Understanding Symbolic Logic (Fifth Edition)

Core Principles of Logic and Argumentation

• Definition of Reasoning: The ability to draw appropriate conclusions from given evidence, effectively expanding knowledge from premises (what is known) to conclusions (what was previously unknown).

• The Verbal Expression of Reasoning: Logic focuses on the publicly ascertainable verbal expression of reasoning, termed an Argument.

• Structure of an Argument:

  • Premises: Sentences or clauses containing evidence or reasons, often preceded by words like "because" or "since."

  • Conclusion: The claim supposed to follow from the premises, often preceded by "therefore," "so," or "thus."

• Arguments vs. Assertions: An assertion is a mere statement of opinion or belief. An argument must provide backing support for that statement.

• Logic as a Normative Enterprise: Logic evaluated arguments as good or bad, correct or incorrect, based on the strength of the evidential relationship between premises and conclusion.

• Grounds of Evaluation: The correctness of an argument depends on the connection between premises and conclusion, not the factual truth of the statements.

Deductive vs. Inductive Logic

• Deductive Argument: An argument where the premises are intended to provide total support for the conclusion. The truth of the premises absolutely guarantees the truth of the conclusion.

• Inductive Argument: An argument where the premises are intended to provide only some degree of support. There is always a "logical gap" where the conclusion could be false even if premises are true.

• Validity in Deduction: A deductive argument is either Valid (absolute support) or Invalid (not absolute).

• Strength in Induction: Inductive goodness is a matter of degree (99\% probable vs. 50\% probable).

Form and Validity in Sentential Logic

• Logical Validity: Depends solely on the consistent form or structure of an argument, rather than the subject matter.

• Form vs. Instance:

  • Form: The general abstract pattern/blueprint (e.g., (p \lor q), \sim q \therefore p).

  • Instance: Particular meaningful examples that exhibit that form using specific sentences (Substitution Instances).

• Definition of a Valid Form: A form is valid if and only if there are no instances of that form in which all premises are true while the conclusion is false.

• Counterexample: An instance of a form with all true premises and a false conclusion. Any form having a counterexample is Invalid.

• Sound Argument: A valid deductive argument in which all premises are actually true.

The Structure of Sentential Logic

• Sentential (Propositional) Logic: The level of logic where complete, simple sentences are treated as unbroken units.

• Simple vs. Compound Sentences:

  • Simple Sentence: One that does not logically contain another complete sentence as a component.

  • Compound Sentence: One that contains at least one other complete declarative sentence as a component (e.g., negated sentences are always compound).

• Sentential Operators: Expressions containing blanks such that when filled with sentences, a new sentence results.

  • Truth-functional Local Operators: The truth value of the resulting compound is completely determined by the truth values of the component parts.

The Five Primary Sentential Operators

Rules of Inference (Linear Proofs)

• Modus Ponens (M.P.): From p \supset q and p, infer q.

• Modus Tollens (M.T.): From p \supset q and \sim q, infer \sim p.

• Hypothetical Syllogism (H.S.): From p \supset q and q \supset r, infer p \supset r.

• Simplification (Simp.): From (p \cdot q), infer p or infer q.

• Conjunction (Conj.): From p and q, infer (p \cdot q).

• Disjunctive Syllogism (D.S.): From (p \lor q) and \sim p, infer q. (Or from \sim q, infer p).

• Dilemma (Dil.): From (p \supset q), (r \supset s), and (p \lor r), infer (q \lor s).

• Addition (Add.): From p, infer (p \lor q).

