Common Logic and Language

The Relationship Between Logic, Language, and Rhetoric

• Inaccuracy of Thought and Expression:

  • As formulated by Augustus De Morgan: "Logic has a tendency to correct, first, inaccuracy of thought, secondly, inaccuracy of expression. Many persons who think logically express themselves illogically, and in so doing produce the same effect upon their hearers or readers as if they had thought wrongly."
  • While the science of logic was developed to establish a sound theory of reasoning, human logical capacity is frequently impeded by language.
  • Natural language and formal logic operate under two distinctly different systems of rules.
  • Linguistic obstacles to logical reasoning arise from factors such as meaning, linguistic context, cultural background knowledge, and the modes of communication utilized in spoken conversation or written text.

• The Classical Trivium and Communication:

  • Historically, the art of communication was divided into three primary categories: logic, rhetoric, and poetry.
  • The boundary separating rhetoric and logic was frequently blurred across historical epochs.
  • During the Renaissance, logic (often termed dialectic) was defined as the study of statements designed to produce valid inferences regarding reality, standing as the primary medium for scientific and scholarly discourse.
  • Rhetoric (distinct from the modern pejorative notion of "empty rhetoric") served as the systematic mode of discourse designed to bridge communication between learned scholars and the general populace.

• Symbolic Metaphors of Fist and Hand:

  • Originating with Zeno and extending through the Middle Ages, logic was symbolized by the closed fist, typifying the dense, rigorous, and tight discourse of the philosopher.
  • In contrast, rhetoric was represented by the open hand, symbolizing the open, accessible discourse conducted between a cultured orator and the general public.
  • Renaissance philosopher and "father of deductive reasoning" Sir Francis Bacon delineated the boundary between the two fields:
  • "It appeareth also that Logic differeth from Rhetoric, not only as the fist from the palm, the one close the other at large; but more is this, that Logic handleth reason exact and in truth, and Rhetoric handleth it as it is planted in popular opinions and manners."

Historical Evolution and Applications of Logic

• Legal and Theological Applications in the Renaissance:

  • Throughout the Middle Ages, formal instruction in logic and rhetoric remained largely restricted to university settings.
  • In 1588, following the release of the first logic text written in the English language, Abraham Fraunce sought to unite English jurisprudence with formal logic by publishing The Lawiers Logike.
  • Fraunce introduced his legal treatise in verse:
  • "I say no more then what I saw, I saw that which I sought, / I sought for Logike in our Law, and found it as I thought."
  • Fraunce's objective was to elevate the practical, bourgeois reputation of legal practice by infusing it with the aristocratic, scholastic prestige of academic philosophy.
  • In 1620, Thomas Granger produced a parallel work for the clergy titled Syntagma Logicum, or The Divine Logike.
  • Granger opened his work with the couplet:
  • "This book's a Garden where doth grow a Tree, / Cal'd Logike, fruitful for Théologie."

• The Port Royal Logic:

  • Published in 1662 by Antoine Arnauld and Pierre Nicole, La Logique, ou L'Art de Penser (Logic, or the Art of Thinking) became one of the most widely read logic textbooks of the seventeenth and eighteenth centuries, sustaining its influence for two hundred years.
  • Arnauld and Nicole belonged to a group of religious reformers and mystics situated near Paris at Port-Royal, a circle that also included Blaise Pascal; consequently, the treatise became commonly known as The Port Royal Logic.
  • The work integrated practical illustrations of reasoning and logical fallacies across diverse fields, including rhetoric, ethics, physics, metaphysics, and geometry.
  • It examined mental operations involved in the generation of ideas, the assignment of linguistic meaning (semantics), and the psychological faculties necessary to evaluate the validity of arguments.
  • Arnauld and Nicole argued that the practical logic operating in everyday life and empirical scientific treatises diverges sharply from academic classroom logic, noting that real-world reasoning is rarely structured into the tidy, universal categorical syllogisms taught to students.

Semantics and Connectives in Natural Language Versus Formal Logic

• The Semantic Intersection:

  • Modern logic overlaps significantly with semantics, the branch of linguistics focused on the formal meaning of words, phrases, and sentential constructions.
  • The seeming simplicity of elementary logical connectives masks substantial communicative complexity.

• Conjunctions and Contrastive Variations:

  • The connective "and" is among the simplest logical operations to manipulate, yet natural language possesses diverse lexical means to express conjunction.
  • Words such as but, although, and occasionally while fulfill the identical truth-functional conjunctive role as "and" while simultaneously signaling a psychological contrast or a shift in expectation:
  • "I would certainly pay you the money I owe you, but I haven't gotten paid yet."
  • "Although I would love to go with you, I can't."
  • "While you've behaved very well today, you haven't cleaned up your room."

