• General Overview
  • Central Method
    • Core thesis: logic reduces ordinary language to abstract classes, then proves conclusions visibly.
    • Normal form: every proposition becomes Quantity, Subject, Copula, Predicate.
    • Three quantities: "Some" asserts existence, "No" denies, "All" combines Some and No.
    • Universe of Discourse: the genus containing both terms supplies missing class attributes.
    • Reality rules: "Some" and "All" assert existence; "No" implies nothing about existence.
    • Translation discipline: everyday sentences must be reshaped without changing logical force.
  • Diagrammatic Logic
    • Biliteral diagram: four cells map x/x′ and y/y′; red marks occupied, grey marks empty.
    • Triliteral diagram: inner square adds middle term m, carrying two premises at once.
    • Marking order: negatives first, then "All" split into its existential and universal parts.
    • Reading results: transfer quarter by quarter; only two greys in a compartment prove emptiness.
    • Visible proof: diagrams show the conclusion emerges jointly from both premises, not one alone.
  • Subscripts and Syllogisms
    • Subscript notation: x₁ means some x exist; x₀ means no x exist.
    • Abstract premises: joined by †; conclusions linked by ¶ in forms like "xm′₀ † my′₀ ¶ xy′₀".
    • Three figures: nullity pairs, nullity plus entity, and existence-forced pairs yield conclusions.
    • Eliminands and retinends: middle terms cancel; retained terms form the conclusion.
    • Named fallacies: like eliminands unasserted, unlike eliminands with entity, and two entity-premises.
  • Sorites Chains
    • Sorites definition: three or more propositions chained through partial conclusions.
    • Separate syllogisms method: pair premises, draw conclusions, repeat until all terms are eliminated.
    • Underscoring method: single and double scores under eliminated letters read directly as a conclusion.
    • Flexible order: any premise may be first; order changes partial results but not the final conclusion.
    • Worked specimens: "Amos Judd loves cold mutton" and "Babies cannot manage crocodiles".
  • Exercises and Answers
    • Progressive drill: from three-premise syllogisms to five-to-ten-premise sorites.
    • Fixed universes: each problem names its domain, e.g. "persons", "birds", or "my boxes".
    • Symbolic keys: classes compress to letters, exposing the logical skeleton.
    • Humor as mnemonic: absurd images and everyday settings fix patterns in memory.
    • Answer-key coverage: AN1–AN9 verify parsing, diagrams, conclusions, fallacies, and sorites.
  • Pedagogical Reforms
    • Existential import: adopted view is that "Some" and "All" assert existence, "No" does not.
    • Textbook fallacies: "two negative premisses prove nothing" is refuted by counterexamples.
    • Diagram critiques: Euler’s and Venn’s methods hide existence assumptions and scaling defects.
    • Carroll’s improvement: closed rectilinear diagrams mark occupied and empty cells unambiguously.
    • Future plans: Part II tackles advanced propositional topics; Part III explores complex syllogisms and problem-construction.
• Deep Dive
  • To Learners – pg001½CHAPTER II
    • Things and Attributes
      • Things: the universe contains Things; examples include I, London, roses, redness.
      • Attributes: properties Things have; examples include large, red, old, received yesterday.
      • Many-to-many: one Thing may have many Attributes, and one Attribute may belong to many Things.
      • Adjunct: any Attribute, or set of Attributes, so we need not repeat “Attribute or Set of Attributes.”
    • Classification
      • Classification: mental process of grouping Things; the group formed is a Class.
      • Three modes: collect all Things, pick from the Class “Things” by an Adjunct, or pick from a narrower Class by an Adjunct.
      • Genus, Species, Differentia: original Class is Genus, selected Class is Species, and its peculiar Adjunct (not possessed by the whole) is Differentia.
      • Real vs. Imaginary: Real if some Thing has the Adjunct; Imaginary if none does.
      • Individual: a one-member Class; any distinguishable Thing may be regarded as such.
        • Example: the Class “towns having four million inhabitants” is the Individual London.
      • Class as a single Thing: a multi-member Class may have an Adjunct no member possesses separately.
        • Example: a regiment may be “formed in square,” though no soldier is.
  • pg003CHAPTER III – pg011CHAPTER II
    • Division and Dichotomy
      • Division: mental process of splitting a class into two or more smaller classes.
      • Codivisional classes: classes obtained by the same division; each is codivisional with itself.
      • Dichotomy: division into two classes whose differentiae are contradictory, e.g. “old” and “not-old”.
