Trivium · the arts of language · art 2 of 7
Logic or dialectic
Dialectica
Definitions, inference, arguments, and objections
- Classical definition
- The art directing the act of reason itself, so that it proceeds in order, easily and without error. (Aquinas, Commentary on the Posterior Analytics, proemium; Commentary on the Peri Hermeneias I, lect. 1.)
- Proper object
- The acts of reason: terms, propositions and arguments (Aquinas assigns all three acts of the mind to logic).
- Place in the order
- Trivium: judging and reasoning · see the whole order
Logic, also called dialectic, is the art of asking what follows from what and how firmly a claim deserves to be held. In the classical order it comes after grammar, because an argument cannot be tested until its words are pinned down. Many claims in health, engineering and the news arrive without their premises, so the practical skill is to reconstruct the argument, define its terms and grade the evidence behind it.
The guiding questionWhat follows from what, and how well do we know it?
The spiral
Three strands, three rounds
Each strand comes back in every round, a level deeper. Finish Round 1 across the arts before Round 2: breadth first, then depth.
Round I Foundations
- ArgumentReconstruct an argumentI can reconstruct a short argument as numbered premises and a conclusion, add its hidden premises, and judge separately whether it is valid and whether it is sound.
- ElenchusTest a definitionI can test a proposed definition by cross-examination: set it against cases that should and should not count, find where it is too wide or too narrow, and revise it or say that it fails.
- StatusGrade a claimI can assign one of the Academy's five statuses to a claim, name the kind of evidence behind it, and say what finding would change the label.
Round II Practice
- ArgumentTest an argument's formI can put an everyday argument into propositional or categorical form and test its validity with a truth table or by building a counterexample.
- ElenchusDefine and refine a conceptI can define a concept by genus and differentia, test it against cases I choose in advance, and revise it until it is neither too wide nor too narrow.
- StatusWeigh rival explanationsI can set out at least three structurally different explanations for an observation, write predictions for each before testing, and re-grade each explanation's status after the result.
Round III Mastery
- ArgumentProve it, and know the limitsI can write a short proof in a stated domain, check it by trying to break it, and state what Gödel's first incompleteness theorem does and does not show.
- ElenchusCross-examine a contested conceptI can run a full cross-examination of a contested concept such as knowledge or fairness, record each definition that failed and why, and state honestly what remains unresolved.
- StatusCalibrate your confidenceI can express a claim as a probability, update it on new evidence, and keep a forecast record whose score shows whether my confidence is calibrated.
Labs
Core reading in the Library
- LibraryProof and Precise Reasoning: From Arguments to TheoremsThe grammar of valid inference: validity and soundness, truth tables, quantifiers, the main proof techniques and counterexamples, framed by the trivium habit of defining, testing and explaining.
- LibraryPeirce on the Fixation of BeliefA guided reading of Charles Sanders Peirce's 1877 essay on four ways of settling belief, with a worked argument reconstruction.
- LibraryObserve, Define, Represent: The First Three MovesHow to separate observation from interpretation, ask what exactly we mean, and choose a representation that exposes structure.
- LibraryHypotheses, Predictions and TestsHow to generate several competing explanations, derive checkable consequences, and commit to predictions before looking at the result.
- LibraryThe Seven Liberal Arts, Hands-On: Logic, Proof and the QuadriviumA practical guide to the trivium and quadrivium, with a practice and a made thing for each art, plus a path through Socratic questioning, Aristotle's logic and Euclid's method.
- LibraryWhat Does It Take to Know Something?A six-text reading path on knowledge and justified belief, from Peirce to Gettier, with a writing prompt for each stop.
Where it is used
- LibraryEvidence-Based Troubleshooting: Separating ExplanationsEach drill asks what an observation really establishes before you choose the next test.2 min glance · 6 min
- LibraryReading Nutrition Evidence: What a Study Can and Cannot ShowPlace a health claim on the ladder of evidence and say what the study cannot show.2 min glance · 4 min
- LibraryTechnical Writing as ReasoningWrite a recommendation so that observation, inference and assumption stay visibly separate.2 min glance · 5 min
Seminars
Read the sources
One primary text for each round, with a question to think through. All 21 seminars →
Logic or dialectic · Round 1
- Plato, Euthyphro, 2a-16a (the whole dialogue)Read the whole dialogue in one sitting. Give most attention to 5d-6e (Socrates asks for the one form that makes every pious act pious), 10a-11b (is the pious loved by the gods because it is pious?) and 15c-16a (the ending). The link is H. N. Fowler's translation on Perseus, which renders the key word (hosion) as 'holy'; many other translations say 'pious'. Benjamin Jowett's translation is also free at Project Gutenberg (ebook 1642).