• Ambiguities in Connectives and Quantifiers:

  • Linguistic ambiguity represents a primary catalyst for deductive errors.
  • The connective "or" is inherently ambiguous in natural discourse, oscillating between an exclusive interpretation ("one or the other, but not both") and an inclusive interpretation ("one, the other, or both").
  • Natural language speakers predominantly favor the exclusive "or," given that alternative options in daily experience are rarely realized simultaneously.
  • The logical quantifier "some" possesses shifting meanings: in formal logic it denotes "at least one and possibly all," whereas in ordinary conversational usage it routinely connotes "some but not all."

Linguistic Interpretations of Predication and Class Inclusion

• Multifunctional Uses of the Copula "Is":

  • Deductive comprehension depends directly on the specific semantic function assumed by the verb "is":
  • Spatial/Locative: "The car is in the garage."
  • Temporal: "The concert is in the evening."
  • Class Inclusion: Aristotle's categorical formulation "Every $A$ is $B$."
  • Identity/Definition: "London is the capital of England" (substituting for "London equals the capital of England").
  • Synonymy/Mutual Entailment: "A duty is a tax," "A liability is a burden," or "Triangles are three-sided polygons."

• Hyponymy and Asymmetric Taxonomic Hierarchy:

  • Class inclusion typically establishes an asymmetric, hierarchical taxonomy:
  • "A poodle is a (type of) dog."
  • "A dog is a (type of) mammal."
  • "A mammal is a (type of) animal."
  • The class of poodles does not equal the class of dogs; rather, the broader class subsumes the narrower one.
  • In linguistics, this lexical inclusion relation is designated as hyponymy (analogous to synonymy and antonymy), where a hyponym represents a subordinate subname. Because the entities designated by "daisy" belong entirely to the category "flower," "daisy" constitutes a hyponym of "flower."

• Syllogistic Misinterpretation and Conversion Errors:

  • In syllogistic tasks, reasoners frequently misinterpret "is" as signifying equivalence or identity ("is identical with" or "is equal to") rather than categorical inclusion ("is included in").
  • Presenting categorical premises in the standard active form "All $A$ are $B$" frequently invites the erroneous assumption that class $B$ cannot be larger than class $A$.
  • Adopting the passive voice—"All $A$ are included in $B$"—substantially clarifies the underlying subset-superset relationship.
  • An even more effective formulation returns to Aristotle's original active phrasing: "$B$ includes all $A$."
  • Explicitly expanding the premise to state "All $A$ are $B$ but some $B$ might not be $A$" significantly reduces illicit conversion errors by directly forewarning reasoners against presuming bidirectional symmetry.

The Pragmatics of Everyday Discourse and Gricean Maxims

• Context, Intent, and Natural Logic:

  • Understanding ordinary communication requires automated contextual processing; listeners typically become conscious of contextual dependencies only when a breakdown in communication occurs.
  • Interpreting utterances within context necessitates making inferences regarding the underlying intentions of the speaker, separating human cognitive flexibility from mechanistic data processors that lack contextual comprehension.
  • Human reasoning frequently follows the conventions of pragmatic reasoning (or "natural logic") founded on practical norms rather than formal logical laws.

• The Gricean Maxims and the Cooperative Principle:

  • In 1967, British philosopher and logician H. Paul Grice established that participants in standard communication operate under an implicit set of conversational maxims:
  • Maxim of Quantity: Be as informative as is required; do not provide more information than is required.
  • Maxim of Quality: Be truthful; do not assert what is believed to be false or that for which adequate evidence is lacking.
  • Maxim of Relation: Be relevant.
  • Maxim of Manner: Be clear; avoid obscurity of expression; avoid ambiguity; be brief; be orderly.
  • Grice synthesized these directives under a singular overarching tenet: the Cooperative Principle ("Be cooperative").
  • Conversation represents an interactive compact where the speaker strives to be clear, truthful, informative, and relevant, and the hearer processes the communication under the explicit presumption of that cooperation.

• Conversational Implicature Versus Logical Definition:

  • Under conversational conventions, speakers are expected to deliver the most comprehensive statement warranted by their knowledge.
  • If a speaker states, "Some parts of the movie were interesting," the listener readily adopts the invited inference that other parts were uninteresting.
  • Although formal logic treats "some" as compatible with "all" ("at least one, and possibly all"), the Cooperative Principle dictates that if the entire movie had been interesting, the speaker would have explicitly said so to satisfy the demand for informativeness.

Enthymemes and Contextual Assumptions in Communication

• Suppression of Premises as Pragmatic Etiquette:

  • In The Port Royal Logic, Antoine Arnauld and Pierre Nicole maintained that communicators naturally restrict their utterances to what is strictly necessary to convey meaning, avoiding extraneous elaboration.
  • Suppressing obvious premises is an essential convention of natural discourse; speakers presume their audience possesses the requisite background to supply missing links.
  • Omission of self-evident propositions is considered conversational good manners, demonstrating respect for the listener's intellect rather than treating them as requiring remedial instruction.

