When most people say ‘logic’, they mean either propositional logic or first-order predicate logic. However, the precise definition is quite broad,and literally hundreds of logics have been studied by philosophers, computer scientists and mathematicians. Any ‘formal system’ can be considered a logic if it has:

- A well-defined syntax
- Well-defined semantics
- A proof-theory

The syntax of a logic defines the syntactically acceptable objects of the language, which are properly called well-formed formulae. The semantics of a logic associate each formula with a meaning. The proof theory is concerned with manipulating formulae according to certain rules.

## What is Propositional Logic?

A proposition is a statement that can be either true or false; it must be one or the other, and it cannot be both. The following are propositions:

A: “It is raining.”

B: “The grass is wet.”

These propositions are represented by capital letters, and their truth values help determine the logical relationships between them. Logical operations are used to combine and manipulate propositions to evaluate their overall truthfulness.

Also Read: Frames in AI

## The Connectives

The connectives we introduce are:

∧ and (& or .)

∨ or (| or +)

¬ not (∼)

⇒ implies (⊃ or →)

⇔ iff

If p and q are arbitrary propositions, then the conjunction of p and q is written,

p ∧ q

and will be true iff both p and q are true.

**Common operators in propositional logic include:**

**Negation (¬):** Represents “not.” For example, ¬A means “it is not raining.”
**Conjunction (∧):** Represents “and.” A ∧ B means “It is raining and the grass is wet.”
**Disjunction (∨):** Represents “or.” A ∨ B means “It is raining or the grass is wet.”
**Implication (→):** Represents “if… then.” A → B means “If it is raining, then the grass is wet.”
**Biconditional (↔):** Represents “if and only if.” A ↔ B means “The grass is wet if and only if it is raining.”

## Syntax** and Semantics**

In propositional logic, syntax refers to the structure of the formulas created using the logical operators and propositions. For example:

**Atomic formulas: **Simple statements like A or B.

**Complex formulas:** Created by combining atomic formulas, such as ¬A ∨ B or A ∧ (¬C → B).

Truth tables help in calculating the true value of complex formulas based on their individual components.

For example, in the case of A→B:

If A is true and B is false, the implication A→B is false.

If A is false, the entire implication is considered true, regardless of B’s value.

Also Read: AdaBoost Algorithm in Machine Learning

To make logical operations more manageable, formulas are often converted into standard forms known as normal forms:

**Conjunctive Normal Form (CNF):** A formula is in CNF if it is a conjunction (AND) of disjunctions (OR) of literals
**Disjunctive Normal Form (DNF): **A formula is in DNF if it is a disjunction (OR) of conjunctions (AND) of literals.

**For example:**

(A∨B)∧(¬A∨C) is in CNF.

(A∧B)∨(¬C∧D) is in DNF.

Converting formulas to CNF or DNF makes them easier to work with in automated reasoning systems.

## Resolution and Proofs

The resolution method is a key inference technique in propositional logic. It involves deriving new clauses by resolving pairs of clauses that contain complementary literals. Resolution is widely used in automated theorem proving, where the goal is to determine whether a formula or a set of formulas is satisfiable.

For instance, given the clauses:

You can resolve them by eliminating A and ¬A, leading to B∨C.

Formal proofs in propositional logic consist of step-by-step derivations of a conclusion (formula G) from a set of assumptions (formula F). The proof system applies logical rules, like “modus ponens” (if A→B and A are true, then B is true).

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**Applications of Propositional Logic in AI**

In AI, propositional logic plays an important role in areas such as:

Knowledge representation: Systems like knowledge bases store facts and rules in a way that can be processed logically. For example, an AI might represent “If it rains, then the ground is wet” using A→B

**Automated reasoning:** AI systems use propositional logic to infer new facts from existing ones. Resolution and entailment help the system make decisions based on logical deductions.

**Model checking:** This involves verifying whether a model satisfies certain properties by checking if all logical statements are true under a given assignment of truth values.

For instance, a model checker might assess whether the formula

A∧(B∨¬C) is valid under a specific set of truth values for A, B, and C.

**Pros and Cons of Propositional Logic**

- Declarative propositional logic holds that syntactic elements match facts.
- As opposed to most databases and data structures, propositional logic permits partial, disjunctive, and negated data.
- Compositional logic derives the meaning of P Q from the meanings of P and Q.
- Propositional logic does not rely on context for meaning, in contrast to natural language, where context determines meaning.
- Unlike natural language, propositional logic has very little expressive power.
- You cannot write one sentence for each square and say, “Pits cause breezes in adjacent squares.”

**Challenges and Limits**

While propositional logic is powerful, it has its limitations. It cannot handle the complexity of scenarios that involve quantifiers (like “for all” or “there exists”) or reasoning about objects and their properties. For such cases, first-order logic (which extends propositional logic) is required.

**Difference between Propositional Logic and Predicate Logic**

A predicate that provides more details about the subject of a sentence is the foundation of predicate logic. It can be called an attribute that establishes the characteristics of the sentence’s subject.

**Conclusion**

Propositional logic serves as the bedrock of many AI systems, providing a clear framework for reasoning about truth and relationships between propositions. Though simple in structure, its formal rules and techniques like resolution and entailment have widespread applications in areas such as decision making, knowledge representation, and automated reasoning in AI. Understanding propositional logic is an essential step in grasping more advanced logical systems used in AI.

**Basis** |
**Propositional Logic** |
**Predicate logic** |

**Definition** |
A declarative statement in propositional logic has a truth value which can either be true or false and can never be both at the same time. |

**Variables** |
Propositional Logic does not consist of variables. |
Variables are present. |

**Logical connectives** |
The logical connectives in propositional logic are AND, OR, NOT, IF-THEN, IF-AND-ONLY-IF. |
Predicate logic uses the same logical connectives as propositional logic, which also include propositional plus quantifiers. |

**Scope analysis** |
Scope analysis is not performed in propositional logic. |
In predicate logic, quantifiers such as universal, existential, and uniqueness qualifiers are used to carry out scope analysis. |

**Representation** |
Propositional logic is a generalised representation. |
Predicate logic is a specialised representation. |

**Truth value** |
In proposition logic, a proposition has a truth value, i.e., true or false. |
In predicate logic, the truth value depends on the value of the variable, |

**Use case** |
Simple logical connections are analysed using propositional logic. |
For a given variable, complex relationships and decisions are expressed using predicate logic. |

FAQs

Propositional logic in AI is considered essential for tasks such as knowledge representation, reasoning, and decision-making. It enables AI systems to model the world and design new information from existing knowledge and make intelligent choices based on logical reasonings.

Contingency. A Contingency is a formula which has both some true and some false values for every value of its propositional variables. As we can see every value of (A∨B)∧(¬A) has both “True” and “False”, it is a contingency.

There are five types in propositional logic:
Negations
Conjunctions
Disjunctions
Conditionals
Biconditionals

The first step is to strike down the possibilities that might affect the process so that the favourable scenarios can be narrowed down. Then the logic can be solved. After that, remember it is very easy to fall into a faulty reasoning.

Predicate logic consists of a predicate that gives further information about a sentence’s subject. It can be referred to as an attribute that determines the properties of the subject in a sentence.