Logic III: Quantification and Its Rules, Axiomatic Systems, Fallacies, and Deduction and Induction
2. The rules of quantification and their restrictions
| Rule | Form | Restriction |
|---|---|---|
| Universal instantiation (UI) | (x)Fx ∴ Fa (or Fy) | None: whatever is true of everything is true of any individual |
| Existential generalisation (EG) | Fa ∴ (∃x)Fx | None: if a particular individual is F, something is F |
| Existential instantiation (EI) | (∃x)Fx ∴ Fa | The constant must be new: it may not occur earlier in the proof (or in the conclusion), since all we know is that something is F, not which thing |
| Universal generalisation (UG) | Fy ∴ (x)Fx | Only from a variable standing for an arbitrarily selected individual; never from a constant, and never from a name introduced by EI (nor within a conditional-proof assumption containing it free) |
| Line | Statement | Justification |
|---|---|---|
| 1 | (x)(Mx ⊃ Px) | Premise |
| 2 | (∃x)(Sx · Mx) | Premise |
| 3 | Sa · Ma | 2 EI (a is new) |
| 4 | Ma ⊃ Pa | 1 UI |
| 5 | Ma · Sa | 3 Com |
| 6 | Ma | 5 Simp |
| 7 | Pa | 4, 6 MP |
| 8 | Sa | 3 Simp |
| 9 | Sa · Pa | 8, 7 Conj |
| 10 | (∃x)(Sx · Px) | 9 EG |
- Order matters: EI is done at line 3, before UI at line 4. Had we instantiated the universal premise to a first (Ma ⊃ Pa), EI at the next step could not use a again, because a would no longer be new, and the proof would stall.
- Why EI needs a new name: from (∃x)Fx and (∃x)Gx one may not infer (∃x)(Fx · Gx). "Something is even and something is odd" does not imply "something is both even and odd". The restriction blocks the bad step, since after Fa from the first premise, EI on the second must use a different constant, Gb, and Fa · Gb gives no common instance.
- Why UG needs an arbitrary individual: from "Socrates is wise" (Ws) it does not follow that everyone is wise. UG is licensed only when the reasoning would have gone through for any individual whatever, which is guaranteed when the variable was introduced by UI from a universal premise and not by EI or as a specific constant.
3. Axiomatic systems: consistency, completeness and related properties
An axiomatic system consists of a formal language, a set of axioms (formulas accepted without proof) and rules of inference; a theorem is any formula derivable from the axioms by the rules. Euclid's geometry is the classical model, and Whitehead and Russell's Principia Mathematica (1910–13) gave an axiomatic system for logic, with five axioms for the propositional calculus (later shown to be reducible to four, as Paul Bernays proved one of them redundant) and the rule of modus ponens. Other systems, such as Hilbert's and Łukasiewicz's, use different axioms but prove the same theorems. What matters philosophically are the properties such a system may have, which are studied in metalogic.
| Property | Definition | Status for classical logic |
|---|---|---|
| Consistency | No formula and its negation are both theorems; equivalently, not every formula is provable | Propositional and first-order logic are consistent |
| Soundness | Every theorem is valid (a tautology, or true in every interpretation) | Both are sound |
| Completeness (semantic) | Every valid formula is a theorem | Propositional logic is complete (Post, 1921); first-order logic is complete (Gödel, 1929–30) |
| Independence | No axiom is derivable from the others | A matter of economy, not of correctness |
| Decidability | There is an effective procedure to tell, of any formula, whether it is a theorem | Propositional logic is decidable (truth tables); first-order logic is undecidable (Church and Turing, 1936) |
4. Formal and informal fallacies
A formal fallacy is an argument whose form is invalid: affirming the consequent and denying the antecedent in propositional logic; four terms, undistributed middle, illicit major or minor, exclusive premises and the existential fallacy in syllogistic. An informal fallacy is an error that lies in the content or context of an argument rather than in its form alone, and is detected by attending to what is said. Copi's classification groups informal fallacies by their source.
| Group | Fallacies |
|---|---|
| Relevance: the premises are not relevant to the conclusion | Appeal to force (ad baculum), to pity (ad misericordiam), to the people (ad populum), against the person (ad hominem, abusive or circumstantial), appeal to inappropriate authority (ad verecundiam), appeal to ignorance (ad ignorantiam), red herring, straw man, irrelevant conclusion (ignoratio elenchi) |
| Defective induction: the premises are too weak | Hasty generalisation (converse accident), false cause (post hoc ergo propter hoc), slippery slope, weak analogy |
| Presumption: an unwarranted assumption is built in | Accident, complex question, begging the question (petitio principii), false dichotomy, suppressed evidence |
| Ambiguity: a shift in meaning | Equivocation, amphiboly, accent, composition (what is true of parts is true of the whole), division (what is true of the whole is true of the parts) |
5. Deductive and inductive logic
| Point | Deductive | Inductive |
|---|---|---|
| Claim | The conclusion follows with necessity | The conclusion is made probable |
| Appraisal | Valid or invalid (no degrees); sound if valid with true premises | Strong or weak (a matter of degree); cogent if strong with true premises |
| Content | Non-ampliative: the conclusion states nothing not already contained in the premises | Ampliative: the conclusion goes beyond the premises |
| Adding premises | Cannot make a valid argument invalid | Can strengthen or weaken the argument |
| Examples | Syllogisms, MP and MT, mathematical proof | Generalisation from samples, analogy, causal inference by Mill's methods (agreement, difference, joint method, residues, concomitant variation) |
Key takeaways
- A: (x)(Sx ⊃ Px); E: (x)(Sx ⊃ ~Px); I: (∃x)(Sx · Px); O: (∃x)(Sx · ~Px). QN: ~(x)Fx ≡ (∃x)~Fx and ~(∃x)Fx ≡ (x)~Fx.