      • Arbitrary rules: needed when ordinary attributes are too loose to decide class boundaries.
      • Repeated dichotomy: subdivide each class again; the number of classes doubles at each step.
    • Names
      • Thing vs. Name: “Thing” conveys an object with no adjunct; a Name adds an adjunct and represents members of a class.
      • Real and unreal names: a Name is real if an existing thing it represents exists, otherwise unreal.
      • Three forms of a Name: all attributes as adjectives; some rolled into a substantive; all rolled into one substantive.
      • Plural names: may represent members separately or the whole class as one thing.
    • Definitions
      • Definition: a name composed of a Genus-name plus the Differentia of the Species.
      • Defining examples: “Treasure” = “valuable Thing”; “Man” = “Animal having two hands and two feet.”
      • Singular definitions: use “the” when only one such thing is known, e.g. “the Town with four million inhabitants.”
    • Propositions Generally
      • Proposition: in normal form asserts that some, no, or all members of the Subject are Members of the Predicate.
      • Normal form parts: sign of quantity, name of subject, copula “are”/“is”, name of predicate.
      • Kinds: “Some” means one or more and gives Particular (I); “No” = Universal Negative (E); “All” = Universal Affirmative (A).
      • Individual subject: treated as universal, since it refers to a one-member class.
      • Equivalent propositions: same information, e.g. “I see John” and “John is seen by me”; arguments require normal form.
    • Propositions of Existence
      • Existence propositions: normal subject is “existing Things”; sign is “Some” or “No”.
      • Reality or imaginariness: they assert the predicate class is real or imaginary, not exact numbers.
      • Equivalent forms: “Some existing Things are honest men” may be “Honest men exist” or “There are honest men.”
  • pg012CHAPTER III – pg026CHAPTER II
    • Propositions of Relation
      • Definition: proposition whose Terms are two Specieses of the same Genus, each Name conveying an Attribute the other lacks.
      • Universe of Discourse: the Genus containing both Terms; supplies the Substantive when a Name is incompletely expressed.
      • Sign of Quantity: "Some", "No", or "All"; states only 0, "one or more", or total members of the Subject.
      • Example: "Some merchants are misers" fits; "Some dogs are setters" does not, because "dogs" adds no distinctive Attribute.
    • Normal Form and Double Proposition
      • Normal form: Sign of Quantity, Subject, Copula, Predicate—arranged in that order.
      • Six rules: identify Subject, make verb "are/is", identify Predicate, supply Univ. for incomplete Names, add Sign, arrange.
      • Incomplete Names: "None but the brave deserve the fair" becomes "No not-brave persons are persons deserving of the fair."
      • "All" is double: "All S are P" is equivalent to "Some S are P" plus "No S are not-P".
      • Example: "All bankers are rich men" = "Some bankers are rich men" and "No bankers are poor men."
    • Reality and Translation
      • "Some": asserts existing Things in both Terms at once; implies each Term is Real.
      • "No": asserts no existing Things in both Terms; implies nothing about Reality of either Term.
      • "All": contains a Some-proposition, so implies each Term is Real; these Reality rules are arbitrary for Part I.
      • Translation to Existence: make "existing Things" the Subject and "Things having all the Attributes of Subject and Predicate" the Predicate.
    • Biliteral Diagram and Counters
      • Diagram as Univ.: whole enclosure stands for the selected Class, e.g. "books".
      • Adjuncts: x/x′ divide North/South; y/y′ divide West/East; the four cells represent intersections like x′y.
      • Cell table: x North, x′ South, y West, y′ East, xy NW, xy′ NE, x′y SW, x′y′ SE.
      • Red Counter in a cell: indicates "This cell is occupied"—at least one Thing exists there.
      • Red Counter on a partition: indicates the two-cell compartment is occupied, but whereabouts are unknown; "sitting on the fence".
      • Grey Counter in a cell: indicates "This cell is empty"—nothing exists there.
  • pg027CHAPTER III – pg043CHAPTER II
    • Biliteral Notation and Counters
      • Uniliteral/biliteral: proposition names count the attribute letters; “Things” is understood.
      • Red counter: marks an occupied compartment — “at least one exists”.
      • Grey counter: marks an empty compartment — “none exist”.
      • Half-empty rule: “No x exist” requires grey counters in both cells; one grey on the partition is insufficient.
      • Universal double: “All x are y” = “Some x are y” + “No x are y′”.
