Logic Basics – Necessary for Exam

Conditional Statements, Necessity & Sufficiency

• A conditional statement (“if … then …”) asserts that the truth of one statement (the antecedent) guarantees the truth of a second (the consequent).
  • Form: \text{If }A, \text{ then }B
  • Antecedent = A (sufficient condition); Consequent = B (necessary condition).
• Sufficient condition: A is sufficient for B when the occurrence of A is all that is required for B to occur.
  • Example: Being a dog is sufficient for being an animal.
• Necessary condition: B is necessary for A when A cannot occur without B.
  • Example: Being an animal is necessary for being a dog.
• Same conditional can be read both ways:
  \text{If X is a dog}\rightarrow\text{X is an animal} expresses that dog ⇒ animal (dog is sufficient).
  \text{If X is not an animal}\rightarrow\text{X is not a dog} expresses that animal is necessary.
• Practical tip: To test sufficiency imagine being told “A holds” inside a closed box; to test necessity imagine being told “B does not hold.”
• Later chapters employ necessity/sufficiency in definitions & causality.

Recognizing Arguments

• Three diagnostic checks:
  1. Presence of indicator words (therefore, since, because, thus, as, consequently…).
  2. Existence of an inferential relationship: conclusion is claimed to follow from premises.
  3. Elimination of common non-arguments.
• Warning: Indicator words alone ≠ argument; always verify the inferential claim.
• In passages w/out indicators, conclusion often appears first; mentally insert “therefore” to test.

Typical Kinds of Nonarguments

• Warnings, statements of belief/opinion, reports, pieces of advice, expository passages, illustrations, explanations, loosely-associated statements, conditional statements.
  • Not mutually exclusive; a passage can function as more than one type.
  • Central skill: decide whether the passage really tries to prove anything.

Deduction vs. Induction

• Every argument expresses an inferential claim with a specific strength.
• Deductive argument: claims that conclusion follows with necessity.
  Definition – Impossible for conclusion to be false if premises are true.
• Inductive argument: claims that conclusion follows only probably.
  Definition – Improbable that conclusion is false if premises are true.

Heuristics for Classifying

1. Special indicator words
   • Deductive: necessarily, certainly, definitely, absolutely.
   • Inductive: probably, likely, plausible, reasonable to conclude.
2. Actual inferential strength (does it truly guarantee or merely support?).
3. Argument form/style.
   • Deductive forms: mathematical reasoning, arguments from definition, categorical syllogism, hypothetical syllogism, disjunctive syllogism.
   • Inductive forms: prediction, argument from analogy, inductive generalization, argument from authority, argument based on signs, causal inference.
4. If factors conflict, prioritize:
   (1) premises confer strict necessity → treat as deductive,
   (2) presence of explicit deductive form,
   (3) presence of recognized inductive form,
   (4) indicator language.

• Caution: many real-world arguments are incomplete; classification may be impossible.

Classic Misconception

• “Deduction = general→particular; Induction = particular→general.”
  • Not reliable: examples exist for every direction.

Evaluating Arguments

Two separate evaluations:

1. Inferential claim quality.
2. Factual correctness of premises.

Deductive Evaluation

• Valid argument: impossible for premises to be true & conclusion false.
• Invalid: possible to have true premises & false conclusion.
• Sound: valid and all premises true ⇒ conclusion must be true.
  Unsound: invalid or at least one false premise.
• Truth table:
  Premises true + Conclusion false → always invalid.

Inductive Evaluation

• Strong argument: improbable that conclusion is false if premises true.
• Weak: otherwise.
• Cogent: strong and all premises true and premises meet Total Evidence Requirement (no overlooked crucial facts).
  Uncogent: weak or has false premise(s) or ignores evidence.
• Strength admits degrees (>50 % probability threshold).

Uniformity of Nature

• Underlies induction: future resembles past; spatial regularities persist.

Argument Forms & Counterexample Method

• Validity is determined by form; any substitution instance (uniform replacement of content terms with others) of a valid form is valid.
• To PROVE INVALIDITY:
  1. Abstract the form (replace content words with letters; keep logical form words: all, no, some, if, then, not, either, or, both, and).
  2. Construct a counterexample: a substitution instance where premises are indisputably true and conclusion is indisputably false.
  3. If such instance exists, form is invalid, thus original argument invalid.
• Recommended term set for categorical syllogisms: cats, dogs, mammals, fish, animals → easily yield clear truth-values.
• For conditional premises, choose antecedent–consequent pairs reflecting necessary connections (e.g., suicide→death).
• Limitation: method only disproves validity; can’t prove validity.

Extended Arguments & Diagramming

• Long passages mix argument chains, subarguments, and non-argumentative material.

Diagramming Conventions

• Number each statement.

- Arrows indicate “supports.”

Vertical pattern (serial reasoning)
4 → 3 → 2 → 1
Horizontal pattern (independent premises)
2 3 4 ⇒ 1 (arrows from each premise separately)

• Conjoint premises: braces show two or more statements that supply joint support; removing one weakens/destroys support.
• Multiple conclusions: a statement can support more than one conclusion (bracket lines splitting into two arrows).

Steps to Analyze

1. Identify ultimate conclusion (often author’s main claim).
   – Look for indicator words; ask “what’s the point?”
2. Identify major reasons/premises for that conclusion; note sub-conclusions.
3. Classify support relations: independent, conjoint, or chained.
4. Omit redundant or purely rhetorical sentences.
5. Draw diagram; use it to clarify evaluation strategy later.

Mathematical & Logical Notation Used

• Conditional: A \to B
• Necessary: A\; \text{only if}\; B (or \lnot B \to \lnot A)
• Sufficient: A \text{ is enough for } B (or A \to B)
• Deductive validity schema (categorical):
  \begin{array}{l} \text{All }A\text{ are }B\ \text{All }B\text{ are }C\ \therefore \text{All }A\text{ are }C \end{array}
• Invalid schema example:
  \begin{array}{l} \text{All }A\text{ are }B\ \text{All }C\text{ are }B\ \therefore \text{All }A\text{ are }C \end{array}
• Counterexample illustration: cats/dogs/animals.

Practical & Ethical Connections

• Understanding necessary vs. sufficient conditions is vital in law (requirements), medicine (diagnostic criteria), ethics (conditions for moral responsibility).
• Distinguishing argument from non-argument prevents manipulation by rhetoric.
• Evaluating deductive vs. inductive strength guides scientific reasoning and everyday decision-making (e.g., medical risk assessments rely on induction).
• Validity ensures no “truth-to-falsehood” inference; critical in legal proofs, mathematical demonstrations.
• Cogency emphasizes total evidence: ethically important for fair policy arguments and scientific integrity.
• Counterexample method teaches intellectual humility: one genuine counterexample refutes universal claims.
• Diagramming assists in civic discourse—clarifies where disagreements lie and locates hidden assumptions.

Study Tips & Common Pitfalls

• Always state premises before conclusion when testing validity/strength.
• Don’t mistake indicator words for genuine support; context matters.
• Be wary of “almost valid” rhetoric—deductive arguments are binary (valid or not).
• In induction, watch for ignored evidence (violates total-evidence requirement).
• When constructing counterexamples, choose everyday truths to avoid controversy.
• Practice diagramming by paraphrasing dense editorials; it reveals argumentative skeleton.
• Remember: a sound argument guarantees truth; a cogent argument guarantees high probability but allows future revision.