Understanding First Order Logic for "Two Purple Mushrooms

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Discussion Overview

The discussion revolves around the representation of the statement "There are exactly two purple mushrooms" in First Order Logic (FOL). Participants explore the components of the FOL expression and clarify the implications of its structure, particularly focusing on the logical relationships involved.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant seeks clarification on the FOL representation, particularly the role of the last part involving the variable z.
  • Another participant explains that omitting the last part would imply there are at least two purple mushrooms, rather than exactly two.
  • A different participant questions why the expression uses an OR ((x=z) v (y=z)), suggesting confusion over its necessity.
  • One participant elaborates that if three purple mushrooms are posited, the last must be identical to one of the first two to maintain the claim of exactly two, thus justifying the OR condition.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the FOL representation, with some agreeing on the necessity of the last part while others seek further clarification. The discussion remains unresolved as participants explore different interpretations and implications of the logical structure.

Contextual Notes

Participants indicate potential misunderstandings related to definitions and the foundational concepts of FOL, which may affect their interpretations of the logical expression.

jamborta
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hi,

could someone explain to me why the sentence - There are exactly two purple mushrooms is represented in FOL like this:
(Ex)(Ey) mushroom(x) ^ purple(x) ^ mushroom(y) ^ purple(y) ^ ~(x=y) ^ (Az) (mushroom(z) ^ purple(z)) => ((x=z) v (y=z))

especially the last part i have problem with. i assume that i misunderstood some of the definitions which are the basis of FOL, that might be the source of the confusion.

thanks for your help
 
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well without the last part with z, you actually stating that there are at least 2 purple mushrooms, while with the last part you stating that there are exactly two.
 
thanks. but i don't understand why it's and OR ((x=z) v (y=z)) which would allow either of them being equal to z.
 
Yes, that's the whole point. If you posit three purple mushrooms, x, y, and z, saying that there are, in fact, only two, the last of them, z, must be the same as either one of the first two. If you start of with two purple mushrooms, x and y, and state they are not the same mushroom, then any third purple mushroom must be the same mushroom as x or y but you don't know which. Yes, definitely, either of them could be equal to z.
 

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