    • Equivalent Trios and Conversion
      • Positive trio: “Some xy exist” = “Some x are y” = “Some y are x”.
      • Negative trio: “No xy exist” = “No x are y” = “No y are x”.
      • Conversion: interchange subject and predicate terms; converses are equivalent.
      • Double relation: “Some x are y and some are y′” uses red counters in both cells of a half.
    • Interpretation of Marked Diagrams
      • Red cell: read the trio “Some xy exist / Some x are y / Some y are x”.
      • Grey cell: read the corresponding “No…” trio.
      • Red on partition: “Some x exist”; two greys in a half: “No x exist”.
      • Red and grey in half: “All x are y”; half fixes subject, red cell fixes predicate.
    • Triliteral Diagram and m
      • Inner square: adds third attribute m; inner square = m, outer border = m′.
      • Locating rule: take attributes in order x, y, m; find compartment, then portion, then portion.
      • Existence with m: red on a partition means occupied; two greys in a compartment mean empty.
      • Relation with m: “Some x are m” = “Some m are x” = “Some x m exist”; similarly for “No”.
      • Universal with m: “All x are m” = “Some x are m” + “No x are m′”.
  • pg050CHAPTER III – The Method of Subscripts. Chapter I
    • Ch. III: Diagram Representation
      • Triliteral diagram: shows two premises, one in x and m, the other in y and m, together.
      • Digit counters: "I" means something exists here; "O" means nothing here.
      • Represent negatives first: "No" premises before "Some" prevents fence-sitting counters.
      • Break up "All": "All m are y" becomes "Some m are y" and "No m are y′".
    • Ch. IV: Diagram Interpretation
      • Transfer quarter by quarter: a "I" in either cell marks the quarter occupied.
      • Mark emptiness only with two "O"s: one empty cell is not enough information.
      • Fence-sitting "I" is unusable: it gives no definite information until resolved.
      • Read result in x and y: choose wording that fits the dictionary, e.g., "No x are y′".
    • Book V: Syllogistic Problems
      • Syllogism: three propositions with codivisional pairs in every two; first two premises, third conclusion.
      • Eliminands and retinends: eliminated terms vanish; retained terms form the conclusion.
      • Validity is formal: depends on proposition relationships, not on actual truth.
      • Method from premises: fix Univ., dictionary, translate, diagram, transfer, read off, retranslate.
      • Checking conclusions: compare consequent conclusion with proposed one; mark complete or incomplete.
      • Unsoundness test: if invented circumstances keep premises true but conclusion false, the syllogism is unsound.
    • Book VI: Subscript Notation
      • Entity: "x₁" means some x exist; "xy₁" means some xy exist.
      • Nullity: "x₀" means no x exist; "xy₀" means no xy exist.
      • † and ¶: "†" means and; "¶" means would, if true, prove.
      • Like/unlike signs: same accent state is like; one accented is unlike.
  • pg071CHAPTER II – Soriteses. Chapter I
    • Representation of Propositions of Relation
      • Some x are y: equivalent to existence proposition "xy exist"; subscript form "xy₁"
      • Converse propositions: same subscript form represents converse, e.g. "Some y are x" → "xy₁"
      • No x are y: equivalent to "xy do not exist"; subscript form "xy₀"
      • All x are y: equivalent to "x exist and xy′ do not exist"; subscript "x₁ † xy′₀"
      • Subscript shorthand: each subscript governs back to the start, so "x₁y′₀" suffices
      • Predicate sign rule: translating "All" changes the predicate's sign in abstract/subscript forms
    • Syllogisms: Representation and Formulæ
      • Subscript syllogism: premisses joined by "†"; conclusion marked by "¶"
      • Example: No x are m′; All m are y. ∴ No x are y′ → "xm′₀ † my′₀ ¶ xy′₀"
      • Formulæ: Diagram-derived subscript patterns solve later syllogisms without redrawing
      • Fig. I: "xm₀ † ym′₀ ¶ xy₀"; two nullities, unlike eliminands, yield nullity; retinends keep signs; existential assertions carry over
      • Fig. II: "xm₀ † ym₁ ¶ x′y₁"; nullity + entity, like eliminands, yield entity; nullity-retinend changes sign
      • Fig. III: "xm₀ † ym₀ † m₁ ¶ x′y′₁"; two nullities, like eliminands asserted to exist, yield entity; both retinends change signs
    • Fallacies
      • Fallacy: pair of propositions proposed as premisses but yielding no conclusion
      • Detection: Method of Diagrams shows no information transferable to the Biliteral Diagram
      • Forms recorded in words: avoids confusing fallacies with subscript Formulæ
