PROPOSITIONAL LOGIC

Trong-Nghia Nguyen

Faculty of DS & AI

Autumn semester, 2026

AI,

Autumn, 2026

Trong-Nghia Nguyen

Content

• Representing problems using logic

• Propositional logic

Trong-Nghia Nguyen

Representing problems using logic

Trong-Nghia Nguyen

Representing problems using logic

Trong-Nghia Nguyen

Representing problems using logic

Trong-Nghia Nguyen

Representing problems using logic

Trong-Nghia Nguyen

Representing problems using logic

Trong-Nghia Nguyen

Content

• Representing problems using logic

• Propositional logic

Trong-Nghia Nguyen

Propositional logic

Trong-Nghia Nguyen

Propositional logic

(ModusTollens rule)

Trong-Nghia Nguyen

Propositional logic

Trong-Nghia Nguyen

Propositional logic

● Atomic sentence: a single clause symbol (or

True/False constant). For example: P, Q, Rain.

● Complex sentence: combined from smaller

sentences using connective. For example: ¬P, P ∧

Q, (P ∨ Q) ⇒ R.

● Literal: an atomic sentence or its negation. For

example: P (positive literal), ¬P (negative literal).

● Clause (clause): collection (∨) of literals. For

example: (P ∨ ¬Q ∨ R)

Sentence → AtomicSentence | ComplexSentence

AtomicSentence → True | False | P | Q | R | …

ComplexSentence → ¬ Sentence

| ( Sentence ∧ Sentence )

| ( Sentence ∨ Sentence )

| ( Sentence ⇒ Sentence )

| ( Sentence ⇔ Sentence )

Trong-Nghia Nguyen

Propositional logic

Trong-Nghia Nguyen

Propositional logic

¬ (Negation) → ∧ (Conjunction) → ∨

(Disjunction) → ⇒ (Implication) → ⇔

(Equivalence)

Trong-Nghia Nguyen

Propositional logic

Trong-Nghia Nguyen

Propositional logic

Model (world / assignment) — interpretation

● A model m is a way of assigning the value True/False to every propositional symbol.

Each model is "a possible world".

● If there are n symbols, there are 2ⁿ models (2ⁿ lines of the truth table).

● For example: with 2 symbols P, Q → there are 4 models:

m₁ = {P=F, Q=F}, m₂ = {P=F, Q=T}, m₃ = {P=T, Q=F}, m

₄

= {P=T, Q=T}.

What does "the sentence α is true in model m" mean?

● Assign values to symbols according to m, then calculate from inside to outside according

to the truth table of each connective (¬, ∧, ∨, ⇒, ⇔) — obtain True or False.

● Notation: m ⊨ α reads "m satisfies α" (α is true in m).

● M(α) = set of all models that make α true.

Trong-Nghia Nguyen

Propositional logic

For example, calculating the value for each model.

Consider the formula α = (P ∨ Q) ∧ ¬Q. Evaluate α across all four models:

→ α is true only in m₃, i.e., M(α) = { m₃ } = {P=T, Q=F}.

Model (world / assignment) — interpretation

Trong-Nghia Nguyen

Propositional logic

Entailment (⊨) — "logically implies"

● KB ⊨ α: "KB entails α," meaning: α is true in every model where KB is true.

● If one accepts KB, one is compelled to accept α.

● In terms of model sets: KB ⊨ α ⟺ M(KB) ⊆ M(α) (every world that satisfies KB also

satisfies α).

How to check for entailment

● Consider all models; only look at models as KB = True ;

● If in all of them, α also = True → KB ⊨ α.

● If there is even a model KB=True but α=False → KB ⊭ α.

Trong-Nghia Nguyen

Propositional logic

Entailment (⊨) — "logically implies"

Example: Given KB = (P

⇒

∧

P, does KB

⊨

● The KB true only at m₄. At m₄, Q = True ⇒ KB ⊨ Q ✓ (this is essentially Modus Ponens

viewed from a semantic perspective).

● Conversely, KB ⊭ ¬Q: because at m₄ (the only model satisfying the KB), ¬Q = False.

Trong-Nghia Nguyen

Propositional logic

Soundness & Completeness

● ⊨ (semantics): KB

⊨

α — "α is true in all models making KB true". This is the truth.

● ⊢ (syntax): KB

⊢

ᵢ α — "α is derived from KB by inference procedure i" (applying rules such as Modus

Ponens, resolution…). This is the calculator that calculates it.

Does procedure

⊢

match truth

⊨

Soundness

● If KB ⊢ α, then KB ⊨ α.

● Meaning: The procedure derives only what is actually true—it never yields an incorrect conclusion.

(⊢ ⊆ ⊨)

Completeness

● If KB ⊨ α, then KB ⊢ α.

● Meaning: the procedure derives everything that is entailed—no truth is missed. (⊨ ⊆ ⊢)

“A procedure that is both sound and complete is

⊢

identical

⊨

: it calculates correctly and has all logical consequences.

That's what we want.”

Trong-Nghia Nguyen

Propositional logic

Soundness & Completeness

Example: SOUND Rule — Modus Ponens

From P ⇒ Q and P, it follows that Q. ("It is raining" ⇒ "the road is wet".)

Check if KB = (P ⇒ Q) ∧ P ⊨ Q?

→ The KB is true in only one row (P=T,

Q=T), and in that row, Q = T. Thus,

every model that satisfies the KB also

satisfies Q → KB ⊨ Q. The rule is sound

✓.

Trong-Nghia Nguyen

Propositional logic

Soundness & Completeness

Example: UNSOUND Rule — Affirming the Consequent

From P ⇒ Q and Q, one might infer P. ("Wet road" ⇒ "raining"?) — it sounds

plausible but is incorrect: the road could be wet due to a street-watering truck, not

rain.

Check if KB = (P ⇒ Q) ∧ Q ⊨ P?

→ There is a model {P=F, Q=T} where

the KB is true but P is False. Thus, KB ⊭

P → the rule is NOT sound ✗. That

model corresponds to the scenario of

"wet ground (Q) but no rain (¬P)."

Trong-Nghia Nguyen

Propositional logic

Trong-Nghia Nguyen

Propositional logic

Trong-Nghia Nguyen

Propositional logic

Trong-Nghia Nguyen

Propositional logic

Trong-Nghia Nguyen

Propositional logic

Trong-Nghia Nguyen

Thank you!