Semantics week 3

LING2100

What is an argument form?

An argument form consists of premises and a conclusion. It is valid if, whenever the premises are true, the conclusion must also be true (truth-preserving)

What is modus ponens?

Modus ponens is a valid argument form:

If p, then q.

p.

∴ q

Example:

If Julian is singing, then Garak is happy.

Julian is singing

∴ Garak is happy.

What is ‘affirming the consequent’, and why is it invalid?

Form:

if p, then q

q

∴ p

This is invalid because q could not be true for reasons other than p.

Example:

If Julian is singing, Garak is happy.

Garak is happy

∴ Julian is singing (INVALID)

What is ‘denying the consequent’ (modus tollens)

Valid form:

If p, then q

Not q

∴ Not p

Example:

If it rained, the lawn is wet.

The lawn is not wet

∴ It didn’t rain.

What is ‘denying the antecedent’?

Invalid form:

If p, then q

Not p.

∴ Not q

Example:

If it rained, the lawn is wet.

it didn’t rain.

∴ The lawn is not wet (INVALID)

What are propositional letters and how are they used?

Symbols like p, q, r used to represent entire propositions (statements that can be true or false). They are the atomic building blocks of propositional logic.

What is a well-formed formula (wff)?

A syntactically correct expression in propositional logic.

Examples:

GOOD: p, ¬p, (p ∧ q)

BAD: ∧p, p¬q

What is the semantic value fo a sentence in propositional logic?

its truth value: either true (T) or false (F). These are assigned by an interpretation function (e.g., l(p) = T

What are the logical connectives in propositional logic?

• ¬: Negation ("not")

• ∧: Conjunction ("and")

• ∨: Disjunction ("or", inclusive)

• →: Material implication ("if...then")

• ↔: Biconditional ("if and only if")

What is the truth table for negation (¬p)?

p ¬p

T F

F T

Negation flips the truth value

What is the truth table for conjunction (p ∧ q)?

p q p ∧ q

T T T

T F F

F T F

F T F

True only if both p and q are true

What is the truth table for disjunction (p ∨ q)

p q p ∨ q

T T T

T F T

F T T

F T F

Inclusive or: true if at least one disjunction is true.

What is exclusive or (XOR), and how does it differ?

p q p ⊕ q

T T F

T F T

F T T

F T F

XOR is true only if exactly one of the inputs is true.