Propositional Logic Cheat Sheet

Overview & Motivation

Key Idea: Logic provides a formal language for representing knowledge and drawing inferences in a truth-preserving way.

Advantages of Logic:

Compact representation of knowledge
Truth-preserving inference rules
Formal reasoning without enumeration
High expressiveness

Three Components of Logic:

Syntax: Valid formulas
Semantics: Meaning (models)
Inference Rules: Derivation rules

1. Syntax

Definition: Syntax defines the set of valid formulas (grammatical expressions).

Building Blocks:

Propositional Symbols (Atoms):

A, B, C, Rain, Wet, Traffic

Atomic formulas representing basic propositions

Logical Connectives:

¬, ∧, ∨, →, ↔

Operators for combining formulas

Recursive Formula Construction:

Connective	Symbol	Formula	Name
NOT	¬	`¬f`	Negation
AND	∧	`f ∧ g`	Conjunction
OR	∨	`f ∨ g`	Disjunction
IMPLIES	→	`f → g`	Implication
IFF	↔	`f ↔ g`	Biconditional

Examples:

Valid: A, ¬A, A ∧ B, (A ∨ B) → C
Invalid: A ¬B, A + B, → A

2. Semantics

Definition: Semantics specifies the meaning of formulas through worlds and models.

World (Possible World):

Definition: A world represents a possible configuration or state of affairs in reality.

Key Concept: A world is a complete description of how things could be. Each world represents a hypothetical scenario where certain propositions are true and others are false.

Example: Different possible worlds:

World 1: It's raining, the ground is wet, it's Monday
World 2: It's not raining, the ground is wet (sprinklers), it's Monday
World 3: It's not raining, the ground is dry, it's Tuesday

Model:

Definition - Model:

A model w in propositional logic is an assignment of truth values to propositional symbols.

Important Note: In logic, the word "model" has a special meaning, quite distinct from how we use it in machine learning (quite an unfortunate collision). A model (in the logical sense) represents a possible state of affairs in the world.

w = {A: 0/1, B: 0/1, C: 0/1, ...}

Model: A mathematical representation of a world - an assignment of truth values (0 or 1) to propositional symbols.

Relationship: Each model w corresponds to exactly one possible world, and vice versa.

Example: All Possible Models

Given: 3 propositional symbols: A, B, C

Number of possible models: 2³ = 8

All 8 possible models w:

                                1. {A:0, B:0, C:0}
                            
                                2. {A:0, B:0, C:1}
                            
                                3. {A:0, B:1, C:0}
                            
                                4. {A:0, B:1, C:1}

                                5. {A:1, B:0, C:0}
                            
                                6. {A:1, B:0, C:1}
                            
                                7. {A:1, B:1, C:0}
                            
                                8. {A:1, B:1, C:1}

General Rule: With n propositional symbols, there are 2ⁿ possible models.

World ↔ Model: We use "world" when thinking about reality and "model" when doing formal logic. They refer to the same thing!

Interpretation Function I(f, w):

Returns true (1) if model w satisfies formula f, false (0) otherwise.

Truth Table for Connectives:

f	g	¬f	f ∧ g	f ∨ g	f → g	f ↔ g
0	0	1	0	0	1	1
0	1	1	0	1	1	0
1	0	0	0	1	0	0
1	1	0	1	1	1	1

Models of a Formula M(f):

M(f) = {w | I(f, w) = 1}

Definition: The set of all models (worlds) that satisfy formula f.

Intuition: M(f) picks out all the possible worlds where formula f is true.

Example: f = Rain ∨ Wet
M(f) includes all worlds/models where Rain=1 or Wet=1 or both.

These are worlds like:

World where it's raining and wet
World where it's not raining but wet (sprinklers)
World where it's raining but somehow not wet (unusual!)

Key Insight: A formula compactly represents a set of possible worlds!

3. Knowledge Base (KB)

Definition: KB is a set of formulas representing their conjunction (all must be true).

M(KB) = ⋂_f∈KB M(f)

Intuition: KB specifies constraints on the world.

M(KB) is the set of all possible worlds satisfying those constraints.

Key Insight: Adding formulas to KB shrinks the set of possible worlds (adds more constraints on reality).

Example:

KB = {Rain, Rain → Wet}
M(KB) = all worlds where both Rain and Wet are true
Interpretation: KB rules out worlds where it's not raining, or where it's raining but not wet
Possible worlds consistent with KB: Only worlds where it's raining AND the ground is wet

Narrowing Down Worlds:

Start: All 2ⁿ possible worlds
Add formula: Eliminate worlds that don't satisfy it
More knowledge → Fewer possible worlds
Goal: Narrow down to the actual world!

