Determine and prove whether an argument in English is valid or invalid. About Prove whether each argument is valid or invalid. First find the form of the argument by defining predicates and expressing the hypotheses and the conclusion using the predicates. If the argument is valid, then use the rules of inference to prove that the form is valid. If the argument is invalid, give values for the predicates you defined for a small domain that demonstrate the argument is invalid. The domain for each problem is the set of students in a class.

Since you haven't provided a specific argument in English, I will describe the general approach you would take to determine if an argument is valid or invalid using the method described.

Answer: To determine if an argument in English is valid or invalid, we need to translate it into logical form, using predicates to abstract the specific contents of the argument into a form that we can manipulate logically. Here are the steps you should follow:

1. Identify the argument and break it down into its constituent propositions. 2. Define predicates that represent the general form of these propositions. A predicate is a function that can return true or false, depending on the input it receives. 3. Express each proposition of the argument in terms of these predicates. 4. Translate the logical structure of the argument into symbolic logic. This will typically involve the use of logical connectives such as "and" ( ∧ ), "or" ( ∨ ), "not" ( ¬ ), "if...then..." ( → ), etc. 5. Use rules of inference to determine if the conclusion logically follows from the premises. Common rules of inference include Modus Ponens, Modus Tollens, Syllogism, etc.

If you can successfully apply rules of inference to show that the conclusion must be true when the premises are true, the argument is valid. If you cannot, the argument may be invalid.

If you believe the argument is invalid, you must demonstrate it by finding a counterexample. This involves providing specific values within the domain (e.g., the set of students in a class) for which the premises are true, but the conclusion is false. If such a counterexample exists, the argument is indeed invalid.

Extra: Validity in logic refers to an attribute of an argument whereby if the premises are true, the conclusion necessarily follows. In other words, a valid argument is one where it is impossible for the premises to be true and the conclusion to be false at the same time.

Invalidity, on the other hand, indicates that there is a possible situation where all the premises can be true while the conclusion is false. Finding this situation, or counterexample, shows that the argument does not hold under all possible circumstances and is therefore invalid.

It is important to note that validity does not concern the actual truth or falsehood of the premises or the conclusion. Even an argument with false premises and a false conclusion can be valid if the conclusion logically follows from the premises. The actual truth is a matter of soundness, not validity. An argument is sound if it is both valid and the premises are actually true.

Understanding and constructing logical arguments is a key element of critical thinking, and these skills are widely applicable in various fields including mathematics, philosophy, law, and computer science. Familiarity with logical notation and rules of inference helps students analyze the structure of arguments and to create clear and cogent arguments of their own.