- UI and EG are unrestricted; EI requires a constant new to the proof; UG only from an arbitrarily selected individual, never from a constant or an EI name. Do EI before UI on the same name.
- Axiomatic system: axioms, rules, theorems (Principia Mathematica). Consistency (no p and ~p), soundness (theorems valid), completeness (valid formulas provable), independence, decidability; propositional logic is consistent, complete and decidable; first-order logic is complete (Gödel 1929–30) but undecidable (Church, Turing 1936).
- Formal fallacies are invalid forms; informal fallacies are of relevance, defective induction, presumption and ambiguity; composition (parts to whole) differs from hasty generalisation (some members to all).
- Deduction claims necessity (valid or invalid, non-ampliative); induction claims probability (strong or weak, ampliative); the "general to particular" definition is inaccurate.
Practice questions (10)
Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.
In quantifier notation, "All S are P" is correctly symbolised as:
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Answer: A — (x)(Sx ⊃ Px)
The universal A form uses the conditional: for everything, if it is S then it is P. (x)(Sx · Px) says everything is both S and P, (∃x)(Sx · Px) is the I form, and (∃x)(Sx ⊃ Px) is true whenever anything is not S.Which restriction applies to existential instantiation (EI)?
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Answer: D — The instantial constant must not have occurred earlier in the proof
From "something is F" we do not know which thing it is, so we must name it with a constant that says nothing else about it: one new to the proof. Using a constant already in play would smuggle in the assumption that the F-thing is that very individual.By quantifier negation, "~(x)Fx" is equivalent to:
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Answer: B — (∃x)~Fx
"Not everything is F" means "at least one thing is not F". (x)~Fx and ~(∃x)Fx both mean "nothing is F", which is stronger.Consider:
1. (∃x)Fx (premise);
2. Fa (1, EI);
3. (x)Fx (2, UG). The step from line 2 to line 3 is:Show answer
Answer: B — Invalid, because UG may not generalise on a name introduced by EI
The name a was introduced by EI to stand for some particular F, not for an arbitrary individual, so UG cannot be applied to it; otherwise "something is F" would prove "everything is F". EI requires a new constant, not an old one, so the last option states the restriction backwards."You cannot accept her argument for tax reform: she was once declared bankrupt." This commits the fallacy of:
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Answer: D — Argument against the person (ad hominem, abusive)
The argument attacks the arguer's character or history instead of the argument, which is the abusive ad hominem. There is no threat (ad baculum), no circular premise (begging the question) and no inference from parts to whole (composition).Assertion (A): Propositional logic is decidable. Reason (R): First-order predicate logic is complete. Choose the correct option.
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Answer: B — Both A and R are true, but R is not the correct explanation of A
Both are true, but R does not explain A. Propositional logic is decidable because the truth-table method settles any formula in finitely many steps. The completeness of first-order logic (Gödel, 1929–30) is a separate result, and first-order logic, though complete, is undecidable.Assertion (A): Deductive arguments always proceed from general premises to particular conclusions. Reason (R): In a valid deductive argument it is impossible for the premises to be true and the conclusion false. Choose the correct option.
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Answer: D — A is false, but R is true
A is false: "all humans are mortal; all Greeks are human; so all Greeks are mortal" is deductive with a general conclusion, so the "general to particular" definition fails. R is true: it states what deductive validity is, which is the real mark of deduction.Which of the following are fallacies of relevance? Select all that apply.
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Answer: B — Straw man; D — Appeal to force (ad baculum)
Ad baculum and straw man offer premises irrelevant to the conclusion (a threat; a distorted version of the opponent's view). Equivocation is a fallacy of ambiguity and complex question a fallacy of presumption.Which of the following properties does classical first-order predicate logic have? Select all that apply.
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Answer: B — Completeness; C — Consistency; D — Soundness
First-order logic is sound, consistent and complete (Gödel's completeness theorem, 1929–30), but it is not decidable: Church and Turing showed in 1936 that no effective procedure decides validity for all first-order formulas.Test the argument "p ⊃ q, r ⊃ s, ~q ∨ ~s ∴ ~p ∨ ~r" by the shorter truth-table method. How many assignments make every premise true and the conclusion false? Type the number.
Numerical answer — type the value.
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Answer: 0
Zero. For the conclusion ~p ∨ ~r to be false, p = T and r = T. Then p ⊃ q true forces q = T, and r ⊃ s true forces s = T. But then ~q ∨ ~s is F ∨ F = F, so the third premise is false. No assignment is a counterexample, so the argument (destructive dilemma) is valid.