EQUIVALENCES and ARGUMENT FORMS and VALIDITY (Updated, 11-10-23) [GROUP 7 REPORT].pptx

COSC 50 Group Report, CvSU Bacoor Campus (GROUP 7)

1. EQUIVALENCES and ARGUMENT FORMS and VALIDITY Presented by: group 7

2. What is Equivalence??? It is a relationship between two statements that have the same truth value. In other words, two statements are equivalent if they are both true or both false.

3. REMEMBER! The main point regarding a valid argument is that it follows from the logical form itself and has nothing to do with the content. When a conclusion is reached using a valid argument, we say the conclusion is inferred or deduced from the premises.

4. WHAT IS MATERIAL EQUIVALENCE? It is a logical relationship between two propositions that have the same truth value for every assignment of truth values to the atomic propositions. In other words, two propositions are materially equivalent if they have the same truth value on every row of a truth table. Material equivalence is also known as biconditional , and it is represented by the symbol "≡".

5. WHAT IS LOGICAL EQUIVALENCE? It is different from material equivalence, although the two concepts are related. Logical equivalence is a relationship between two logical statements that have the same truth value in every model. Two statements are logically equivalent if they always produce the same truth value .

6. REMEMBER! An argument is valid if and only if in every case where all the premises are true, the conclusion is true. Otherwise, the argument is invalid. To determine the validity of an argument, we can use truth tables or recognize certain common forms of arguments that are valid or invalid.

7. WHAT IS A TRUTH TABLE?

8. TRUTH TABLE It is a tabular representation used in logic and mathematics to list all possible combinations of truth values (usually "true" or "false") for a given logical expression, Boolean function, or set of logical propositions. It provides a systematic way to illustrate how the truth values of the variables involved in a logical expression or function affect the overall truth value of that expression.

9. A typical truth table consists of the following elements: Variables: The variables involved in the logical expression are listed as columns at the top of the table. Each column represents a different variable, and the variables can take on one of two possible truth values: "true" (T) or "false" (F). Rows: Each row of the truth table represents a unique combination of truth values for the variables. The number of rows in the table is determined by the number of variables, with 2^n rows for n variables. Logical Expression/Function Column: The rightmost column in the truth table is dedicated to the logical expression or Boolean function that you are analyzing. It shows the resulting truth value of the expression for each combination of input values. Evaluation: To fill in the truth values in the logical expression column, you evaluate the expression using the logical operators (e.g., AND, OR, NOT) based on the truth values of the variables in each row.

10. Here's a simple example for a truth table with two variables (A and B) and an AND operation: In this example, you can see all possible combinations of truth values for A and B, and the resulting truth values for the expression A AND B. You can construct truth tables for more complex expressions by following the same principles.

11. Argument Forms: Proving Invalidity Constructing proofs is an effective way to demonstrate that an argument of the propositional calculus is valid. If an argument happens to be invalid, of course, it would be impossible to construct a proof of its validity. But that's not a very good method of spotting invalid arguments, since our inability to devise an appropriate proof on a particular afternoon is as likely to result from our own limitations as from the general impossibility of doing so.

12. Proving Invalidity Of course, we could always fall back on truth-tables as a method of proving invalidity. We simply inspect the truth-table columns for all of the premises and the conclusion; if there is any line on which all of the premises are true while the conclusion is false, then the argument is invalid (and if not, it is valid). In this sense, truth-tables are a decision procedure: in a finite number of steps, they will provide us with evidence of the validity or invalidity of any argument. But that finite number of steps can become extremely large. (With seven propositional variables and five premises, we would have to fill in 1,664 individual truth-values.) Fortunately, we can often short-cut the process significantly when we suspect that an argument may be invalid. Remember, it only takes one line with true premises and a false conclusion to establish the invalidity of an invalid inference. So we don't really need to look at every line of the argument's truth-table; we can concentrate on our effort to find just the right one. Consider, for example, the following argument:

