be conjugations of the same set of variables, like ${tex`(((x))(y))`} or ${tex`((x)(y))`}. If *a* and *b* are marked individually, as they are on the right hand side (the *“not AND”*-part) of *XOR*, by application of *C1*[^4]*:* ${tex`((a)) = a`}, they will dissolve into any of the terms *x*, *y*, (*x*) or (*y*). > From here, there are two possible simplifications: Either some of the terms are equal, then by the *law of calling* they merge into one. Or we might have the same variable in two terms, but one of them is marked (like *x* and (*x*) ). In this case, by *C2*[^5]*:* ${tex`(ab)b = (a)b`} we can degenerate the *x* from (*x*) leaving us with just the mark, which will dominate every other term. Consequentially, the right hand term of *XOR* will simply cancel out, leaving us with ${tex`((ab)) = ab`} by *C1*. > In general, if there is a difference in both conjunction terms *a* and *b* and they both contain the exact same set of variables, the *“not AND”*-part of *XOR* will always cancel out and *XOR* will simplify to *OR*. However, this is not the case if *a* and *b* are both the same terms, because then *XOR* will (by definition) always yield 0, while *OR* will condense both FORMs into one.`33:T50f,univHTML = { const FORM = Object.values(univ2ui_ab).reduce((acc,curr) => acc+curr); const PADDING = {top: 10, bottom: 10, left: 10, right: 10}; const TYPE = prefs.REPR; const id = DOM.uid(); yield md`## 4-valued Universes > *(Click the link behind the formula for a better view of the complete FORM.)*

${`${FORM}`}

`; const graph = formform.graph.createGraph(TYPE,FORM,{parentId: id.id, styleClass: prefs.STYLECLASS, padding: PADDING}); const parent = d3.select(graph.parent._groups[0][0]); parent.style('margin', '0'); //parent.style('margin', '0 0 1rem 0'); if (TYPE === 'pack') { const svg = d3.select(graph.svg._groups[0][0]); const scale = 0.32; const g = svg.select('g'); const str = g.attr('transform').split(','); const cx = g.node().getBBox().width/2; const cy = g.node().getBBox().height/2; g.attr('transform',`matrix(${scale}, 0, 0, ${scale}, ${scale*cx}, ${scale*cy})`); svg.style('width',`${Math.floor(width)}px`) .style('height',`${Math.floor(g.node().getBBox().height * scale)}px`); } else { parent.style('overflow','scroll'); } }34:T5e5,md`As you can see, the formula gets incredibly huge even for just two variables (although I used double marks here for grouping, but still) which just reminds us of the complexity involved when dealing with decisions in the undetermined compared to a 2-valued calculus which would only have to deal with four constituents/decisions:

