Problem 1 Given This Statement For All Planets In The Solar ✓ Solved

Problem 1given This Statementfor All Planets In The Solar System The

Given this statement: For all planets in the solar system, there exists a continent on the planet, such that for all jungles on the continent, there exists a species in the jungle with no teeth or some hair.

a. Write the negation of the statement. (It may be useful to work the problem symbolically, but write your final answer in words. 6 points.)

b. Unless the original statement is a paradox, either the original statement or its negation is true. Explain which you believe is true and why. (Your answer will be evaluated based on the logic, not the biology. 2 points.)

c. Consider the statement: For all jungles on the continent, there exists a species in the jungle with no teeth or some hair. On a continent that has no jungles, can this statement be declared either true or false? Why? (2 points.)

Sample Paper For Above instruction

The logical analysis of complex universal and existential statements is fundamental in understanding the structure of logical propositions. Here, we analyze a statement concerning planets, continents, jungles, and species, applying principles of formal logic to dissect and interpret the implications of negation and the truth values in different contexts.

Part a: Negation of the Original Statement

The original statement can be formalized as follows: "For all planets in the solar system, there exists a continent such that for all jungles on the continent, there exists a species with no teeth or some hair." Symbolically:

∀p (Planet(p) → ∃c (Continent(c, p) ∧ ∀j (Jungle(j, c) → ∃s (Species(s, j) ∧ (NoTeeth(s) ∨ Hair(s))))))

The negation of this statement involves reversing the quantifiers and the scope of the statement, leading to: "There exists at least one planet in the solar system such that, for every continent on that planet, there exists at least one jungle on the continent where, for every species in that jungle, all species either have teeth or do not have hair." In words, the negation is:

There exists a planet such that, for every continent on that planet, there exists a jungle where, for every species, every species either has teeth or lacks hair.

Part b: Is the Original Statement or Its Negation True?

Generally, in the absence of definitive biological evidence and considering the logical nature of the statement, the original appears plausible. It states that on every planet, there is a continent containing jungles with species that either lack teeth or have hair—conditions common among various species on Earth. Thus, it is reasonable to believe the original statement holds, making its negation less likely true. Logically, unless proven paradoxical, one of these statements must be true, and given general biological diversity, the original seems more probable.

Part c: Validity of the Statement on a Continent with No Jungles

The statement in question is: "For all jungles on the continent, there exists a species with no teeth or some hair." If a continent has no jungles, then the statement's antecedent (all jungles on the continent) is vacuously true, since there are no jungles to refute it. In logic, universal statements with an empty domain are considered true because there are no counterexamples. Therefore, on a continent with no jungles, the statement can be declared true because there are no jungles that would violate it.

References

• Barwise, J., & Etchemendy, J. (1992). The Language of First-Order Logic. CSLI Publications.