Symbolic Logic And The Game Of Logic

1. Logic as Mental Recreation and Fallacy Detection

Mental recreation is a thing that we all of us need for our mental health; and you may get much healthy enjoyment, no doubt, from Games, such as Back-gammon, Chess, and the new Game “Halma”.

Beyond mere games. Lewis Carroll, or Charles Lutwidge Dodgson, presents Symbolic Logic not just as an academic discipline but as a delightful mental recreation, akin to a game. He argues that while traditional games offer temporary amusement, mastering logic provides lasting intellectual benefits and a valuable skill set applicable to all aspects of life. This pursuit offers a unique blend of enjoyment and practical utility, making it a worthwhile endeavor for anyone seeking mental stimulation.

Cultivating clear thought. The study of Symbolic Logic sharpens one's ability to think clearly, navigate complex puzzles, and organize ideas systematically. It equips individuals with the power to dissect arguments, identify inconsistencies, and expose the flimsy, illogical reasoning often encountered in daily life. This critical thinking skill is invaluable, whether reading books, newspapers, listening to speeches, or even sermons, protecting one from being easily deluded by flawed arguments.

A fascinating art. Carroll passionately believes that Symbolic Logic is one of the most fascinating mental recreations available, accessible even to intelligent children. He invested years of hard work to popularize this subject, hoping it would be adopted in schools and families as a healthful mental exercise. The reward for mastering this "machinery" is an absorbing intellectual occupation that yields tangible results in improved clarity of thought and analytical prowess.

2. The Universe of Discourse and its Divisions

‘Classification,’ or the formation of Classes, is a Mental Process, in which we imagine that we have put together, in a group, certain Things.

Defining the scope. At the heart of Carroll's logical system is the concept of the 'Universe of Discourse' (Univ.), which is the overarching class of "Things" being considered in an argument. This mental enclosure allows for a clear boundary within which all subsequent classifications and propositions operate. For instance, if discussing "books," the entire diagram represents all books, setting the stage for further logical divisions.

Hierarchical organization. Within this Universe, "Things" possess 'Attributes' (Adjuncts), which are used to form 'Classes'. These classes are organized hierarchically: a 'Genus' is a larger class, and a 'Species' is a smaller class picked out from it, distinguished by a 'Differentia' (a peculiar attribute). This process of 'Classification' allows for precise grouping, whether forming 'Real' classes (existing things) or 'Unreal' (imaginary) ones, like "towns paved with gold."

Dichotomy for clarity. 'Division' is the mental process of breaking a class into smaller, 'codivisional' classes. The most fundamental form is 'Dichotomy,' where a class is split into two contradictory parts (e.g., "old" and "not-old" books). This ensures that every member of the original class falls into exactly one of the two new classes, providing a clear and exhaustive partition essential for logical analysis.

3. Propositions: The Core Statements of Logic

A ‘Proposition,’ when in normal form, asserts, as to certain two Classes, which are called its ‘Subject’ and ‘Predicate,’ either (1) that some Members of its Subject are Members of its Predicate; or (2) that no Members of its Subject are Members of its Predicate; or (3) that all Members of its Subject are Members of its Predicate.

Standardizing statements. Carroll emphasizes the importance of reducing all statements to a 'Normal form' for logical analysis. This standard structure consists of four parts: a 'Sign of Quantity' ("some," "no," or "all"), the 'Name of Subject,' the 'Copula' ("are" or "is"), and the 'Name of Predicate.' This normalization ensures clarity and consistency, allowing for unambiguous interpretation of any proposition.

Categorizing propositions. Propositions are categorized based on their quantity and quality. 'Particular' propositions begin with "Some" (e.g., "Some apples are not ripe"), referring to a part of the subject. 'Universal' propositions refer to the whole subject, either 'Universal Negative' ("No," e.g., "No lambs are accustomed to smoke cigars") or 'Universal Affirmative' ("All," e.g., "All bankers are rich men"). Individual propositions, like "John is ill," are treated as Universal because they refer to the entirety of a one-member class.