4. Relationships: Entailment, Contradiction, Contingency

Entailment

KB ⊨ f

Definition: M(f) ⊇ M(KB)

f adds no information (already known)

Example:
Rain ∧ Snow ⊨ Snow

Contradiction

KB ⊨ ¬f

Definition: M(KB) ∩ M(f) = ∅

f contradicts KB

Example:
Rain ∧ Snow
contradicts ¬Snow

Contingency

Neither

Definition: ∅ ⊊ M(KB) ∩ M(f) ⊊ M(KB)

f adds non-trivial information

Example:
Rain and Snow

Important Relationship: KB contradicts f ⟺ KB entails ¬f

5. Knowledge Base Operations

Tell[f] → KB

Purpose: Add new information

Responses:

KB ⊨ f: Already knew that
KB ⊨ ¬f: Don't believe that
Contingent: Learned something new (update KB)

Ask[f] → KB

Purpose: Query for information

Responses:

KB ⊨ f: Yes
KB ⊨ ¬f: No
Contingent: I don't know

6. Satisfiability & Model Checking

Satisfiability: KB is satisfiable if M(KB) ≠ ∅

Reduce Ask[f] to Satisfiability:

Check Entailment (KB ⊨ f):
Is KB ∪ {¬f} unsatisfiable?

Check Contradiction (KB ⊨ ¬f):
Is KB ∪ {f} unsatisfiable?

Complexity: SAT is NP-complete!

Model Checking as CSP:

Propositional Logic	⇔	CSP
Propositional symbol	⇔	Variable
Formula	⇔	Constraint
Model	⇔	Assignment

Algorithms:

DPLL: Backtracking search + pruning
WalkSat: Randomized local search

7. Inference Rules

Inference Rule: Syntactic pattern for deriving new formulas

Premises
───────────
Conclusion

Key Idea: Inference rules operate directly on syntax, not semantics!

Classic Inference Rules:

Modus Ponens:

p, p → q
─────────
q

"If p implies q and p is true, then q is true"

Modus Tollens:

p → q, ¬q
─────────
¬p

"If p implies q and q is false, then p is false"

Forward Inference Algorithm:

Repeat until no changes to KB:
Choose formulas f₁, ..., fₖ ∈ KB
If matching rule exists: f₁,...,fₖ / g
Add g to KB

Derivation (KB ⊢ f): f is eventually added to KB by applying inference rules

8. Soundness & Completeness

"The truth, the whole truth, and nothing but the truth"

Soundness

"Nothing but the truth"

{f : KB ⊢ f} ⊆ {f : KB ⊨ f}

Meaning: Everything derived is true. If we can derive f using inference rules, then f must be entailed.

Completeness

"The whole truth"

{f : KB ⊢ f} ⊇ {f : KB ⊨ f}

Meaning: All truths can be derived. If f is entailed, we can derive it using inference rules.

Visual Analogy:

Glass = Set of true formulas {f : KB ⊨ f}

Water = Set of derived formulas {f : KB ⊢ f}

Sound: Water never overflows the glass
Complete: Water fills the glass to the brim

Checking Soundness Example:

Is Rain, Rain → Wet / Wet sound?

Check: M(Rain) ∩ M(Rain → Wet) ⊆ M(Wet)

Result: ✅ Yes, this is sound!

9. Quick Reference Table

Notation	Meaning	Description
World	Possible World	A complete possible configuration of reality
`w`	Model	Mathematical representation of a world; assignment of truth values to propositional symbols
`I(f, w)`	Interpretation	Returns 1 if world w satisfies formula f, 0 otherwise
`M(f)`	Models of f	Set of all possible worlds (models) that satisfy f
`M(KB)`	Models of KB	Set of all possible worlds consistent with the knowledge base
`KB ⊨ f`	Entailment	f is true in all possible worlds where KB is true
`KB ⊢ f`	Derivation	f can be derived from KB using inference rules
`SAT(KB)`	Satisfiability	KB has at least one satisfying world (model)

10. Key Formulas & Relationships

Remember: M(f) represents the set of all possible worlds where formula f is true. All formulas below are about sets of worlds!

Model Operations (Operations on Sets of Worlds):

M(KB) = ⋂_f∈KB M(f)

The worlds satisfying KB are exactly those worlds that satisfy all formulas in KB

M(f ∧ g) = M(f) ∩ M(g)

Worlds where both f and g are true

M(f ∨ g) = M(f) ∪ M(g)

Worlds where at least one of f or g is true

M(¬f) = All worlds \ M(f)

Worlds where f is false

Relationship Tests:

KB ⊨ f ⟺ M(f) ⊇ M(KB)

KB contradicts f ⟺ M(KB) ∩ M(f) = ∅

KB contradicts f ⟺ KB ⊨ ¬f

Satisfiability Tests:

KB ⊨ f ⟺ KB ∪ {¬f} is unsatisfiable

SAT(KB) ⟺ M(KB) ≠ ∅

Summary

Three Views of Logic:

Syntax: Manipulation of symbols according to rules
Semantics: Truth in possible worlds (models)
Inference: Deriving new knowledge from existing knowledge

Model Checking:

Uses semantics
Enumerate and test models
Guaranteed correct
Can be slow (NP-complete)

Inference Rules:

Uses syntax
Apply rules to formulas
Can be efficient
Need soundness & completeness