Logic

Categorical syllogism rules and fallacies

A categorical syllogism is a deductive argument made of exactly three categorical propositions and exactly three terms. The major term is the predicate of the conclusion, the minor term is its subject, and the middle term appears in both premises but not in the conclusion. The premise that contains the major term is the major premise, and the premise that contains the minor term is the minor premise. Each proposition is classified by quality (affirmative or negative) and quantity (universal or particular) into four types: A (All S are P), E (No S are P), I (Some S are P), and O (Some S are not P). The mood of a syllogism is the sequence of these letters in the order major premise, minor premise, conclusion. For example, 'All mammals are vertebrates; some pets are mammals; therefore some pets are vertebrates' has the mood AII. The figure, by contrast, is fixed by the position of the middle term, so mood and figure must be identified separately. Validity depends on distribution. A term is distributed when the proposition refers to every member of its class: the subject of A and E propositions and the predicate of E and O propositions are distributed, while the remaining terms are not. The standard rules are as follows. (1) The middle term must be distributed at least once. (2) A term distributed in the conclusion must be distributed in the premise in which it occurs. (3) Two negative premises yield no valid conclusion. (4) A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise. (5) Two affirmative premises yield an affirmative conclusion. (6) Two particular premises yield no conclusion, and a particular premise yields a particular conclusion. The fallacy of undistributed middle occurs when the middle term is undistributed in both premises, so the two premises never connect the other two classes. 'All cats are animals; some dogs are animals; therefore some dogs are cats' is invalid for this reason. The fallacy of illicit major (or illicit minor) occurs when the major (or minor) term is distributed in the conclusion but not in its premise, as in 'All dogs are mammals; no cats are dogs; therefore no cats are mammals.'

Propositional logic rules of inference

Propositional logic evaluates arguments whose validity depends on truth-functional connectives: negation \sim, conjunction \cdot, disjunction \lor, and material implication \supset. A rule of inference is a step that is valid whatever truth values its letters take. The core rules are modus ponens (p \supset q, p; therefore q), modus tollens (p \supset q, \sim q; therefore \sim p), hypothetical syllogism, disjunctive syllogism (p \lor q, \sim p; therefore q), simplification (p \cdot q; therefore p), conjunction, addition, and constructive dilemma. Disjunctive syllogism is often used in short derivations: from 'it rains or the match is cancelled' and 'it does not rain', one infers that the match is cancelled. When a disjunction and a denial are combined with a conditional, the chain can be continued by modus ponens to reach a further conclusion. Replacement rules are equivalences that let a part of a formula be rewritten. The main ones are De Morgan's theorem (\sim(p \cdot q) \equiv \sim p \lor \sim q), material implication (p \supset q \equiv \sim p \lor q), transposition (p \supset q \equiv \sim q \supset \sim p), double negation, and commutation. A negated conditional is especially important: \sim(p \supset s) \equiv p \cdot \sim s, so a denied conditional gives its antecedent as true and its consequent as false. For example, from A \supset (B \cdot C) and A, modus ponens gives B \cdot C, and simplification then gives B and C. Long derivations apply these rules in sequence. To reach \sim q from \sim(p \supset (q \lor r)), one can rewrite the conditional by material implication, apply De Morgan's theorem to obtain p \cdot \sim(q \lor r), apply De Morgan's theorem again to obtain p \cdot \sim q \cdot \sim r, and finally simplify. Each step must be a valid rule or equivalence, and different orders of application may reach the same result. Two invalid forms must be avoided: affirming the consequent (p \supset q, q; therefore p) and denying the antecedent (p \supset q, \sim p; therefore \sim q). Scope also matters when a definite description occurs in a negated sentence. In Russell's theory of descriptions, a description has primary occurrence when it takes wide scope over the negation, so the sentence asserts that there is exactly one such object and that it lacks the predicate. It has secondary occurrence when the negation takes wide scope, so the sentence denies that there is exactly one such object with the predicate. If no such object exists, the primary reading is false and the secondary reading is true. The biconditional is also useful for testing philosophical theses. In Sartre's account, human freedom is what brings nothingness into the world, so affirming nothingness amounts to affirming freedom. With N for 'nothingness exists' and F for 'human freedom exists', this is N \equiv F, and the two claims stand or fall together. Bad faith, by contrast, is a self-deceptive denial of one's own freedom. Being and nothingness are not self-evidently exclusive for Sartre, because nothingness is found within being, so an exclusion between them must be argued rather than assumed.

Mill's methods of induction

John Stuart Mill's methods are rules for inferring causal connections from observed instances. They are inductive, so their conclusions are probable and depend on whether the relevant circumstances have been properly controlled. The five methods are the Method of Agreement, the Method of Difference, the Joint Method of Agreement and Difference, the Method of Concomitant Variation, and the Method of Residues. The Method of Agreement holds that if several instances of a phenomenon share only one antecedent circumstance, that circumstance is the cause or a part of the cause. An argument that draws a conclusion from a feature common to several cases, such as observed symptoms matched with a known disease, is classed under Agreement. The Method of Difference compares two instances that are identical except for one circumstance: if the phenomenon occurs in one and not in the other, that circumstance is its cause. The Joint Method combines the two, using agreement among positive instances and difference between positive and negative instances. The Method of Concomitant Variation holds that when a change in one phenomenon is regularly accompanied by a change in another, the two are causally connected. Arguments that rest on a long record of two quantities rising and falling together, and that infer a link from that covariation, belong to this method. The Method of Residues subtracts from a complex phenomenon the parts already known to be the effects of known antecedents. The remaining part is attributed to the remaining antecedent. If every known cause of a condition has been excluded, the leftover cause is inferred to be the new factor. Students should remember which method uses which logical test: agreement looks for a shared circumstance, difference looks for a single contrasting circumstance, concomitant variation looks for co-variation, and residues looks for subtraction.