Propositions of Relation. While 'Propositions of Existence' assert the reality or imaginariness of a class, 'Propositions of Relation' assert a relationship between two species of the same genus (the Universe of Discourse). Crucially, a proposition beginning with "All" is considered a 'Double Proposition,' conveying two pieces of information: "Some Members of the Subject are Members of the Predicate" and "No Members of the Subject are Members of the Class whose Differentia is contradictory to that of the Predicate."

4. The Existential Import of Propositions

A Proposition of Relation, beginning with “Some”, is henceforward to be understood as asserting that there are some existing Things, which, being Members of the Subject, are also Members of the Predicate; i.e. that some existing Things are Members of both Terms at once.

Existence is key. Carroll takes a firm stance on the "Existential Import" of propositions, a point of contention with traditional logicians. He dictates that any proposition beginning with "Some" or "All" explicitly asserts the actual existence (reality) of its Subject and Predicate. For example, "Some rich men are invalids" implies that both "rich men" and "invalids" are real classes, and that "rich invalids" exist.

"No" implies no existence. In contrast, a proposition beginning with "No" does not imply the reality of its terms. "No mermaids are milliners" asserts that no existing things are "mermaid-milliners," but it makes no claim about whether mermaids or milliners actually exist. This distinction is vital for avoiding fallacies and ensuring logical consistency within his system, preventing absurd conclusions from non-existent subjects.

Refuting alternative views. Carroll rigorously defends his position, arguing against interpretations where propositions do not assert existence. He demonstrates that such views lead to logical contradictions, invalidating accepted syllogisms like Darapti and even the basic process of conversion for 'I' propositions. His system prioritizes a clear, consistent understanding of existence, making it a foundational principle for his method.

5. Visualizing Propositions with Diagrams and Counters

Let us agree that a Red Counter, placed within a Cell, shall mean “This Cell is occupied” (i.e. “There is at least one Thing in it”).

The Biliteral Diagram. Carroll introduces a simple, intuitive diagrammatic method to represent propositions. The 'Biliteral Diagram' is a square divided into four cells, representing the four possible combinations of two attributes and their contradictories (e.g., xy, xy', x'y, x'y'). This visual space, designated as the 'Universe of Discourse,' allows for a clear mapping of logical statements.

Counters for truth. Information about these classes is conveyed using 'Counters'. A Red Counter ('1') signifies that a cell is 'occupied' (meaning "there is at least one Thing in it"). A Grey Counter ('0') indicates that a cell is 'empty' ("there is nothing in it"). If a Red Counter is placed on a division line, it means the compartment is occupied, but it's unknown which specific cell within it holds the occupants—a state Carroll humorously calls "sitting on the fence."

The Triliteral Diagram. For more complex arguments involving three attributes, Carroll expands to the 'Triliteral Diagram.' This involves drawing an inner square within the Biliteral Diagram, creating eight cells to represent combinations of three attributes (e.g., xym, xym', etc.). This visual tool allows for the simultaneous representation of multiple propositions, laying the groundwork for solving syllogisms by observing the resulting patterns of occupied and empty cells.

6. Syllogisms: Unveiling Hidden Conclusions

When a Trio of Biliteral Propositions of Relation is such that (1) all their six Terms are Species of the same Genus, (2) every two of them contain between them a Pair of codivisional Classes, (3) the three Propositions are so related that, if the first two were true, the third would be true, the Trio is called a ‘Syllogism’.

The essence of deduction. A 'Syllogism' is a trio of propositions where two 'Premisses' logically lead to a 'Conclusion.' The key is that the premisses share a 'Middle Term' (or 'Middle Terms')—an attribute that links the other two terms, called 'Retinends.' This middle term is then 'eliminated' in the conclusion, which expresses a new relationship between the retinends. The validity of a syllogism depends solely on the logical relationship between its propositions, not their actual truth.