      • Like Eliminands not asserted to exist: two nullities such as "xm₀ † ym₀" yield nothing
      • Unlike Eliminands with an Entity-Premiss: e.g. "xm₀ † ym′₁" yields nothing
      • Two Entity-Premisses: "xm₁ † ym₁" or "xm₁ † ym′₁" yield nothing
    • Method of Proceeding with a Given Pair
      • Classify premisses: pair of nullities, nullity + entity, or pair of entities
      • Nullities, unlike eliminands: case of Fig. I, with existential variants
      • Nullities, like eliminands: Fig. III if an eliminand is asserted to exist, else first fallacy
      • Nullity + entity, like eliminands: case of Fig. II; unlike eliminands → second fallacy
      • Pair of entities: always the third fallacy
    • Soriteses: Introductory
      • Sorites: set of three or more propositions chained by partial conclusions, with final conclusion tacked on
      • Terms: eliminands drop out; retinends remain in the complete conclusion
      • Conclusion relation: depends entirely on relationship of propositions, not their actual truth
      • Specimen: No a are b′; All b are c; All c are d; No e′ are a′; All h are e′ ∴ All h are d
      • Alternative orders: rearranging premisses yields different partial conclusions, e.g. order 41523
      • Nine partial conclusions: the specimen Sorites contains nine discoverable partial conclusions
  • pg087CHAPTER II
    • The Sorites Problem
      • Sorites: given three or more Propositions of Relation, ascertain what conclusion, if any, follows.
      • Scope: limited for now to problems workable by the formulæ of Fig. I.
      • Two methods: separate syllogisms, or underscoring.
      • Universe of Discourse: name it before translating premisses.
      • Dictionary: let a, b, c represent the terms.
      • Subscript form: translate propositions into nullities and existentials, e.g. "All k are l" → k₁l′₀.
    • Method of Separate Syllogisms
      • Process: subscript the premisses, select a pair, draw conclusion, repeat until all are used.
      • Pairing: choose premisses containing a pair of codivisional classes, workable by Fig. I.
      • Partial conclusions: each intermediate result feeds the next syllogism.
      • Complete conclusion: the last conclusion, put into concrete form.
      • Worked example: from policemen and poets, conclude "Amos Judd loves cold mutton."
      • Freedom: you may begin with any premiss.
    • Method of Underscoring
      • Elimination shorthand: from xm₀ † ym′₀, underscore m and m′, then read the pair directly as xy₀.
      • Underscore convention: single score under the first eliminated letter, double under the second.
      • Subscripts omitted: copy premisses without subscripts; 0s are implied, 1s checked only for the final conclusion.
      • Worked example: same Amos Judd problem yields a₁e′₀, "All a are e".
      • Five-premiss model: concludes "My dog is not satisfied with anything I give him."
      • Any starting point: begin with any premiss; the complete conclusion still emerges.
  • Chapter I
    • Exercises in Diagrammatic Logic (Chapter I · I)
      • Propositions of Relation
        • Normal form: reduce everyday relational propositions into abstract x/y/m notation.
        • Translation practice: ordinary sentences must be restated without changing logical force.
        • Exceptive phrases: forms like “none but” and “unless” require exact symbolic handling.
      • Triliteral Diagrams
        • Abstract pairs: represent two propositions, one in x/m and one in y/m, on the same triliteral diagram.
        • Marked diagrams: read given triliteral diagrams back into conclusions about x and y.
        • Double representation: a single diagram carries both premisses before any conclusion is drawn.
      • Finding Conclusions
        • Abstract premiss pairs: derive the necessary conclusion from unmaterialized x/y/m propositions.
        • Concrete premiss pairs: repeat the same inference work with real-world statements.
        • Transfer of method: abstract and concrete exercises exercise the same diagrammatic reasoning.
      • Examining Syllogisms
        • Abstract trios: test whether the proposed conclusion follows from two premisses.
        • Concrete trios: apply the test to everyday arguments, exposing invalid leaps.
        • Verdict required: each syllogism is to be examined, not merely solved.
      • Sorites Chains
        • Abstract sets: eliminate intermediate terms across multiple premisses to reach a complete conclusion.
        • Concrete sets: translate narrative chains into symbols, then perform the same elimination.
        • Flexible order: any premiss may be taken first; §8 yields 129 variants and §9 yields 273.
