Lecture 4 - Inference Laws

Vocabulary flashcards covering key concepts from the lecture notes on logic, arguments, inference rules, and quantified statements.

Argument

A sequence of propositions (hypotheses) followed by a final proposition (conclusion).

Hypothesis

A proposition that serves as a premise in an argument.

Conclusion

The final proposition derived from the hypotheses in an argument.

Internal Argument

An everyday reasoning process structured as hypotheses leading to a conclusion.

Premises

The hypotheses or assumptions that support the conclusion in an argument.

Validity

A property of an argument where, if all premises are true, the conclusion must be true (p1 ∧ … ∧ pn → c is a tautology).

Invalid

An argument where the conclusion can be false even when all premises are true.

Tautology

A formula that is true in every possible interpretation (a universally valid implication).

Truth table

A table listing all possible truth values of propositions and the truth value of a formula for each combination.

Modus ponens

If p is true and p→q is true, then q is true.

Modus tollens

If p→q is true and q is false, then p is false.

Hypothetical syllogism

If p→q and q→r, then p→r.

Disjunctive syllogism

If p∨q and ¬p, then q.

Addition

From p, infer p∨q (for any q).

Simplification

From p∧q, infer p (and also q).

Conjunction

From p and q, infer p∧q.

Resolution

From p∨q and ¬p, infer q (and the symmetric form).

Logical proof

A sequence of steps where each step is a proposition with a justification; the last step is the conclusion.

Proposition

A declarative statement that can be true or false; a basic unit in logic.

Justification

The reason or rule used to derive a step in a proof.

Quantified statements

Statements that use quantifiers like for all (∀) and exists (∃) to express properties about elements of a domain.

Domain

The set of individuals under consideration (e.g., the students in this class).

Predicate

A property or relation P(x), Q(x) that can be true or false when applied to elements of the domain.

Existential Instantiation

From ∃x P(x), infer P(c) for some new constant c.

Universal Instantiation

From ∀x P(x), infer P(a) for any element a in the domain.

Existential Generalization

From P(c) holds for some element c, infer ∃x P(x).

Universal Generalization

From P(x) holds for an arbitrary element c, infer ∀x P(x).

P(x)

A predicate example; e.g., P(x): 'x studied hard'.

Q(x)

Another predicate; e.g., Q(x): 'x failed the class'.