Analyzing Logical Statements and Conclusions

The question asks us to evaluate a given statement and determine which of the provided conclusions logically follows from it. This involves understanding the relationship between the statement and each potential conclusion.

Understanding the Given Statement

The statement is:

"All hardworking worker in the factory have been promoted in the last five year"

This statement tells us something specific about a group of workers: the ones who are hardworking and are in the factory. It says that every single worker within this group received a promotion within the specified time frame (the last five years).

Let's break it down:

• Subject: Hardworking workers in the factory
• Predicate: Have been promoted in the last five years

The statement essentially means that the set of "hardworking workers in the factory" is completely contained within the set of "workers promoted in the last five years".

Evaluating Each Conclusion Option

Now, let's look at each conclusion provided and see if it must be true based on the original statement.

Option 1: Some worker who were promoted in the last five year are hardworking

This statement says that among the group of workers who were promoted in the last five years, some of them are hardworking workers. Based on our original statement, we know that *all* hardworking workers were promoted. If all hardworking workers were promoted, then the group of promoted workers *must* include all the hardworking workers. Therefore, it must be true that some of the promoted workers are hardworking (specifically, all the hardworking ones). This conclusion logically follows.

Option 2: There are no worker in the factory who are not hardworking

This statement claims that every single worker in the factory is hardworking. Our original statement only gives information about what happened to the *hardworking* workers (they got promoted). It says nothing about the workers who are *not* hardworking. There could be workers in the factory who are not hardworking. They might not have been promoted, or perhaps they were promoted for other reasons. The original statement gives us no information to confirm or deny this conclusion. Therefore, this conclusion does not logically follow.

Option 3: Hardworking is the only criteria for promotion

The original statement says that *all* hardworking workers were promoted. It does *not* say that *only* hardworking workers were promoted. It's possible that there were other criteria for promotion besides being hardworking, such as seniority, specific skills, or performance reviews. Workers who were not hardworking could potentially have been promoted based on these other criteria. The statement doesn't rule this out. Therefore, this conclusion does not logically follow.

Option 4: After promotion, some worker stopped working hard

The original statement deals with the promotion of hardworking workers over the last five years. It describes a relationship that existed *up to* the point of promotion. It provides no information whatsoever about the behavior or work ethic of workers *after* they have been promoted. Whether promoted workers continue to be hardworking or stop being hardworking is outside the scope of the original statement. Therefore, this conclusion does not logically follow.

Conclusion that Logically Follows

Based on the analysis of each option, only the first conclusion is necessarily true if the original statement is true.

Therefore, the conclusion that logically follows from the statement "All hardworking worker in the factory have been promoted in the last five year" is that some workers who were promoted in the last five years are hardworking.

Revision Table: Key Logical Concepts

Additional Information: Understanding Conversions in Logic

In logic, a conversion is a type of immediate inference where the subject and predicate of a proposition are interchanged. However, not all conversions are logically valid for all types of statements.

• For a universal affirmative statement ("All A are B"), the simple conversion ("All B are A") is *not* valid. For example, from "All dogs are animals," we cannot logically conclude "All animals are dogs."
• The valid conversion for "All A are B" is the conversion by limitation (or per accidens), which is "Some B are A". If every single member of group A is in group B, then looking at group B, at least some members of group B must be from group A (specifically, all of group A). This is valid assuming group A is not empty, which is usually a reasonable assumption in these types of problems unless stated otherwise.

In our question, the statement "All hardworking workers (A) have been promoted (B)" is a universal affirmative. Option 1, "Some promoted workers (B) are hardworking workers (A)", is the valid conversion by limitation. This is why it logically follows.

Understanding these basic logical rules helps in solving problems that require drawing conclusions from given statements.