Proving Propositional Logic with Quantifiers: Copi's Symbolic Logic Problem

  • MHB
  • Thread starter agapito
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In summary, propositional logic is a branch of mathematical logic that uses symbols and logical operators to determine the truth value of complex statements. Quantifiers, such as universal and existential, are used to specify the quantity of objects in a statement. Copi's Symbolic Logic Problem is a problem that involves proving the validity of a logical argument using propositional logic and quantifiers. To prove propositional logic with quantifiers, one must translate the argument into symbolic notation and apply logical rules and inference. This process is important for determining the validity of arguments and formalizing reasoning in various fields.
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agapito
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Formally prove the following using only propositional logic + quantifiers (problem from Copi "Symbolic Logic")

∀x ∃y (Kx /\ Ly) Premise

∃y ∀x (Kx /\ Ly) Conclusion

I'm having a hard time with the strictures of Universal Quantification and Existential Instantiation. Thanks for all help.
 
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  • #2
1. ∀x (Kx → ∃y Ly) Premise2. ∀x Kx Assumption [A]3. ∃y Ly Universal Instantiation [1, 2]4. ∃y ∀x (Kx /\ Ly) Existential Generalization [3]5. Kx Universal Generalization [2]6. Kx /\ Ly Conjunction Introduction [5, 3]7. ∀x (Kx /\ Ly) Universal Generalization [6]8. ∀x ∃y (Kx /\ Ly) Premise9. ∃y ∀x (Kx /\ Ly) Modus Ponens [7, 8]10. ∃y ∀x (Kx /\ Ly) Conditional Proof [A–9]
 

FAQ: Proving Propositional Logic with Quantifiers: Copi's Symbolic Logic Problem

What is propositional logic?

Propositional logic is a branch of mathematical logic that deals with the study of propositions or statements and their logical relationships. It uses symbols to represent propositions and logical operators to express the relationships between them.

What are quantifiers in propositional logic?

Quantifiers are symbols used to express the quantity or scope of a statement. In propositional logic, there are two types of quantifiers: universal quantifiers (∀) and existential quantifiers (∃). Universal quantifiers express that a statement is true for all elements in a given domain, while existential quantifiers express that a statement is true for at least one element in a given domain.

What is Copi's Symbolic Logic Problem?

Copi's Symbolic Logic Problem refers to a challenge posed by Irving Copi, a philosopher and logician, to prove the validity of a certain type of argument in propositional logic with quantifiers. The problem involves proving the validity of an argument that contains both universal and existential quantifiers.

How is Copi's Symbolic Logic Problem solved?

Copi's Symbolic Logic Problem can be solved by using a combination of rules and principles from propositional logic and quantifier logic. This includes using the rules of inference, such as modus ponens and universal instantiation, as well as the principles of quantifier negation and quantifier exchange.

Why is Copi's Symbolic Logic Problem important?

Copi's Symbolic Logic Problem is important because it tests the understanding and application of key concepts in propositional logic and quantifier logic. It also demonstrates the power and limitations of formal logic in proving the validity of arguments. Solving this problem can help improve logical reasoning skills and understanding of symbolic logic.

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