In a syllogism, the conclusion is a statement that is derived from the premises using the rules of logical inference.

A categorical proposition is a statement that makes a general claim about a category of things. In the example given, the statement "All dogs are mammals" makes a general claim about the category of dogs.

In a categorical proposition, the subject term refers to the category of things that the proposition is making a claim about.

In a categorical proposition, the predicate term refers to the property or characteristic that is being attributed to the subject term.

A universal affirmative proposition is a statement that affirms that all members of a category have a certain property. In the example given, the statement "All dogs are mammals" affirms that all members of the category of dogs have the property of being mammals.

A universal negative proposition is a statement that denies that any members of a category have a certain property. In the example given, the statement "No cats are fish" denies that any members of the category of cats have the property of being fish.

A particular affirmative proposition is a statement that affirms that some members of a category have a certain property. In the example given, the statement "Some birds can fly" affirms that some members of the category of birds have the property of being able to fly.

A particular negative proposition is a statement that denies that some members of a category have a certain property. In the example given, the statement "No cats are fish" denies that some members of the category of cats have the property of being fish.

In a syllogism, the major premise is the premise that contains the major term, which is the term that appears in the conclusion.

In a syllogism, the minor premise is the premise that contains the minor term, which is the term that appears in the conclusion.

In a syllogism, the middle term is the term that appears in both premises but not in the conclusion.

A valid syllogism is a syllogism in which the conclusion follows logically from the premises. In the example given, the conclusion "Socrates is mortal" follows logically from the premises "All men are mortal" and "Socrates is a man".

An invalid syllogism is a syllogism in which the conclusion does not follow logically from the premises. In the example given, the conclusion "All dogs are carnivores" does not follow logically from the premises "All dogs are mammals" and "Some mammals are carnivores".

A sound syllogism is a syllogism that is both valid and has true premises. In the example given, the syllogism "All men are mortal. Socrates is a man. Therefore, Socrates is mortal." is both valid and has true premises, so it is a sound syllogism.

An unsound syllogism is a syllogism that is either invalid or has false premises. In the example given, the syllogism "All dogs are mammals. Some mammals are carnivores. Therefore, all dogs are carnivores." is invalid, so it is an unsound syllogism.