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Lesson_Nine_and_Test

# Lesson_Nine_and_Test - LESSON 9 Quantificational Logic Part...

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Section 18: Symbolism of Quantification , Barker, pp. 110-115. ***Read the section, then the following additional comments. In this section we learn to combine what we learned in the earlier part of the course, categorical logic, with the subject matter that has occupied us for the last little while, truth-functional logic. From truth-functional logic we continue to utilize a logical system based on statements in various relationships. From categorical logic we adopt the use of quantifiers. We call this new system quantificational logic. The kind of statements we use in quantificational logic are more complex than the ones we worked with in truth-functional logic. The statements involve identifying a subject and a particular function or property which the subject has. Jane has a ball. The frog is green. Water boils at 100 degrees. become Having a ball—attributed to Jane. Being green—attributed to the frog. Boiling at 100 degrees—attributed to water. To shorten our expressions even further, we use capital letters to express the function, small letters for the subject: Lj Gf Bw These small letters are called constants or names. They refer to specifically identified entities. The quantifiers help us translate less definite statements. Something is wrong. Everything must come to an end. Someone stole Jane’s ball. Now we cannot supply a name since there is no specific entity referred to. Instead we need to come back to our old friends, “all” and “some,” and use them as quantifiers. We still use small 1 LESSON 9 Quantificational Logic, Part 1

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letters as subjects of the functions, but now we have to consider them as variables, viz. as open-ended general collectors of unspecified content or magnitude. Think of variables as containers, empty buckets perhaps. To what extent they are filled—and with what—remains to be seen. Let us slowly work through a translation of the first sample sentence. We go from Something is wrong. to There is something such that this something is wrong. to There is some x, such that x is wrong. to For some x, Wx to ( 5 x)Wx The symbol, ( 5 x), is called the existential quantifier, and it translates as “some” or a cognate expression. Be sure to look carefully at the chapter in the text to catch all of the slight variations on the translation of the existential quantifier.
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Lesson_Nine_and_Test - LESSON 9 Quantificational Logic Part...

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