${`${'((a)b)(a(b))((a)(b))(ab)'}`} → 10, 01, 11, 00

> *See my earlier Observable [“Universes of decision FORMs”](https://observablehq.com/@formsandlines/universes-of-decision-forms) for some experimental *binary* universes (“binary” because their constituents are not clearly separated for undetermined and imaginary values, just for marked/unmarked).* Of course, since *FORM logic* is what Ralf Peyn called a *Erkenntnislogik* (logic of cognition), this is not an *ontologically real* universe like a universe of things, classes or facts obtainable in the world. FORM logic presupposes no actual universe. It would be more like a *universe of possibilities* to focus our attention between *n* different identities (which are just designation products and thus results of the same operations of the mind that apply to all FORMs). I guess such a universe would be more in line with the *continuum* concept of Charles Sanders Peirce (which is inherently vague/undetermined) than Boole's universe of infinite, but distinct class divisions.`35:T412,viewof footnotes = footlist([ {ref: '1', text: `See Ralf Peyn: uFORM iFORM *(engl. edition, 2019)*, p. 24–27 for reference on ((*a*)*b*) FORMs.`}, {ref: '2', text: `See ibid., p. 37–46 for reference on self-equivalent Re-entry FORMs and how they relate to even and odd numbers of marks/Re-entries. Also see my tweets on [{2r|a}](https://twitter.com/diagramaniac/status/1138009177127866368) and [{a}](https://twitter.com/diagramaniac/status/1138024034795499520) (aka {2r+1|a}) if you do not know how to derive these FORMs from Ralfs explications.`}, {ref: '3', text: `See ibid., p. 122f for reference on the commutative Relation of undetermined FORMs and a look into its SelFi evolution.`}, {ref: '4', text: `See George Spencer Brown: Laws of Form *(Bohmeier Verlag, reprint 2011)*, p. 23 where *C1* is a consequence of “The primary algebra” which can also be applied to *FORM logic*.`}, {ref: '5', text: `See ibid., p. 25 where *C2* is also a consequence of “The primary algebra”.`}, ], {separator: false, title: false})36:[] 2f:{"title":"Determination in the undetermined","cells":[{"id":0,"value":"md`# Determination in the undetermined`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1298,"value":"metadata({autor:\"Peter Hofmann\", created:\"25.06.2019\", fstPublish:\"12.07.2020\", lastUpdate:\"13.07.2020\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":122,"value":"viewof prefs = userPrefs.form","pinned":false,"hidden":false,"mode":"ojs"},{"id":349,"value":"mdfn`Ralf Peyn has shown in [*uFORM iFORM*](https://uformiform.info) that we form decisions by operating ${tex`((a)b)`} FORMs and that this is essential to our ability to create concepts. His logic of cognition called *FORM logic* operates with attention/focus, forming signs to produce identities. We need ${tex`((a)b)`} for our orientation in the undetermined, it is already actualized when we set a mark to observe *something* instead of everything else: ${tex`(a)`}.[^1]`","pinned":false,"hidden":false,"mode":"ojs"},{"id":384,"value":"{ \n const PADDING = {top: 10, bottom: 10, left: 10, right: 10};\n const TYPE = prefs.REPR;\n \n const id1 = DOM.uid();\n const id2 = DOM.uid();\n \n yield md`

…and of course:

`;\n\n formform.graph.createGraph(TYPE,'((a)b)',{parentId: id1.id, styleClass: prefs.STYLECLASS, padding: PADDING});\n formform.graph.createGraph(TYPE,'{a,b}',{parentId: id2.id, styleClass: prefs.STYLECLASS, padding: PADDING});\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":138,"value":"md`## FORMs to determine single values for one variable\n\nThe following four FORMs are determined/marked for exactly one interpretation while they are empty/unmarked for everything else.\n\n> *Note:* To check the value table for each FORM, click on the link under its formula. It will take you to an online tool called [FORM tricorder](https://formform.formsandlines.eu/tricorder/) that I created where you can view, calculate and analyze 4-valued FORMs. If you click on “Show explanations” you can also see a reference on my paranthesis notation for undetermined FORMs.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":8,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":232,"value":"mdfn`My attempt to interpret these FORMs is that 1=1 and 0=1 is *determination by relating the undetermined*. They work exactly because ${tex`\\{2r|a\\}`} or ${tex`f=((f)a)`} results in an *even* number and ${tex`\\{a\\}`} or ${tex`f(f)=(fa)`} results in an *odd* number of marks and ${tex`ui=m`}.[^2] It seems that we need the imaginary to relate to the undetermined to obtain new determination when confronted with ambiguous results.\n\nIn contrast, I would interpret the FORMs 2=1 and 3=1 as *decision **for** the undetermined/imaginary*. Essentially, it is a relation of decisions for/against *a* in the context of either *undetermined* or *imaginary* FORMs. As you can see, in 2=1 *a* is marked in the context of {2r|a} *(uFORM)* and in 3=1, *a* is marked in the context of {a} (which is equivalent to {2r+1|a}) *(iFORM)*.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":151,"value":"md`## FORMs to determine values for two and more variables\n\nEach of these four primitives can be combined using a *conjunction FORM* ${tex`((a)(b))`} to fix interpretation values for two and more variables. Try for yourself:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":524,"value":"viewof userConst1 = form(html``)","pinned":false,"hidden":false,"mode":"ojs"},{"id":55,"value":"{ \n const PADDING = {top: 10, bottom: 10, left: 10, right: 10};\n const TYPE = prefs.REPR;\n const FORM = combinedVals1;\n \n const id01 = DOM.uid();\n \n \n yield md`*Example for ${userConst1.a}${userConst1.b}=1:*\n

