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**Formal Criteria for Evaluating Arguments**

Validity and Soundness

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**The Form Content Distinction**

Some students are athletes. Some athletes are political. Some students are political. 1. Some young adults are in high school. Some high school students are in the 10th grade. Some young adults are in the 10th grade. Both arguments have the following form: Some A are B. Some B are C. Some A are C. However, the arguments differ in content.

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**Formal Evaluation of Arguments**

The informal evaluation of an argument has to do with determining whether the argument commits an informal fallacy, which is an error in reasoning that has to do with the arguments form, content, and context. In general, an informal fallacy cannot be determined solely in virtue of the form of an argument. The formal evaluation of an argument has to do solely with the form of the argument, and not with the content or context of the argument. While there are disagreements as to what the form of a particular argument is. Once the form has been determined and the system for evaluation determined, assessing the argument requires using techniques of formal logic.

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Validity The central criteria for assessing arguments formally is validity. An argument form is valid when it is impossible for the premises to be true and the conclusion false. An argument form is invalid when it is possible for the premises to be true and the conclusion false. It is impossible for an argument form to have true premises and a false conclusion just in case there is no instance of the form in which there are true premises and a false conclusion. It is possible for an argument form to have true premises and a false conclusion just in case there is an instance in which the premises are true and the conclusion is false.

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**Invalidity Invalid argument with true premises and a true conclusion.**

Some politicians are conservative. (true) Some conservatives are against public health care. (true) Some politicians are against public health care. (true) NOT AN INVALIDATING CASE Invalid argument with true premises and a false conclusion. Some even numbers are whole numbers. (true) Some whole numbers are odd numbers. (true) Some even numbers are odd numbers. (false) INVALIDATING CASE

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**Variations on Validity**

Since validity is about form, and not content, there are various combinations of true premises, false premises, true conclusion, and false conclusion that can actually occur in a valid or invalid argument. Premises Conclusion Validity All True True Valid or Invalid Some True, Some False False Only Invalid

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**Validity and the Counterexample Method**

When thinking about validity think about the form of the argument, not the actual premises and conclusion of the argument. Ask yourself the question: If the premises were true, could the conclusion be false? If your intuition is that this could occur, try to come up with an example in which the premises are true and the conclusion is false. If the example you come up with is coherent, then you have a counterexample that proves the invalidity of the form of the argument.

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**An Example of the Counterexample Method**

Original Argument: All democrats are people that might vote. All liberals are people that might vote. All democrats are liberals. Sounds okay, maybe you don’t know if all the premises are true. But it looks safe. Now think about the form, and delete the content. What form of reasoning is the person using? Counterexample: All A are B. All C are B. All A are C. Now put in new terms to make true premises and a false conclusion. A = even numbers B = whole numbers C = odd numbers

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Soundness Although formal evaluation of arguments has to do primarily with the form of the arguments, there are occasions when we need to pay attention to the content as well. The occasions on which this is true, is when we are assessing soundness. An argument is sound when it has a valid form and true premises. From the definition of soundness it follows that all sound arguments have a true conclusion. If the form of the argument is such that it is impossible for the premises to be true and the conclusion false, because the form is a valid form, and then we add actually true premises, it has to be the case that the conclusion is also true.

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**Variations on Soundness**

Since soundness includes validity, and validity is about the form of an argument, there are variations of how these two ideas can interact. Basically, if an argument is not valid, it cannot be sound. Validity Soundness Yes No

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**Two Cases to Keep in Mind**

True premises, True conclusion, invalid argument, and unsound. Some students are philosophy majors. Some philosophy majors are double majoring in political science. Some students are double majoring in political science Not a good argument because it is unsound because it is invalid. False premise, False conclusion, valid argument, and unsound. All cats are dogs. All dogs are rats. All cats are rats. Valid Form: All A are B. All B are C. All A are C.

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Goodness ≠ Soundness Many people often confuse the goodness of an argument with its soundness. One cannot infer from the validity or soundness of an argument that it is good. Anand is the teacher of this class. So, Anand is the teacher of this class. The argument is valid, since its form P, therefore P is valid, because one and the same statement P cannot be both true and false. The argument is sound, since it is true that Anand is the teacher of this class. However, the argument is clearly not good. It is circular.

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**Formal Evaluation by Formal Methods**

Although it is possible to evaluate the form of an argument by thinking about it and trying to come up with a counterexample, the method has limitations. It does not follow from the fact that Adam cannot come up with a counterexample to argument form A, that A is valid. The failure to find a counterexample to an argument does not prove that it is valid. (sounds like appeal to ignorance –can’t find it so it doesn’t exist, therefore valid.) Thus, we need a formal method involving abstract reasoning and methods of proof to test for validity.

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Translation One of the best ways to test to see whether an argument is valid is to translate it into another language. Just as one can translate English into French, most arguments which occur in natural languages such as English, can be translated into a formal language, such as the language of propositional logic, categorical logic, predicate logic, modal logic, or tense logic. Our goal is to learn to translate arguments into the language of propositional logic for the purposes of applying some formal system of proof for checking validity.

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**Checking for Validity Steps from here:**

Learn the syntax and semantics of propositional logic. Learn to translate propositional arguments into the language of propositional logic. Learn how to use a truth-table to check for various properties of statements and groups of statements. Learn how to construct a truth-tree to check for various properties of statements and groups of statements. Learn how to construct a natural deduction proof for propositional arguments. Learn how to use conditional and indirect proof techniques.

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10/17/2015 Prepared by Dr.Saad Alabbad1 CS100 : Discrete Structures Proof Techniques(1) Dr.Saad Alabbad Department of Computer Science

10/17/2015 Prepared by Dr.Saad Alabbad1 CS100 : Discrete Structures Proof Techniques(1) Dr.Saad Alabbad Department of Computer Science

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