Diagrammatic solution. Carroll's method for solving syllogisms involves representing the two premisses on a Triliteral Diagram. By carefully placing '1' and '0' counters according to the rules of each proposition, the diagram visually encodes the combined information. The next step is to transfer the relevant information (concerning only the retinends) to a simpler Biliteral Diagram, from which the conclusion can be directly "read off."

Concrete to abstract. To apply this method, concrete propositions (e.g., "All cats understand French") are first translated into abstract form using letters (e.g., "All m are x"), and a 'Universe of Discourse' is established. This abstraction simplifies the process, allowing the logical structure to be analyzed independently of specific content. Carroll provides numerous examples, demonstrating how to systematically derive conclusions or identify when no valid conclusion exists.

7. The Method of Subscripts: A Concise Algebraic Notation

Let us agree that “x1” shall mean “Some existing Things have the Attribute x”, i.e. (more briefly) “Some x exist”; also that “xy1” shall mean “Some xy exist”, and so on.

Symbolic shorthand. To streamline logical analysis, Carroll introduces the 'Method of Subscripts,' a concise algebraic notation for propositions. An 'Entity' (assertion of existence) is denoted by a subscript '1' (e.g., "x1" for "Some x exist," "xy1" for "Some xy exist"). A 'Nullity' (assertion of non-existence) is denoted by a subscript '0' (e.g., "x0" for "No x exist," "xy0" for "No xy exist"). The symbol "†" means "and," and "¶" means "would, if true, prove."

Translating propositions. Propositions of relation are easily translated into this subscript form. "Some x are y" becomes "xy1." "No x are y" becomes "xy0." The more complex "All x are y" is represented as a double proposition: "x1 † xy'0" (Some x exist and no xy' exist), or more concisely, "x1y'0." A helpful rule for "All" propositions is that the predicate changes sign (from positive to negative or vice versa) when translated into subscript form.

Efficiency in analysis. This method allows for rapid representation and manipulation of logical statements without the need for diagrams, especially useful for experienced practitioners. By converting concrete problems into abstract subscript forms, the underlying logical structure becomes immediately apparent, facilitating quicker identification of syllogism types and potential conclusions.

8. Formulæ for Syllogisms: Simplifying Deduction

Two Nullities, with Unlike Eliminands, yield a Nullity, in which both Retinends keep their Signs.

Universal rules for deduction. Carroll distills the vast array of syllogistic forms into three fundamental 'Formulæ,' or "Figures," each with a simple rule for deriving conclusions. These figures categorize pairs of premises based on whether they are Nullities (no existence) or Entities (some existence) and whether their Eliminands (middle terms) are 'Like' or 'Unlike' in sign.

The three figures:

• Fig. I (xm0 † ym'0 ¶ xy0): Two Nullities with Unlike Eliminands yield a Nullity where Retinends keep their signs. This is the most common form, often with variants where Retinends are asserted to exist.
• Fig. II (xm0 † ym1 ¶ x'y1): A Nullity and an Entity with Like Eliminands yield an Entity where the Nullity-Retinend changes its sign.
• Fig. III (xm0 † ym0 † m1 ¶ x'y'1): Two Nullities with Like Eliminands asserted to exist (m1) yield an Entity where both Retinends change their signs.

Streamlined problem-solving. Once these formulæ are memorized, they provide a powerful shortcut for solving syllogisms. Instead of drawing diagrams every time, one can simply translate the premises into subscript form, identify the figure, and instantly deduce the conclusion. This systematic approach makes complex logical deductions accessible and efficient, moving beyond the "exasperating Rules" of traditional logic.

9. Detecting Fallacies: Guarding Against Illogical Arguments

Any argument which deceives us, by seeming to prove what it does not really prove, may be called a ‘Fallacy’.