13. Geese are migratory waterfowl, so they fly south for the winter. This argument is missing a premise: Migratory waterfowl fly south for the winter. The argument can now be rephrased to make its form apparent: All geese are migratory waterfowl. All migratory waterfowl are birds that fly south for the winter. Therefore, all geese are birds that fly south for the winter. The form of the argument is All A are B. All B are C. All A are C. Consider the following argument: This form is valid, and it captures the reasoning process of the argument. If we assume that the As (whatever they might be) are included in the Bs, and that the Bs (whatever they might be) are included in the Cs, then the As must necessarily be included in the Cs. This necessary relationship between the As, Bs, and Cs is what makes the argument valid. This is what we mean when we say that the validity of a deductive argument is determined by its form. Since validity is determined by form, it follows that any argument that has this valid form is a valid argument. Th us, we might substitute “daisies” for A, “flowers” for B, and “plants” for C and obtain the following valid argument: All daisies are flowers. All flowers are plants. Therefore, all daisies are plants. Any argument such as this that is produced by uniformly substituting terms or statements in place of the letters in an argument form is called a substitution instance

14. Proving Invalidity This diagram suggests that we can prove the form invalid if we can find a substitution instance having actually true premises and an actually false conclusion. In such a substitution instance the As and the Cs would be separated from each other, but they would both be included in the Bs. If we substitute “cats” for A, “animals” for B, and “dogs” for C, we have such a substitution instance:

15. What is the Counterexample method? - SIMPLIFIED DEFINITION It is a powerful way of exposing what is wrong with an argument that is invalid. If we want to proceed methodically, there are two steps: 1) Isolate the argument form 2) Construct an argument with the same form that is obviously invalid. - FLUENT DEFINITION It consists of isolating the form of an argument and then constructing a substitution instance having true premises and a false conclusion. This proves the form invalid, which in turn proves the argument invalid.

16. Counterexample method The counterexample method can be used to prove the invalidity of any invalid argument, but it cannot prove the validity of any valid argument. Thus, before the method is applied to an argument, the argument must be known or suspected to be invalid in the first place. Let us apply the counterexample method to the following invalid categorical syllogism: This argument is invalid because the employees who are not social climbers might not be vice presidents. Accordingly, we can prove the argument invalid by constructing a substitution instance having true premises and a false conclusion. We begin by isolating the form of the argument: Next, we select three terms to substitute in place of the letters that will make the premises true and the conclusion false. The following selection will work:

17. Counterexample method The substitution instance has true premises and a false conclusion and is therefore, by definition, invalid. Because the substitution instance is invalid, the form is invalid, and therefore the original argument is invalid. In applying the counterexample method to categorical syllogisms, it is useful to keep in mind the following set of terms: “cats,” “dogs,” “mammals,” “fish,” and “animals.” Most invalid syllogisms can be proven invalid by strategically selecting 1 three of these terms and using them to construct a counterexample. Because everyone agrees about these terms, everyone will agree about the truth or falsity of the premises and conclusion of the counterexample. Also, in constructing the counterexample, it often helps to begin with the conclusion. First, select two terms that yield a false conclusion, and then select a third term that yields true premises. Another point to keep in mind is that the word “some” in logic always means “at least one.” For example, the statement “Some dogs are animals” means “At least one dog is an animal”—which is true. Also note that this statement does not imply that some dogs are not animals. (Continuation of previous slide)

18. Counterexample method Being able to identify the form of an argument with ease requires a familiarity with the basic deductive argument forms. The first task consists in distinguishing the premises from the conclusion. Always write the premises first and the conclusion last. The second task involves distinguishing what we may call “form words” from “content words.” To reduce an argument to its form, leave the form words as they are, and replace the content words with letters. For categorical syllogisms, the words “all,” “no,” “some,” “are,” and “not” are form words, and for hypothetical syllogisms the words “if,” “then,” and “not” are form words. Additional form words for other types of arguments are “either,” “or,” “both,” and “and.” For various kinds of hybrid arguments, a more intuitive approach may be needed.

19. Thank you for your attention in our group report!!! - Group 7