    • Sorites Practice and Symbolic Translation (Chapter I · II)
      • Progressive Exercise Sets
        • Three-premise core: items 7–25 drill the basic syllogistic form with one middle term pair.
        • Four-premise links: items 26–36 add an extra premise and require two connecting steps.
        • Longer sorites: items 37–60 expand into chains of five to ten premises, building gradual complexity.
      • Logical Vocabulary in Premises
        • Standard categorical forms: premises use “No”, “All”, and “None but”, ready for symbolic translation.
        • Conditional phrasing: “unless”, “if”, and “when” require conversion to categorical propositions before solving.
        • Exclusive markers: “only” and “except” signal inverted term relationships in the symbolic dictionary.
      • Universes and Dictionaries
        • Explicit universes: every problem fixes its domain, e.g., “persons”, “things”, “birds”, or “my boxes”.
        • Symbolic key: each class is compressed to a letter, making the logical skeleton visible.
        • Narrowed worlds: universes like “cheques received by me” bound the puzzle to a specified context.
      • Humor as Mnemonic
        • Absurd images: lace-collared ducks, skipping-ropes, and quadruple somersaults fix the pattern in memory.
        • Everyday domains: shops, gardens, kitchens, and households give abstractions concrete settings.
        • Playful tone: whimsical premises turn the exercise set into a varied, memorable practice.
  • pg125CHAPTER II
    • Proposition Parsing (§1 Answers)
      • Canonical template: every proposition is reduced to "Quantity | Subject | Copula | Predicate".
      • Quantity markers: "All," "No," and "Some" consistently head the parsed subject.
      • Negated classes: answers freely use complements such as "not-'John'" and "not-brave persons".
    • Diagrammatic Results (§2–§3 Answers)
      • Diagrammatic readings: diagrams are transcribed as existential or universal statements, e.g. "Some x y exist".
      • No information: empty or ambiguous diagrams are honestly marked as giving no conclusion.
      • Equivalent forms: pairs such as "All x′ are y′" and "All y are x" express the same diagram.
    • Syllogistic Conclusions and Fallacies (§4–§7 Answers)
      • Valid conclusions: answers give precise forms like "Some x are y′" or "All y are x′".
      • Like Eliminands not asserted to exist: recurring ground for refusing a syllogistic conclusion.
      • Unlike Eliminands with an Entity-Premiss: another named fallacy blocking inference.
      • Corrected verdicts: §6–§7 mark each conclusion right or supply the true conclusion.
    • Translating Symbols into English (§5 Answers)
      • Ordinary-language renderings: formulas become natural sentences such as "John is ill".
      • Fallacy flags: many items receive "No Concl." with the governing fallacy named.
      • Real-world flavor: exercises test logic with greyhounds, brides, lodgers, and cats.
    • Abstract Notation and Sorites (§8–§9 Answers)
      • §8 shorthand: results are compressed into symbols like "a₁b₀" or "d₁a₀".
      • §9 rendering: sorites answers collapse chains into one statement, e.g. "Babies cannot manage crocodiles".
      • Answer-key design: AN1–AN9 verify the exercises uniformly from parsing to full sorites.
  • pg134CHAPTER III
    • Solving Syllogisms by Diagram and Subscript (pg134CHAPTER III · I)
      • Reducing Propositions to Normal Form
        • Universe of discourse: every exercise fixes its Univ. — persons, things, laws — before assigning classes.
        • Normal form: every proposition becomes Quantity | Subject | Copula | Predicate, e.g. "All | I's | are | persons who have been out for a walk".
        • Individuals as classes: "I" forms the one-member class of "I's", so its quantity is "all".
        • Verbs as predicates: replace "have been", "can", "wear" with "are persons who..." to expose subject and predicate classes.
        • Phrases as attributes: "if not well-cooked" and "what is difficult" become class attributes, e.g. "not well-cooked dishes".
        • Contraries as subjects: "none but John" or "not-pale" form complementary classes used with "No" or "All".
      • Diagrammatic Solutions
        • Middle term m: choose m for the class common to both premisses, then x and y for the terms of the conclusion.
        • Two premisses: encode each as empty or occupied regions of the x-y-m diagram.
        • Conclusion: read off the region forced empty or occupied; if neither, no conclusion follows.
        • Example: No m are x; Some m are y′ → Some x′ are y′.
      • Subscript Method
        • Subscript notation: x m′ 0 means "No x are m′"; m y 1 means "Some m are y".