${`${FORM}`}

\n\nAs you can see, this one is constructed from ${userConst1.a}=1 for *a* and ${userConst1.b}=1 for *b*.`;\n\n formform.graph.createGraph(TYPE,FORM,{parentId: id01.id, styleClass: prefs.STYLECLASS, padding: PADDING});\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":177,"value":"md`\n## How to construct very specific FORMs\n\nTo assign a value other than the marked state, we just have to combine that value with our previous FORM using yet another conjunction FORM. This works, because the marking of the *m*-state (the result of our previous FORM) unmarks it so that we *focus/decide* on the other value instead. Let's try it out:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":573,"value":"viewof userInterpr1 = form(html``)","pinned":false,"hidden":false,"mode":"ojs"},{"id":163,"value":"{ \n const FORM = productTerm1;\n const PADDING = {top: 10, bottom: 10, left: 10, right: 10};\n const TYPE = prefs.REPR;\n \n const id01 = DOM.uid();\n \n \n yield md`*Example for ${userConst1.a}${userConst1.b}=${userInterpr1.v}:*\n

${userConst1.a}${userConst1.b}=${userInterpr1.v}: ${`${FORM}`}

`;\n\n formform.graph.createGraph(TYPE,FORM,{parentId: id01.id, styleClass: prefs.STYLECLASS, padding: PADDING});\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":603,"value":"md`Look at how the value table exhibits only your desired output:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":591,"value":"{ \n const vals = formform.form.calcAll(productTerm1); \n return valueTableWizard(vals);\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1248,"value":"md`Now let's try to combine the *ab*-term with another term for the variables *b* and *c:*`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1098,"value":"viewof userConst2 = form(html``)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1139,"value":"viewof userInterpr2 = form(html``)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1082,"value":"{ \n const FORM = productTerm2;\n const PADDING = {top: 10, bottom: 10, left: 10, right: 10};\n const TYPE = prefs.REPR;\n \n const id01 = DOM.uid();\n \n \n yield md`*Term 2 (${userConst2.b}${userConst2.c}=${userInterpr2.v}):*\n

${userConst2.b}${userConst2.c}=${userInterpr2.v}: ${`${FORM}`}

`;\n\n formform.graph.createGraph(TYPE,FORM,{parentId: id01.id, styleClass: prefs.STYLECLASS, padding: PADDING});\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1168,"value":"mdfn`We have to be careful though if we combine product terms, because the operation of relating terms in FORM logic is an *inclusive OR,* hence it also yields *m* when both terms are *m* (or one is *u* and one *i*).[^3] Observe how the values from one term leak to the other:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1411,"value":"md`*Results of the commutative relation of Term 1 and 2 (excluding 0 results):*`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1188,"value":"{ \n const vals = formform.form.calcAll(combinedTerms12_false); \n return valueTableWizard(vals,{search: [1,2,3], filterByVals: true});\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1197,"value":"md`But we want to keep term results isolated from each other. To do this, we could transform the *OR*-structure into a *XOR*-structure which is really complicated and messy just for a simple relation like \\`\\`ab\\`\\`, as you can see:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1281,"value":"{ \n const FORM = '((ab) ( (((a)(b))) ))';\n const PADDING = {top: 10, bottom: 10, left: 10, right: 10};\n const TYPE = prefs.REPR;\n \n const id01 = DOM.uid();\n \n \n yield md`

${`${FORM}`}

`;\n\n formform.graph.createGraph(TYPE,FORM,{parentId: id01.id, styleClass: prefs.STYLECLASS, padding: PADDING});\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1276,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":1425,"value":"md`So to continue with our FORM construction, we only need to explicitly negate the variables that are excluded from the respective term:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1203,"value":"{ \n const FORM = productTerm1_fixed;\n const PADDING = {top: 10, bottom: 10, left: 10, right: 10};\n const TYPE = prefs.REPR;\n \n const id01 = DOM.uid();\n \n \n yield md`*Fixed term 1 by adding 0=1 for c:*\n

a(${userConst1.a})b(${userConst1.b})c(0) = ${userInterpr1.v}: ${`${FORM}`}

`;\n\n formform.graph.createGraph(TYPE,FORM,{parentId: id01.id, styleClass: prefs.STYLECLASS, padding: PADDING});\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1212,"value":"{ \n const FORM = productTerm2_fixed;\n const PADDING = {top: 10, bottom: 10, left: 10, right: 10};\n const TYPE = prefs.REPR;\n \n const id01 = DOM.uid();\n \n \n yield md`*Fixed term 2 by adding 0=1 for a:*\n

