{"id":265952,"date":"2026-07-17T18:07:35","date_gmt":"2026-07-17T15:07:35","guid":{"rendered":"https:\/\/1kitap1.com\/en\/logic-and-philosophy-a-modern-introduction-alan-hausman\/"},"modified":"2026-07-17T18:07:35","modified_gmt":"2026-07-17T15:07:35","slug":"logic-and-philosophy-a-modern-introduction-alan-hausman","status":"publish","type":"post","link":"https:\/\/1kitap1.com\/en\/logic-and-philosophy-a-modern-introduction-alan-hausman\/","title":{"rendered":"Logic And Philosophy A Modern Introduction – Alan Hausman"},"content":{"rendered":"
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Then the sentence \u201cAll who study will pass the test\u201d is false (since some who study do not pass), whereas the professor\u2019s sen- tence, \u201cOnly those who study will pass the test\u201d turns out to be true. Hence, the symbol- ization (x)(Sx \u0001 Px) cannot be the correct symbolization of the professor\u2019s statement. Another way to look at the problem is as follows. To say that only those who study will pass is to say that anyone who does not study will not pass.<\/p>\n

Because the sentence \u201cAll who do not study will not pass the test\u201d is correctly symbolized as (x)(\u0002 Sx \u0001 \u0002 Px), the equivalent sentence (1) \u201cOnly those who study will pass the test\u201d also is correctly sym- bolized in this way. And since (x)(\u0002 Sx \u0001 \u0002 Px) is equivalent to (x)(Px \u0001 Sx), as will be proved in later chapters, it follows that (1) is correctly symbolized as (x)(Px \u0001 Sx).<\/p>\n

Sentences containing the phrase \u201cnone but\u201d are handled in a similar fashion. Thus, the sentences \u201cOnly the good die young\u201d and \u201cNone but the good die young\u201d are equivalent and can each be symbolized as (x)(Yx \u0001 Gx). The phrase \u201cnone but\u201d in a sentence of this kind means roughly the same thing as the phrase \u201cnone except.\u201d For instance, the sentence \u201cNone but the ignorant are happy\u201d means the same thing as the sentence \u201cNone, except the ignorant, are happy.\u201d<\/p>\n

Therefore, both of these sentences are symbolized as (x)(Hx \u0001 Ix). English usage allows us to use the word \u201cunless\u201d to produce sentences that are equiv- alent to the ones we\u2019ve just been discussing. For instance, instead of saying \u201cOnly those who study will pass\u201d or \u201cNone but those who study will pass,\u201d we can say, \u201cNo one will pass unless they study.\u201d<\/p>\n

(We use \u201cthey\u201d here to avoid the masculine pronoun \u201che.\u201d Some object to this usage as ungrammatical. An alternative is \u201cNo one will pass unless he or she studies.\u201d) \u201cWithout\u201d sometimes performs a similar function in English, as in \u201cNo one will pass without studying.\u201d All these sentences say roughly the same thing, and hence for our purposes can be symbolized in the same way\u2014namely, (x)(Px \u0001 Sx).<\/p>\n

Remember that \u201cunless\u201d also often functions as a truth-functional connective, as explained in Chapter Two. So a sentence such as \u201cNo one will get dessert unless everyone quiets down\u201d should be symbolized as \u0002 (x)(Px \u0001 Qx) \u0001 \u0002 (\u2203x)(Px \u22c5Dx). The word \u201cif\u201d has a similar dual usage.<\/p>\n

It can function as a truth-functional connective as in \u201cIf everyone in the class studies, no one in the class will fail\u201d symbolized as (x)(Cx \u0001 Sx) \u0001 \u0002 (\u2203x)(Cx \u22c5Fx)\u2014and can also be used to make generalizations, as in \u201cIf H2O freezes, it expands\u201d\u2014symbolized as (x)[(Hx \u22c5Fx) \u0001 Ex)]. Examples 1. Only celebrities can be elected president.<\/p>\n

(x)(Ex \u0001 Cx) or \u0002 (\u2203x)(Ex \u22c5\u0002 Cx) 2.<\/p>\n

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(u)(. . . u . . .) :: ~ (\u2203u) ~ (. . . u . . .) (\u2203u)(. . . u . . .) :: ~ (u) ~ (. . . u . . .) (u) ~ (. . . u . . .) :: ~ (\u2203u)(. . . u . . .) (\u2203u) ~ (. . . u . . .) :: ~ (u)(. .<\/p>\n

. u . . .) Rule ID: (. . . u . . .) (. . . u . . .) u = w \/\u2234(. . . w . . .) w = u \/\u2234(. . . w . . .) Rule IR: \/\u2234(x)(x = x) Provided: 1. (. . . w . . .) results from replacing each occur- rence of u free in (. . . u . . .) with a w that is either a constant or a variable free in (.<\/p>\n

. . w . . .) (making no other changes). Provided: 1. w is not a constant. 2. w does not occur free previously in the proof. 3. (. . . w . . .) results from replacing each occur- rence of u free in (. . . u . . .) with a w that is free in (.<\/p>\n

. . w . . .) (making no other changes). Provided: 1. u is not a constant. 2. u does not occur free previously in a line obtained by EI. 3. u does not occur free previously in an assumed premise that has not yet been discharged. 4. (. . . w .<\/p>\n

. .) results from replacing each occur- rence of u free in (. . . u . . .) with a w that is free in (. . . w . . .) (making no other changes) and there are no additional free occurrences of w already contained in (. . . w . . .). Provided: 1. (. . . w . . .) results from replacing at least one occurrence of u, where u is a constant or a vari- able free in (.<\/p>\n

. . u . . .) with a w that is free in (. . . w . . .) (making no other changes) and there are no additional free occurrences of w already contained in (. . . w . . .).<\/p>\n<\/blockquote>\n

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.<\/em><\/p>\n

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