        • Eliminand pairing: combine premisses by matching m or m′; then cancel the middle term to derive a relation between x and y.
        • Fig. I–III: resulting forms show "No x are y", "Some x are y", etc., and carry the conclusion.
        • Double conclusions: some pairs yield two forms, such as "All x are y and all y′ are x′" [Fig. I (β)].
      • Fallacies and Invalidity
        • Like Eliminands not asserted to exist: same eliminand in both premisses, but neither asserts it exists → no conclusion.
        • Unlike Eliminands with Entity-Premiss: when one eliminand is negated and an entity-premiss asserts its opposite, no conclusion follows.
        • No conclusion: solution lists mark these explicitly rather than forcing a conclusion.
      • Checking Proposed Conclusions
        • Working backward: assign m, x, y to the proposed argument, then encode premisses and see if the conclusion really follows.
        • Right conclusions: e.g. "No doctors are enthusiastic; You are enthusiastic" yields "You are not a doctor".
        • Wrong conclusions: e.g. the epicure/uncle argument yields "Some epicures are not uncles of mine", not "My uncles are not epicures".
        • No conclusion: some valid-looking pairs prove nothing, e.g. "Sugar-plums are sweet; Some sweet things are liked by children".
    • Judging Syllogisms and Solving Sorites (pg134CHAPTER III · II)
      • Invalid Premise Patterns
        • Unlike eliminands with an entity-premiss: named fallacy; verdict is "No Conclusion".
        • Like eliminands not asserted to exist: same eliminand lacks an existence-mark; verdict is "No Conclusion".
        • No Conclusion discipline: refusing to infer is a legitimate solution, not a failure.
        • Zero/one notation: 0 marks empty classes, 1 existing classes; verdicts rest on both.
        • Entity-premiss lure: an existential premise tempts a universal conclusion the form does not support.
      • Correcting Wrong Conclusions
        • Verdicts are corrective: wrong conclusions are replaced by the exact right one, not merely marked false.
        • Overstrong claim: "All these bonbons are chocolate-creams; all are delicious" supports only "Some chocolate-creams are delicious".
        • Wrong direction: "His songs are never tedious" is wrong; right is "Some tedious songs are not his".
        • Wrong scope: no "All railways are profitable" from "Railways are never ill-managed"; the right result concerns other businesses.
        • Missing complement: "No puppies are wasps" needs "Wasps are not puppies" to be complete.
        • Too-broad denial: "No emperor is dreaded by children" overreaches; the valid result is existential.
      • Valid Syllogism Verdicts
        • Right verdicts: examples such as "Oysters are not fossils" show valid derivation when eliminands are properly paired.
        • Figure labels: each valid form is tagged with its syllogistic figure, e.g. Fig. I, II, III.
        • Content independence: absurd content, like fossils crossed in love, still obeys formal rules.
        • Existence flow: a valid entity-premiss can carry existential force into the conclusion.
      • Sorites Solution Chains
        • Chained elimination: each premise cancels a term against a matching opposite in the next.
        • Surviving relation: the two terms left form the conclusion, often with an existence marker like a 1.
        • English rendering: symbolic results are translated into plain sentences, e.g. "Babies cannot manage crocodiles".
        • Long chains: the method scales to ten-premise examples without changing the rules.
        • Complete workflow: every solution lists premises, cancellations, and final translation, making the reasoning checkable.
  • pg164NOTES
    • The Objection
      • Petitio principii: critics charge that syllogism begs the question, since the conclusion lies in a premise.
      • Diagrams refute this: they show the conclusion emerges from both premises together, each contributing its share.
    • Fig. I — Two negative premisses
      • Premiss (x m_0): empties the inner cell of the N.W. quarter.
      • Premiss (y m_0): empties the outer cell of the same quarter.
      • Combined force: both premisses jointly empty the whole N.W. quarter, proving (xy_0).
    • Fig. II — One negative, one positive premiss
      • Premiss (x m_0): empties the inner cell of the N.W. quarter.
      • Premiss (y m_1): shows the inner W. half is occupied, but leaves the exact cell uncertain.
      • Combined force: the empty upper cell authorizes placing the ‘I’ in the lower cell, proving (x'y_1).
    • Fig. III — Existence plus two negative premisses
      • Premiss (m_1): only puts an ‘I’ somewhere in the inner square.
      • Premiss (x m_0): drives it out of the N. half.
      • Premiss (y m_0): drives it out of the W. half.