Identifying flawed reasoning. A crucial application of logic is detecting 'Fallacies'—arguments that appear valid but are not. Carroll distinguishes between 'Fallacious Premisses' (when the premises themselves yield no conclusion) and 'Fallacious Conclusion' (when the drawn conclusion is incorrect or incomplete). This skill is paramount for navigating the often-illogical arguments encountered in everyday life.

Forms of fallacies. Carroll identifies specific "Forms of Fallacies" that consistently yield no conclusion:

• Like Eliminands not asserted to exist: Both premises are Nullities, and their Eliminands are Like (e.g., xm0 † ym0).
• Unlike Eliminands with an Entity-Premiss: One premise is a Nullity, the other an Entity, and their Eliminands are Unlike (e.g., xm0 † ym'1).
• Two Entity-Premisses: Both premises are Entities (e.g., xm1 † ym1).
  These forms, when represented diagrammatically, show no transferable information to derive a conclusion.

Beyond traditional limitations. Carroll criticizes traditional logicians for their "morbid dread of negative Attributes" and their limited system, which often misidentifies valid arguments as fallacies. His broader system, encompassing all possible forms, provides a more robust framework for fallacy detection, allowing for a more accurate assessment of logical soundness.

10. Sorites: Building Chains of Reasoning

When a Set of three or more Biliteral Propositions are such that all their Terms are Species of the same Genus, and are also so related that two of them, taken together, yield a Conclusion, which, taken with another of them, yields another Conclusion, and so on, until all have been taken, it is evident that, if the original Set were true, the last Conclusion would also be true.

Extended logical chains. A 'Sorites' is an extended form of argument consisting of three or more propositions, where conclusions are drawn in a series. Two initial premises yield a 'Partial Conclusion,' which then combines with another premise to form a new partial conclusion, and so on, until all premises are used, resulting in a 'Complete Conclusion' that relates the final 'Retinends' (terms not eliminated).

Methods of solution. Carroll offers two primary methods for solving Sorites:

• Method of Separate Syllogisms: This involves systematically pairing premises, finding their conclusions using the syllogism formulæ, and then using those conclusions as new premises until a final conclusion is reached.
• Method of Underscoring: A more streamlined approach where eliminated terms are simply marked (underscored) in the subscript form of the premises, allowing the final conclusion to be read directly from the remaining, un-underscored terms.

Practical application. By allowing premises to be stated in any order and including propositions in 'E' (Universal Negative), Carroll's approach to Sorites is far more flexible and practical than the "childish simplicity" of traditional Aristotelian or Goclenian forms. This method empowers the reader to tackle complex, multi-premise arguments, revealing the ultimate logical consequence of a series of statements.

11. Carroll's Modern and Practical Approach to Logic

Of all the strange things, that are to be met with in the ordinary text-books of Formal Logic, perhaps the strangest is the violent contrast one finds to exist between their ways of dealing with these two subjects [Syllogisms and Sorites].

Simplifying complexity. Lewis Carroll's "Symbolic Logic" is a revolutionary attempt to simplify and popularize formal logic, making it accessible and enjoyable. He critiques the "exasperating Rules" and limited scope of traditional logic, which he found unnecessarily complex and incomplete. His system streamlines the nineteen traditional syllogism forms into just three universal figures, each with a straightforward rule, significantly reducing the learning curve.

A comprehensive system. Carroll's method embraces propositions and forms that traditional logic often ignored, particularly those involving negative attributes (e.g., "All not-x are y"). He champions a consistent interpretation of existential import and refutes common fallacies in traditional teachings, such as the notion that "two Negative Premisses prove nothing." This broader, more consistent framework allows for the analysis of a wider range of arguments.

Empowering the learner. Through engaging language, playful analogies, and intuitive diagrams, Carroll transforms logic from a dry academic subject into a captivating mental game. His goal is to equip readers with practical tools for clear thinking and fallacy detection, skills he believes are essential for navigating the complexities of the real world. His work stands as a testament to making profound intellectual pursuits both rigorous and delightful.

Last updated: January 9, 2026