• The Classical Enthymeme:

  • Arnauld and Nicole defined the enthymeme as:
  • "…a syllogism perfect in the mind, but imperfect in the expression, since some one of the propositions is suppressed as too clear and too well known, and as being easily supplied by the mind of those to whom we speak. This way of reasoning is so common in conversation and in writing, that it is rare, on the contrary, to express all the propositions, since there is commonly one of them clear enough to be understood, and since the nature of the human mind is rather to prefer that something be left it to supply, than to have it thought that it needs to be taught everything. Thus, this suppression flatters the vanity of those to whom we speak, in leaving something to their intelligence, and by abbreviating conversation, render it more lively and effective."
  • Ordinary discourse depends on substantial shared cultural, factual, and contextual foundations, leaving many operational assumptions unspoken.

• Information Processing: Formal Logic Versus Pragmatic Discourse:

  • Formal logic demands total cognitive compartmentalization: external real-world knowledge must be quarantined, and the reasoner must accept solely the minimum semantic commitment necessitated by the stated premises.
  • Everyday pragmatic reasoning operates in reverse: it synthesizes all accessible background knowledge, contextual cues, and heuristic associations available to the thinker.

Conditional Inferences and Contextual Suppression Effects

• Conditional Interpreted as Biconditional:

  • The Cooperative Principle explains why material conditionals are routinely interpreted as biconditionals in natural language.
  • If an individual states, "If you shovel the snow, I'll give you ten dollars," the listener invariably infers: "I will give you ten dollars if and only if you shovel the snow."
  • The hearer assumes the speaker is truthful, relevant, and concise; thus, the promise is not interpreted as leaving open the possibility of receiving ten dollars regardless of whether the snow is shoveled.

• Blocking Fallacies via Alternative Conditionals:

  • Empirical research indicates that fallacious invited inferences—such as denying the antecedent—can be blocked by introducing alternative conditions capable of producing the same consequent.
  • Consider the set of premises:
  • "If Lisa met her friend, then she went to a play."
  • "If Lisa met her brother, then she went to a play."
  • "Lisa did not meet her friend."
  • By making explicit an alternative route to attending the play (meeting her brother), reasoners resist the fallacious conclusion that Lisa did not go to the play.

• Suppression of Valid Deductions via Additional Conditionals:

  • While alternative conditionals suppress fallacies, introducing additional conditionals (prerequisite background conditions) inadvertently blocks valid deductive inferences such as Modus Ponens.
  • Consider the premises:
  • "If Lisa met her friend, then she went to a play."
  • "If Lisa had enough money, then she went to a play."
  • "Lisa met her friend."
  • Although the premise "Lisa met her friend" strictly entails that she went to the play under classical Modus Ponens, participants fail to draw this valid deduction.
  • The explicit mention of monetary constraints prompts reasoners to integrate practical common sense, leading them to infer that both necessary conditions must be satisfied concurrently.

Cognitive Perspectives: Natural Logic, Heuristics, and Commonsense Reasoning

• Martin Braine’s Theory of Natural Logic:

  • New York University psycholinguistics professor Martin Braine argued that human reasoning errors stem from the intrusion of practical conversational habits into tasks that demand formal logical execution.
  • Formal logic demonstrates an intractable misalignment with natural language connectives, most notably regarding if/then, or, and some.
  • Braine contended that human divergence from formal systems reflects distinct heuristic processes and interpretive comprehension strategies rather than an intrinsic absence of logical capacity.

• Heuristic Reasoning Versus Algorithmic Logic:

  • Martin Gardner emphasized that formal logic must not be confounded with heuristic reasoning—the intuitive, informal strategies that human minds deploy when resolving real-world problems.
  • Human beings do not function as mechanistic logic engines.

• Keith Devlin on Machine Versus Human Cognition:

  • Keith Devlin, executive director of the Stanford University Center for the Study of Language and Information, noted that computational systems lack the commonsense knowledge central to human reasoning.
  • Humans are uniquely capable of negotiating mutual understanding across cultural and linguistic barriers in ways that automated machines cannot replicate.
  • Spoken natural languages function merely as external manifestations of an internal, shared mental language.
  • As formulated by Devlin: "Human logical thought and our use of language almost certainly involve more than the mechanistic application of rules. But that does not mean that there are no rules."

Societal and Epistemological Importance of Logical Deduction

• Foundational Role Across Disciplines:

  • While individual non-specialists may dismiss formal logic as irrelevant—much like advanced algebra or the rules of chess—the evaluation of whether statements necessarily follow from premises forms the bedrock of metaphysics, epistemology, science, mathematics, and ethics.
  • Mastering deductive inference rules remains mandatory for scientific education, carrying heightened practical significance given the dominance of computing technology in contemporary society.

• Mary Henle on the Social Necessity of Logic:

  • Psychologist Mary Henle highlighted that shared logical reasoning is the indispensable prerequisite for social cohesion:
  • If individuals lacked an underlying logical capacity and derived completely erratic, divergent conclusions from identical premises, human beings would be fundamentally unable to comprehend one another's thoughts, follow continuous arguments, arrive at collective agreements, or engage in cooperative endeavors.