      • Combined force: both premisses force the ‘I’ into the inner S.E. quarter, proving (x'y'_1).
    • The Lesson
      • No fallacy: each premiss supplies indispensable information; neither alone yields the conclusion.
      • Visible logic: the diagrams make clear that the conclusion is genuinely drawn from the premisses together.
  • Addressed to Teachers
    • Existential Import and Diagram Methods (Addressed to Teachers · I)
      • Existential Import of Propositions
        • Author's rule: writers may fix word meanings as they choose, if consistent with logic and fact.
        • Three conceivable views: I and A assert; E and A assert; or none assert existence.
        • Adopted view: “some” and “all” assert existence; “no” does not assert it.
        • Second view absurd: universal negatives asserting existence produces inconvenient real-world rules, as Jones’s club shows.
        • Third view refuted: non-existential interpretation invalidates Darapti and the conversion of I-propositions.
      • The Copula and Negative Premisses
        • Not placement: “is-not” versus “is not-” is taste, not logical right and wrong.
        • Affirmative preference: “Some men are Gentiles” is better than “Some men are-not Jews.”
        • Negative attributes welcomed: dichotomy’s two sides are treated alike, enabling useful syllogisms.
        • “Two negative premisses prove nothing”: a logicians’ craze, refuted by concrete counterexamples.
      • Euler’s Method of Diagrams
        • Euler’s circles: two circles show inclusion and exclusion between classes.
        • Hidden assumption: every Euler diagram asserts “Some not-x are not-y,” which may be false.
        • Insufficient for particulars: particular propositions need at least three diagrams; “some not-x are not-y” needs all four.
      • Venn’s Method of Diagrams
        • Venn’s advance: one fixed diagram, with shaded empty regions and + for occupied regions.
        • Shared infinite area: only seven closed compartments hold eight classes; the eighth roams infinite space.
        • Evasion: “No x′ are y′” is handled by not troubling to shade the outside.
        • Scaling defect: diagrams for five and six letters give 31 and 62 closed compartments, not full 2ⁿ subdivisions.
      • My Method of Diagrams
        • Closed universe: a finite area encloses every class, ending the infinite-region anomaly.
        • Rectilinear cells: squares and oblongs replace curves; I marks occupied, O marks empty.
        • Elementary range: two-letter and three-letter diagrams use four and eight cells.
        • Eight-letter extension: lattices inside cells yield 128 cells, supporting larger symbolic work.
    • Syllogistic Methods and Diagrammatic Reform (Addressed to Teachers · II)
      • Scaling Diagrams Up to Ten Letters
        • Octoliteral diagram: eight letters fill 256 cells; each corner holds 16 class-combinations.
        • Nine letters: put two Octoliteral diagrams side by side, assigned to m and m′, giving 512 cells.
        • Ten letters: arrange four Octoliteral diagrams in a square, assigned to mn, mn′, m′n, m′n′, giving 1024 cells.
      • One Syllogism, Six Methods
        • Ordinary method: as stated, both premisses are negative, so no conclusion; obverting the minor yields Fresison, reducible to Ferio.
        • Symbolic representation: universe “persons”; x=philosophers, m=conceited, y=gamblers; syllogism becomes “No x are m; Some m are y′; ∴ Some y′ are x′.”
        • Euler’s diagrams: nine combinations, after ignoring m, support only “Some y′ are x′”; he assumed such propositions always true.
        • Venn’s diagrams: mark the m y′ constituents as saved, erase x m; surviving m y′ x′ eliminates m, yielding y′ x′.
        • Carroll’s diagrams: mark x m empty and m y′ occupied; the only x-y information left is that x′ y′ is occupied.
        • Subscript notation: from x m₀ and m y′₁, infer x′ y′₁.
      • Reforming Syllogisms and Sorites
        • Textbook Sorites: only two forms are recognized—Aristotelian (A-propositions, chain) and Goclenian (same chain reversed).
        • Carroll’s freedom: admits E propositions and random premise order; reader arranges them into regular syllogism series.
        • Three forms suffice: textbook nineteen syllogisms plus ignored ones all fall under three forms; only question is Fig. I, II, or III.
        • Sixteen orders: five-premise Aristotelian sorites has 16 valid orders; only first and last received names.
        • Goclenius jab: inverting premise order is like discovering 4×5 = 5×4, a glimpse of the obvious.
      • Plans for Parts II and III
        • Part II: discusses existential import, negative copula, “two negatives prove nothing,” alternatives, multi-term propositions, Hypotheticals and Dilemmas.
