Help with Logic Proof: Establishing 7.

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    Logic Proof
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SUMMARY

The forum discussion focuses on a logic proof from Copi's "Symbolic Logic," specifically addressing the establishment of line 7, ¬(∀x) Fx, using lines 1, 2-6, and the Existential Instantiation (EI) rule. The proof begins with the assumptions that there exists an x such that Fx is not true and that for all x, Fx is true. The contradiction arises when the assumption of universal truth (line 3) leads to a specific instance (line 4), which contradicts the initial existential assumption (line 1). Thus, the conclusion of line 7 is valid as it follows from the established premises and rules of inference.

PREREQUISITES
  • Understanding of first-order logic and quantifiers
  • Familiarity with Copi's "Symbolic Logic" and its notation
  • Knowledge of rules of inference, particularly Existential Instantiation (EI) and Modus Tollens (MT)
  • Ability to analyze logical proofs and identify contradictions
NEXT STEPS
  • Study the rules of inference in detail, focusing on Existential Instantiation (EI) and Modus Tollens (MT)
  • Review the concepts of existential and universal quantifiers in first-order logic
  • Practice constructing and deconstructing proofs using Copi's "Symbolic Logic" as a reference
  • Explore examples of contradictions in logical proofs to strengthen understanding
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Students of logic, philosophers, and anyone interested in formal proof techniques and the intricacies of first-order logic.

agapito
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Consider the following proof from Copi's "Symbolic Logic", p. 109:

1. (∃x) ¬Fx Assumption

2. ¬Fy Assumption

3. (∀x) Fx Assumption

4. Fy 3, UI

5. (∀x) Fx ⇒ Fy 3-4, CP

6. ¬(∀x) Fx 5,2 MT

7. ¬(∀x) Fx 1,2-6, EI

8. (∃x) ¬Fx ⇒ ¬(∀x) Fx 1-7, CP

9. (∀x) Fx ⇒ ¬(∃x) ¬Fx 8, Trans, DN

I cannot understand how 7.- is established, everything else is clear. Can someone explain how 1,2-6, EI results in 7. ?
 
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Don't (1), "There exist x such that Fx is not true", and (3), "for all x Fx is true", contradict one another?
 

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