a(0)b(${userConst2.b})c(${userConst2.c}) = ${userInterpr2.v}: ${`${FORM}`}

`;\n\n formform.graph.createGraph(TYPE,FORM,{parentId: id01.id, styleClass: prefs.STYLECLASS, padding: PADDING});\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1155,"value":"{ \n const FORM = combinedTerms12; \n \n yield md`Combination of terms ${tex`(a(${userConst1.a}) \\land b(${userConst1.b}) \\land c(0)=${userInterpr1.v}) \\lor (a(0) \\land b(${userConst2.b}) \\land c(${userConst2.c})=${userInterpr2.v})`}: ${`${FORM}`} now yields our desired output:`;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1200,"value":"{ \n const vals = formform.form.calcAll(combinedTerms12); \n return valueTableWizard(vals,{search: [1,2,3], filterByVals: true});\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":190,"value":"md`Using this method, every conceivable FORM can be constructed by relating the distinct constituents that represent each of its interpretation results. This way, it becomes possible to algorithmically generate *very* complex FORMs that are at the same time completely distinct and precise in their results. If you have any application at all for this madness, please let me know, I would love to hear about it! \n\nMoreover, if you construct a FORM out of all ${tex`4^n`} constituents for *n* variables, what you get is some sort of *universe of all possible decisions or determinations in the undetermined* (between *n* variables) similar to the binary universe of class relations conceived by George Boole in his 1854 *“An Investigation of the Laws of Thought”*.\n\nHere is what it looks like with all 16 constituent FORMs between two variables:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":212,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":333,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":1549,"value":"md`---\n\n### Footnotes`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1304,"value":"$35","pinned":false,"hidden":false,"mode":"ojs"},{"id":206,"value":"md`---\n### Appendix`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1116,"value":"univ1ui = x => ({v0: `({(${x})}{2r|(${x})})`, v1: `({${x}}{2r|${x}})`, v2: `(({(${x})}${x})({2r|${x}}(${x})))`, v3: `(({${x}}(${x}))({2r|(${x})}${x}))`})","pinned":false,"hidden":false,"mode":"ojs"},{"id":2,"value":"univ2ui = (x,y) => {\n const univ1ui_x = univ1ui(x);\n const univ1ui_y = univ1ui(y);\n return {v00: `((${univ1ui_x.v0})(${univ1ui_y.v0}))`,\n v01: `((${univ1ui_x.v0})(${univ1ui_y.v1}))`,\n v02: `((${univ1ui_x.v0})(${univ1ui_y.v2}))`,\n v03: `((${univ1ui_x.v0})(${univ1ui_y.v3}))`, \n v10: `((${univ1ui_x.v1})(${univ1ui_y.v0}))`,\n v11: `((${univ1ui_x.v1})(${univ1ui_y.v1}))`,\n v12: `((${univ1ui_x.v1})(${univ1ui_y.v2}))`,\n v13: `((${univ1ui_x.v1})(${univ1ui_y.v3}))`,\n v20: `((${univ1ui_x.v2})(${univ1ui_y.v0}))`,\n v21: `((${univ1ui_x.v2})(${univ1ui_y.v1}))`,\n v22: `((${univ1ui_x.v2})(${univ1ui_y.v2}))`,\n v23: `((${univ1ui_x.v2})(${univ1ui_y.v3}))`,\n v30: `((${univ1ui_x.v3})(${univ1ui_y.v0}))`,\n v31: `((${univ1ui_x.v3})(${univ1ui_y.v1}))`,\n v32: `((${univ1ui_x.v3})(${univ1ui_y.v2}))`,\n v33: `((${univ1ui_x.v3})(${univ1ui_y.v3}))`};\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":6,"value":"univ1ui_a = univ1ui('a') // ({v0: `({(a)}{2r|(a)})`, v1: `({a}{2r|a})`, v2: `(({(a)}a)({2r|a}(a)))`, v3: `(({a}(a))({2r|(a)}a))`})","pinned":false,"hidden":false,"mode":"ojs"},{"id":4,"value":"univ1ui_b = univ1ui('b') // ({v0: `({(b)}{2r|(b)})`, v1: `({b}{2r|b})`, v2: `(({(b)}b)({2r|b}(b)))`, v3: `(({b}(b))({2r|(b)}b))`})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1133,"value":"univ1ui_c = univ1ui('c')","pinned":false,"hidden":false,"mode":"ojs"},{"id":499,"value":"combinedVals1 = `(( ${univ1ui_a['v'+userConst1.a]} )( ${univ1ui_b['v'+userConst1.b]} ))`","pinned":true,"hidden":false,"mode":"ojs"},{"id":1114,"value":"combinedVals2 = `(( ${univ1ui_b['v'+userConst2.b]} )( ${univ1ui_c['v'+userConst2.c]} ))`","pinned":true,"hidden":false,"mode":"ojs"},{"id":593,"value":"productTerm1 = `((${userInterpr1.v})(${combinedVals1}))`","pinned":true,"hidden":false,"mode":"ojs"},{"id":1091,"value":"productTerm2 = `((${userInterpr2.v})(${combinedVals2}))`","pinned":true,"hidden":false,"mode":"ojs"},{"id":1174,"value":"combinedTerms12_false = `${productTerm1} ${productTerm2}`","pinned":true,"hidden":false,"mode":"ojs"},{"id":1176,"value":"combinedVals1_fixed = `(( ${univ1ui_a['v'+userConst1.a]} )( ${univ1ui_b['v'+userConst1.b]} )( ${univ1ui_c['v0']} ))`","pinned":true,"hidden":false,"mode":"ojs"},{"id":1181,"value":"combinedVals2_fixed = `(( ${univ1ui_a['v0']} )( ${univ1ui_b['v'+userConst2.b]} )( ${univ1ui_c['v'+userConst2.c]} ))`","pinned":true,"hidden":false,"mode":"ojs"},{"id":1183,"value":"productTerm1_fixed = `((${userInterpr1.v})(${combinedVals1_fixed}))`","pinned":true,"hidden":false,"mode":"ojs"},{"id":1185,"value":"productTerm2_fixed = `((${userInterpr2.v})(${combinedVals2_fixed}))`","pinned":true,"hidden":false,"mode":"ojs"},{"id":1084,"value":"combinedTerms12 = `${productTerm1_fixed} ${productTerm2_fixed}`","pinned":true,"hidden":false,"mode":"ojs"},{"id":1124,"value":"univ2ui_ab = univ2ui('a','b')","pinned":true,"hidden":false,"mode":"ojs"},{"id":201,"value":"{\n mutable DEFAULTS = ({repr: 'pack', style: 'basic', padding: 20});\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":124,"value":"userPrefs.scripts(prefs)","pinned":true,"hidden":false,"mode":"ojs"},{"id":10,"value":"CSS","pinned":false,"hidden":false,"mode":"ojs"},{"id":11,"value":"valueTableCSS","pinned":false,"hidden":false,"mode":"ojs"},{"id":1307,"value":"footnoteCSS","pinned":false,"hidden":false,"mode":"ojs"},{"id":1793,"value":"LANG = 'en'","pinned":false,"hidden":false,"mode":"ojs"},{"id":14,"value":"import {formTric, padding, valueTableWizard, valueTableCSS, CSS, userPrefs, mutable DEFAULTS} from '793523caac58b657@119'","pinned":false,"hidden":false,"mode":"ojs"},{"id":437,"value":"import {form} from '@mbostock/form-input@158'","pinned":false,"hidden":false,"mode":"ojs"},{"id":1779,"value":"import {metadata} with { LANG as setLang } from 'b537017f986f4d7c@266'","pinned":false,"hidden":false,"mode":"ojs"},{"id":1294,"value":"import {mdfn, footlist, footnoteCSS} with { footnotes as footnotes, LANG as setLang } from '4a01c6bf448dded8@1125'","pinned":false,"hidden":false,"mode":"ojs"},{"id":916,"value":"d3 = require('d3@5.16.0')","pinned":true,"hidden":false,"mode":"ojs"},{"id":1767,"value":"formform = require('formform@0.8.0') // 0.7.3","pinned":true,"hidden":false,"mode":"ojs"}],"files":"$W36","stdlib":"1","titling":"auto"}