1
Set out an argument
Rewrite a passage as numbered premises and one conclusion, strike the sentences that do no work, and say what the rest are offered in support of.
Course · Philosophy
Arguments and Validity
Tell a valid argument from a persuasive one. Set out a passage as premises and a conclusion, test it by looking for a counterexample row, decide whether a set of beliefs can all be true, and refute a bad form with a better case.
valid ⟺ no row: premises T, conclusion F
¬(p ∧ q) ≡ ¬p ∨ ¬q De Morgan
p → q ≡ ¬q → ¬p contrapositive
inconsistent: no row makes all true
∀x ∃y is not ∃y ∀x order matters
What this course is about
- Subject
- Philosophy
- Lessons
- 12
- Format
- Interactive
Overview
What you will be able to do
By the end you can say what an argument claims, test whether its conclusion follows, and say what rejecting that conclusion commits you to giving up.
2
Test validity by counterexample
Build the table, find or rule out a row with true premises and a false conclusion, and keep validity apart from soundness.
3
Rewrite without changing the claim
Decide equivalence by comparing columns, negate with De Morgan, and give a conditional's contrapositive without its converse.
4
Test a set of beliefs
Decide whether sentences can all be true, and find the smallest subset that cannot.
5
Test categorical arguments
Read All, No and Some as claims about regions, test a syllogism, and state what changes when existential import is assumed.
6
Read quantifier order, and refute a fallacy
Read quantifiers over a finite universe in the right order, and answer an invalid form with an argument of the same form that has true premises and a false conclusion.
Lessons
12 lessons
An argument is set out first, then tested by tables, then by regions, then with quantifiers, and the last lesson turns the test into a method of refutation.
Premises, Conclusions and Standard FormFind the claim a passage defends, and lay out what is offered for it.Not marked
Validity and SoundnessTwo questions kept apart: could the premises be true with the conclusion false, and are the premises in fact true?Not marked
Truth Values and the ConnectivesBuild the table of a compound sentence and find the row where the two kinds of or part.Not marked
The ConditionalFill the four rows of if-then and translate only if, unless, necessary and sufficient.Not marked
Validity by Truth TableLay premises and conclusion in one table, and find or rule out the counterexample row.Not marked
Equivalence, De Morgan and ContrapositionDecide whether two sentences say the same thing by comparing their columns.Not marked
Consistency and Belief SetsDecide whether a set of beliefs can all be true, and find the smallest part that cannot.Not marked
Categorical Statements and Immediate InferenceRead All, No and Some as claims about regions, and test one-step inferences on them.Not marked
The Square of Opposition and Existential ImportFind out which of the traditional inferences need the assumption that a term has members.Not marked
Syllogisms Tested by Venn RegionsTest a three-term argument by listing the region patterns its premises allow.Not marked
Quantifiers and Their OrderRead every and some over a finite universe, and see why their order cannot be swapped.Not marked
Fallacies and the Counterexample MethodRefute an invalid form by writing an argument of the same form with true premises and a false conclusion.Not marked
Recommended background
No logic is assumed, and school arithmetic is enough.
Practice suggestions
Work forward. “Validity by Truth Table” assumes you can fill the table of a conditional, and “Consistency and Belief Sets” assumes you can test an argument by table. The second half uses the first throughout.
Use the labs to look for the row. Almost every lab here lists the cases and marks the ones that matter. Edit a premise and watch the verdict move; a minute spent trying to break an argument teaches more than reading its verdict.
Keep two questions apart. Every lab in this course decides whether a conclusion follows from premises. None decides whether the premises are true, and “Validity and Soundness” is where that line is drawn. The rest of the Subject asks that second question.
Say each verdict in words. After the lab reports a counterexample, state in a sentence the case it describes. If you cannot, you have a result and not yet an understanding.
What it does not cover
Proofs and derivations. Validity here is decided by truth table and by countermodel, never by deriving a conclusion step by step. Logic and Proof on the Discrete Mathematics path teaches proof technique.
Validity in full first-order logic. Quantified sentences are read over a finite universe you build, so the lab computes truth in that universe and does not search every possible one for a countermodel.
Informal fallacies and rhetoric. Ad hominem, straw men and the like are failures of relevance and fairness, and there is no table to check them against. They are left to the reader's judgement.
The history of logic. The syllogism is taught as a way of testing arguments, not as Aristotle's own system, and nothing here is about who said what.