Euthyphro is certain he knows what piety is, yet each answer he gives fails under Socrates' questions. What does the questioning accomplish, and would you call the ending a failure?
- At 6d-e Socrates asks for the one form that makes every pious act pious, not a list of pious acts. Why is a list of examples not a definition?
- At 10a Socrates asks whether the gods love the pious because it is pious, or whether it is pious because they love it. Why does the order of the explanation matter for the definition?
- At 11b-d Euthyphro complains that whatever they propose moves about and will not stay where they put it, and each speaker suggests the other is the Daedalus who sets it moving. Whose fault is it: the definitions', the speaker's or the questioner's?
Logic or dialectic · Round 2
- Aristotle, Prior Analytics, Book I, I.1-7 (24a10-29b28); the definition of a syllogism is at 24b18-20, the perfect syllogism at 24b22-26, and the first figure begins at I.4, 25b26Translation by A. J. Jenkinson in The Works of Aristotle, vol. 1, ed. W. D. Ross (Oxford, 1928), on Wikisource, which prints Bekker numbers in the margin. The same translation is on the MIT Internet Classics Archive (classics.mit.edu/Aristotle/prior.1.i.html), divided into numbered parts that match the chapters but without Bekker numbers. Chapter 3 (conversion of modal premises) can be read lightly. The weight falls on chapters 1, 2 and 4-7: the definitions, conversion of premises, the three figures and the reduction of every syllogism to the universal syllogisms of the first figure.
Aristotle says that in a syllogism, certain things being stated, something other than what is stated 'follows of necessity' (24b18-20). What does it mean for a conclusion to follow of necessity, and how could you tell, in an argument you meet today, that it does?
- In chapter 4 Aristotle states the first-figure arguments with letters (A, B, C) instead of examples. What do letters let him do that examples cannot?
- In chapter 4 he shows that some pairs of premises yield no syllogism by giving two sets of terms that make the same premises true: one where the first term belongs to all of the last (animal, man, horse) and one where it belongs to none (animal, man, stone). How does this compare with the counterexample method on the Proof and Precise Reasoning page?
- In chapter 1 (24b22-26) a perfect syllogism needs nothing beyond what has been stated to make the conclusion plain, and in chapter 7 every syllogism is reduced to the universal syllogisms of the first figure. Why might a less obvious argument need to be made perfect before you trust it?
Logic or dialectic · Round 3
- David Hilbert, Mathematical Problems (address to the International Congress of Mathematicians, Paris, 1900; English translation by Mary Winston Newson, Bulletin of the American Mathematical Society 8, 1902), The introductory passage on the solvability of every mathematical problem (ending 'in mathematics there is no ignorabimus'), and Problem 2, 'The compatibility of the arithmetical axioms'David Joyce's page at Clark University hosts the full Newson translation. Read the introduction and Problem 2 only. Hilbert calls it a conviction, 'which no one has as yet supported by a proof', that every definite mathematical problem can be settled, either by an answer or by a proof that it cannot be solved.
- Kurt Gödel, On formally undecidable propositions of Principia Mathematica and related systems I (Monatshefte für Mathematik und Physik 38, 1931), Section 1, the informal outline of the proof (pp. 173-176 of the original)Martin Hirzel's English translation (2000), posted by the translator, covers sections 1 and 2 only, omits the footnotes and gives the original page numbers in the margin. Read section 1 only. The outline in section 1 assumes that every provable formula is true; the exact proof in section 2 replaces this with the weaker, purely formal condition of omega-consistency, and Rosser (1936) later showed that plain consistency suffices.
Hilbert states a conviction, which he admits no one has yet proved, that every definite mathematical problem can be settled, and that in mathematics there is no ignorabimus. Gödel's introduction sketches a statement that a formal system can neither prove nor refute, and then says it has been decided by other means. Does Gödel contradict Hilbert?
- Gödel says the undecidable statement 'has hence been decided by meta-mathematical considerations' (p. 176). From what standpoint is it decided, and does that standpoint lie inside or outside the system?
- Gödel notes a close kinship between his undecidable statement and the liar antinomy (p. 175). What separates an undecidable statement from a paradox?
- Hilbert's Problem 2 asks for a proof that the arithmetical axioms are consistent, and Gödel's introduction points ahead to surprising results about consistency proofs in section 4 (p. 176). What would count as success for Problem 2 after 1931?