Logic And Philosophy A Modern Introduction – Alan Hausman

Then the sentence “All who study will pass the test” is false (since some who study do not pass), whereas the professor’s sen- tence, “Only those who study will pass the test” turns out to be true. Hence, the symbol- ization (x)(Sx
Px) cannot be the correct symbolization of the professor’s 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.

Because the sentence “All who do not study will not pass the test” is correctly symbolized as (x)( Sx
 Px), the equivalent sentence (1) “Only those who study will pass the test” also is correctly sym- bolized in this way. And since (x)( Sx
 Px) is equivalent to (x)(Px
Sx), as will be proved in later chapters, it follows that (1) is correctly symbolized as (x)(Px
Sx).

Sentences containing the phrase “none but” are handled in a similar fashion. Thus, the sentences “Only the good die young” and “None but the good die young” are equivalent and can each be symbolized as (x)(Yx
Gx). The phrase “none but” in a sentence of this kind means roughly the same thing as the phrase “none except.” For instance, the sentence “None but the ignorant are happy” means the same thing as the sentence “None, except the ignorant, are happy.”

Therefore, both of these sentences are symbolized as (x)(Hx
Ix). English usage allows us to use the word “unless” to produce sentences that are equiv- alent to the ones we’ve just been discussing. For instance, instead of saying “Only those who study will pass” or “None but those who study will pass,” we can say, “No one will pass unless they study.”

(We use “they” here to avoid the masculine pronoun “he.” Some object to this usage as ungrammatical. An alternative is “No one will pass unless he or she studies.”) “Without” sometimes performs a similar function in English, as in “No one will pass without studying.” All these sentences say roughly the same thing, and hence for our purposes can be symbolized in the same way—namely, (x)(Px
Sx).

Remember that “unless” also often functions as a truth-functional connective, as explained in Chapter Two. So a sentence such as “No one will get dessert unless everyone quiets down” should be symbolized as  (x)(Px
Qx)
 (∃x)(Px ⋅Dx). The word “if” has a similar dual usage.

It can function as a truth-functional connective as in “If everyone in the class studies, no one in the class will fail” symbolized as (x)(Cx
Sx)
 (∃x)(Cx ⋅Fx)—and can also be used to make generalizations, as in “If H2O freezes, it expands”—symbolized as (x)[(Hx ⋅Fx)
Ex)]. Examples 1. Only celebrities can be elected president.

(x)(Ex
Cx) or  (∃x)(Ex ⋅ Cx) 2.

(u)(. . . u . . .) :: ~ (∃u) ~ (. . . u . . .) (∃u)(. . . u . . .) :: ~ (u) ~ (. . . u . . .) (u) ~ (. . . u . . .) :: ~ (∃u)(. . . u . . .) (∃u) ~ (. . . u . . .) :: ~ (u)(. .

. u . . .) Rule ID: (. . . u . . .) (. . . u . . .) u = w /∴(. . . w . . .) w = u /∴(. . . w . . .) Rule IR: /∴(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 (.

. . 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 (.

. . 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 .

. .) 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 (.

. . 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 . . .).

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.

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