Replacement Rules (Logical Equivalence)

• Double Negation (D.N.): p \equiv \sim \sim p

• Commutation (Comm.): (p \lor q) \equiv (q \lor p) and (p \cdot q) \equiv (q \cdot p)

• Association (Assoc.): ((p \lor q) \lor r) \equiv (p \lor (q \lor r)) and ((p \cdot q) \cdot r) \equiv (p \cdot (q \cdot r))

• Duplication (Dup.): p \equiv (p \lor p) and p \equiv (p \cdot p)

• De Morgan's (DeM.): \sim (p \lor q) \equiv (\sim p \cdot \sim q) and \sim (p \cdot q) \equiv (\sim p \lor \sim q)

• Contraposition (Contrap.): (p \supset q) \equiv (\sim q \supset \sim p)

• Biconditional Exchange (B.E.): (p \equiv q) \equiv ((p \supset q) \cdot (q \supset p))

• Conditional Exchange (C.E.): (p \supset q) \equiv (\sim p \lor q)

• Exportation (Exp.): ((p \cdot q) \supset r) \equiv (p \supset (q \supset r))

• Distribution (Dist.): (p \cdot (q \lor r)) \equiv ((p \cdot q) \lor (p \cdot r)) and (p \lor (q \cdot r)) \equiv ((p \lor q) \cdot (p \lor r))

Conditional and Indirect Proofs

• Conditional Proof (C.P.): If by assuming p you can derive q, you may infer p \supset q. Everything between the assumption and the conclusion falls within the Scope of that assumption.

• Indirect Proof (I.P.): Also known as Reductio ad Absurdum. Assume the opposite of what you want to prove (p). If a contradiction (q \cdot \sim q) follows, infer the negation of the assumption (\sim p).

• Restrictions:

  1. Every assumption made must be eventually discharged.

  2. Once discharged, neither the assumption nor any step derived within its scope can be used again.

  3. Nested assumptions must be discharged in reverse order (scope markers must not cross).

Monadic Predicate Logic

• Singular Sentence: A sentence asserting a property of a particular named individual (e.g., "John is happy," symbolized as Hj).

• Individual Constant: Lowercase letter (a, b, c, j\dots) standing for a specific name.

• Individual Variable: Lowercase letter (x, y, z\dots) serving as a placeholder.

• Propositional Function: An expression with a variable (e.g., Hx for "x is happy") that is not a sentence and has no truth value until a constant is substituted or it is quantified.

Quantifiers and Categorical Propositions

• Universal Quantifier: (x) expresses "For every x."

• Existential Quantifier: (\exists x) expresses "For some x" (defined as "at least one").

• The Four Categorical Propositions:

  • A: Universal Affirmative - "All S are P" symbolized by (x)(Sx \supset Px).

  • E: Universal Negative - "No S are P" symbolized by (x)(Sx \supset \sim Px).

  • I: Particular Affirmative - "Some S are P" symbolized by (\exists x)(Sx \cdot Px).

  • O: Particular Negative - "Some S are not P" symbolized by (\exists x)(Sx \cdot \sim Px).

• Quantifier Negation (Q.N.) Rules:

  • \sim (x)\phi x \equiv (\exists x)\sim \phi x

  • \sim (\exists x)\phi x \equiv (x)\sim \phi x

Higher Quantifier Rules and Identities

• Universal Instantiation (U.I.): From (x)\phi x, infer \phi a (any instance).

• Existential Instantiation (E.I.): From (\exists x)\phi x, infer \phi a provided a is a new letter successfully flagged.

• Universal Generalization (U.G.): From \phi a, infer (x)\phi x provided a is in a flagged subproof and meets restrictions.

• Existential Generalization (E.G.): From \phi a, infer (\exists x)\phi x.

• Identity Rules:

  • Identity Reflexivity (I. Ref.): Infer a = a from any premise.

  • Identity Symmetry (I. Sym.): (a = b) \equiv (b = a).

  • Identity Substitution (I. Sub.): From (a = b) and \phi a, infer \phi b.

• Numerical Statements using Identity:

  • "At least two Gs": (\exists x)(\exists y)(Gx \cdot Gy \cdot x \neq y)

  • "At most one G": (x)(y)((Gx \cdot Gy) \supset x = y)

  • "Exactly one G": (\exists x)(Gx \cdot (y)(Gy \supset x = y))

Semantic Methods for Invalidity

• Natural Interpretation Method: Assigning a domain and meanings to predicates to find true premises and a false conclusion.

• Model Universe Method: Proving invalidity by testing restricted, finite domains ({a}, {a, b}) and rewriting quantified statements as conjunctions (for universal) or disjunctions (for existential).