Compare predicate logic and propositional logic. | Previous Year Questions Solution(MCSE003)

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Question 1 : Compare predicate logic and propositional logic. Write De Morgan’s law for both. How will you verify that a given formula is valid or invalid ? What do you understand by validity and consistency of any well form formula ? 

Answer :

 Comparison between Predicate Logic and Propositional Logic:

Predicate Logic and Propositional Logic are two formal systems used in mathematical logic to reason and analyze statements. Here are the key differences between them:

Scope: Propositional Logic deals with propositions or statements as atomic units, without considering their internal structure or meaning. Predicate Logic, on the other hand, allows for the representation of more complex statements by introducing variables, quantifiers, and predicates.

Expressiveness: Predicate Logic is more expressive than Propositional Logic because it allows for the representation of relationships, properties, and quantification over variables. Propositional Logic, in contrast, is limited to the manipulation of simple truth values and logical connectives.

Quantification: Predicate Logic introduces quantifiers, such as universal quantifier (∀) and existential quantifier (∃), to express statements about all or some elements in a given domain. Propositional Logic does not include quantifiers.

Variables: Predicate Logic uses variables to represent unspecified elements or objects, allowing for general statements and reasoning over a range of instances. Propositional Logic does not have variables.

Atomic vs. Compound Statements: Propositional Logic deals with atomic statements, combining them using logical connectives (AND, OR, NOT) to form compound statements. Predicate Logic has atomic statements, but it also allows for the representation of compound statements through the use of quantifiers, variables, and predicates.

De Morgan's Laws:

De Morgan's Laws describe the relationship between negation and conjunction (AND) or disjunction (OR) in both Predicate Logic and Propositional Logic:

For Propositional Logic:

De Morgan's Law for Conjunction (AND):

¬(P ∧ Q) ≡ (¬P ∨ ¬Q)

De Morgan's Law for Disjunction (OR):

¬(P ∨ Q) ≡ (¬P ∧ ¬Q)

For Predicate Logic:

De Morgan's Law for Conjunction (AND):

¬(∀x P(x)) ≡ (∃x ¬P(x))

De Morgan's Law for Disjunction (OR):

¬(∃x P(x)) ≡ (∀x ¬P(x))

Validity and Consistency:

Validity: A formula or statement is valid if it is true in all possible interpretations or truth assignments. In other words, a formula is valid if there is no possible situation where it can be false.

Consistency: A set of formulas is consistent if there is at least one interpretation or truth assignment in which all the formulas in the set can be true simultaneously. In other words, a set of formulas is consistent if it is possible for all the formulas to be true together.

Verification of Validity or Invalidity:

To verify if a given formula is valid or invalid, you can use various methods, including:

Truth Tables: Construct a truth table that exhaustively lists all possible combinations of truth values for the atomic propositions in the formula. If the formula evaluates to true in all rows of the truth table, it is valid. If there is at least one row where the formula evaluates to false, it is invalid.

Logical Inference: Use logical inference rules and proof techniques, such as modus ponens, modus tollens, universal instantiation, existential instantiation, and rules of inference specific to Predicate Logic, to demonstrate the validity or invalidity of a formula based on the given premises or axioms.

Semantic Methods: Use semantic models and interpretations to evaluate the truth value of a formula. If the formula is true under all possible interpretations, it is valid. If there is at least one interpretation where the formula is false, it is invalid.

By employing these methods, you can determine whether a formula is valid (always true), invalid (not always true), or contingent (true in some interpretations and false in others) in both Predicate Logic and Propositional Logic.