With regard to exercise 3-6 (assigned for today) and 3-7 and 3-8 (assigned for Friday)… Notice that your textbook is constructing tables using sentence forms rather than sentences. I will continue the practice I began on Monday, using sentences (e.g., A Ú B) rather than sentence forms (e.g., p Ú q). Please do the same in your exercises and on the test. Also please remember to construct the guide columns in your truth tables in the order in which the sentence constants occur in the sentences you are given. For example, if “A” and “B” both occur in your sentence but “B” occurs first before any occurrence of “A,” your guide column for “B” should be to the left of your guide column for “A.”
[3.4.] “Logical Equivalences.”
In this section we will construct truth tables that specify the truth values of more than one compound sentence at a time. The task will be to determine whether two sentences are logically equivalent:
logically equivalent (df.) two sentences are logically equivalent if and only if it is impossible for them to have different truth values.
For example, "A · B" and "B · A" are logically equivalent. If one is true, so is the other; and if one is false, so is the other.
We can determine whether two sentences are logically equivalent using the tabular truth table method: two sentences are logically equivalent if and only if they have the same truth values on every line of a truth table.
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EXAMPLES: p.71
Exercise 3-7 (p.72)
§ [do one or two even-numbered in class]
§ complete all of these for next time; check your even answers and we’ll look at the odds at the beginning of the next class [Watch out for #7, it’s especially tricky]
[3.5.] “Truth Table Test of Validity.”
We will now apply the tabular truth table method in a different way: to determine whether arguments are valid or invalid.
Recall these definitions from the first week of class:
argument (df.): a set of statements some of which (the premises) are intended to serve as evidence or reasons for thinking that another statement (the conclusion) is true. In the words of the textbook: “A series of sentences, one of which (the conclusion) is claimed to be supported by the others (the premises).” (p.16)
validity (df.): A valid argument is one in which
1. the truth of the premises would guarantee the truth of the conclusion;
2. it is impossible for the premises to be all true and the conclusion to be false at the same time;
3. if the premises were true, then the conclusion would have to be true as well.
(These are three ways of defining validity. They all mean the same thing.)
Applied to arguments in sentential logic, this definition of validity means that an argument is valid if and only if there is no interpretation on which all its premises are true and its conclusion false.
So, we can tell whether an argument in sentential logic, such as
A, ~A Ú B, /\B
is valid or invalid by constructing a truth table and checking for an interpretation on which all premises are true and the conclusion is false:
§ if there is such an interpretation, the argument is invalid
§ if there is not such an interpretation, the argument is valid
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There is only one row (interpretation) in which both premises ("A" and "~A Ú B") are true. On that row (interpretation), the conclusion ("B") is also true. So this argument is valid.
Exercise 3-8 (pp.74-75)
§ complete this exercise for next time; check your answers to the even-numbered questions; we will review the odd-numbered problems at the beginning of class next time.
§ Make sure you indicate which interpretations show the argument to be valid or invalid
§ This is the most time-consuming set of problems you’ve been assigned up to this point
***This is the end of the material that will be covered on your first test. The remainder of today’s lecture notes will not be covered on the first test. It is background material you will need to master in order to begin constructing proofs, which we will cover beginning in the next chapter.
Stopping point for Wednesday January 30. For next time, complete exercises 3-7 and 3-8 and begin studying for your test (which is Monday). Come to class with any questions you have about the material we’ve covered so far.