Specialization-generalization(mathematical logic)

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In summary, specialization is defined as a concept where every instance of B is also an instance of A, but there are instances of A that are not instances of B. This means that A cannot be a special case of B, as it would contradict the definition and create inconsistency. This logic is important for consistent scientific theories, although there may be some established theories that do not fulfill this requirement.
  • #1
RockyMarciano
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Specialization as defined in Wikipedia:
"Concept B is a specialisation of concept A if and only if:

  • every instance of concept B is also an instance of concept A; and
  • there are instances of concept A which are not instances of concept B"
We then call B as special case of A, it seems evident from the definition that in no case, given this definition, can A be simultaneously a special case of B, because it would be in contradiction and that conceptual system would be inconsistent.

Should scientific theories(mathematically founded) follow this logic or can you think of any example that wouldn't necessarily? In other words is this a requirement for consistent scientific theories?
 
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  • #2
yes :smile:
 
  • #3
BvU said:
yes :smile:
Thanks, I supposed so but I wanted to confirm it, people would be surprised by certain established theories that don't fulfill this consistency check.
 
  • #4
RockyMarciano said:
Thanks, I supposed so but I wanted to confirm it, people would be surprised by certain established theories that don't fulfill this consistency check.
Perhaps the issue is not so clear when the special cases happen to be also degenerate cases, but then I can see that in itself as a symptom of possible inconsistency.
 

Related to Specialization-generalization(mathematical logic)

1. What is the difference between specialization and generalization in mathematical logic?

Specialization and generalization are two concepts in mathematical logic that refer to the relationship between more specific and more general statements or theories. Specialization is the process of creating a more specific statement or theory from a more general one, while generalization is the process of creating a more general statement or theory from a more specific one.

2. How are specialization and generalization used in mathematical proofs?

In mathematical proofs, specialization and generalization are often used to break down complex statements into simpler ones that are easier to prove. Specialization allows for more specific and concrete statements to be proven, while generalization allows for broader and more abstract statements to be proven.

3. What are some examples of specialization in mathematical logic?

One example of specialization in mathematical logic is the process of creating a more specific theorem from a more general one by adding additional assumptions or conditions. For example, from the general statement "All triangles have three sides," we can specialize to the statement "All equilateral triangles have three sides." Another example is the process of specializing a function to a particular value, such as specializing the function f(x) to f(2) to evaluate it at x=2.

4. How does generalization help in mathematical modeling?

In mathematical modeling, generalization is used to create abstract and simplified models that can be applied to a wide range of situations. By generalizing, we can identify common patterns and relationships between different phenomena and create more universal and robust models.

5. Can specialization and generalization be used together in mathematical logic?

Yes, specialization and generalization are often used together in mathematical logic. In the process of specialization, we often make use of previously generalized statements or theories as starting points. Similarly, in the process of generalization, we may specialize certain aspects or parameters to create more refined and applicable models. Both concepts are important in developing and refining mathematical theories and proofs.

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