        • Part II goal: cover textbook ground and equip readers to solve examination problems harder than those now set.
        • Part III: deals with out-of-the-way topics: analysis of propositions, numerical/geometrical problems, problem construction, and complex syllogisms and Sorites.
        • Eliminands vs Retinends: a complete conclusion states all relations among retained terms that are deducible from the premisses.
      • Eight Closing Challenge Problems
        • Challenge set: eight problems close the section as a taste of Part II; readers may submit complete conclusions, symbols or not.
        • Everyday scenarios: monitor’s schoolroom notes, pork-chop logicians, Froggy’s waistcoat, and M.P. suitability.
        • Formal puzzles: walking-party couples, condiment rules, brotherly traits, kinship chains, and Jack Sprat as a Sorites.
        • Double-proposition clue: “All x are y” is equivalent to “Some x are y, and none are y′,” needed for the first problem.
  • pg195NOTES TO APPENDIX – Advice to Writers
    • Refuting the Textbook Contradictory
      • Textbook view: "All xy are z" and "Some xy are not-z" are treated as Contradictories, so one must always be true.
      • Forced conclusion: because both allegedly imply "Some x are y", that proposition would be always true — an absurdity.
      • Four states: the z/not-z membership of class xy yields four cases; exactly one is true, three false.
      • State equivalents: state (2) = "All xy are z"; state (3) = "All xy are not-z"; state (4) = "No xy exist".
      • Correct contradictory: "All xy are z" is contradicted by the alternative "Some xy are not-z or no xy exist".
      • Textbook flaw: reducing this alternative to "Some xy are not-z" alone drops the "no xy exist" disjunct.
    • Rival Views on Existential Import
      • Compatibility reading: "Some x are y" may mean only that the attributes x and y are compatible, not that x exists.
      • Jones reductio: telling Jones "Some of your brothers are swindlers" would not be softened by calling it mere conceivability.
      • Ambiguous A: "All x are y" sometimes implies existence of x, sometimes not; concrete wording decides.
      • Common-usage support: the ambiguity view fits ordinary speech, though it is hard to make precise.
      • Part II promise: the subtleties are too formidable for beginners and are postponed.
    • Appendix Notes and Answers
      • Note C conclusions: "No conceited child of mine is greedy"; "None of my boys could solve this problem"; "Some unlearned boys are not choristers".
      • Self-checking aid: these three conclusions let readers verify the syllogisms of §4.
      • Expository caution: Part I avoids advanced existential-import disputes to stay accessible to beginners.
• Core Conclusion and Practical Takeaways
  • Core Logical Framework
    • Universe of discourse: fix the class you reason about before assigning symbols.
    • Normal form: every proposition reduces to Quantity | Subject | Copula | Predicate.
    • Quantity signs: "Some" asserts existence; "No" asserts emptiness; "All" combines both.
    • Codivisional pairs: syllogisms cancel matching eliminands and keep the retained relation.
    • Validity is formal: conclusions depend on proposition form, not on content or truth.
  • Diagramming Practices
    • Biliteral diagram: four cells for x/x′ and y/y′; red marks occupied, grey marks empty.
    • Triliteral diagram: inner square adds m; mark negative premisses before existential ones.
    • Break up universals: "All m are y" becomes "Some m are y" plus "No m are y′".
    • Transfer quarters: move occupied or empty quarters to the biliteral diagram, then read x/y.
    • Subscript shorthand: x₁ means existing, xy₀ means empty; formulæ Fig. I–III replace redrawing.
  • Practical Methods
    • Syllogism workflow: set universe, dictionary, translate, diagram, transfer, read off, retranslate.
    • Sorites by underscoring: cancel matched eliminands across premises; surviving terms form the conclusion.
    • Check conclusions backward: encode the proposed argument and see whether the premises really force it.
    • Name the fallacy: no conclusion when like eliminands lack existence or unlike eliminands meet an entity-premiss.
    • Existence flow: existential force carries into the conclusion only when a premise asserts "some".
  • Mindset Shifts
    • No conclusion is a verdict: refusing to infer is legitimate, not failure.
    • Two negative premisses can prove: the textbook "craze" is refuted by concrete diagram counterexamples.
    • Diagrams refute petitio principii: each premise contributes its share; the conclusion emerges jointly.
    • Logician's freedom: writers may fix word meanings consistently, provided they respect logic and fact.
    • Same rules scale: the method runs unchanged from two letters to ten-letter diagrams.