A Profile of Mathematical Logic by Howard DeLong

Author:Howard DeLong
Language: eng
Format: epub, pdf
ISBN: 9780486139159
Publisher: Dover Publications
Published: 2012-10-23T04:00:00+00:00

then second

(∀z)(((1 < 1) · (1 < z)) ⊃ (1 < z))·

(∀z)(((1 < 2) · (2 < z)) ⊃ (1 < z))·

(∀z)(((2 < 1) · (1 < z)) ⊃ (2 < z))·

(∀z)(((2 < 2) . (2 < z)) ⊃ (2 < z)),

and finally,

(((1 < 1) · (1 < 1)) ⊃ (1 < 1))·

(((1 < 1) · (1 < 2)) ⊃ (1 < 2))·

(((1 < 2) · (2 < 1)) ⊃ (1 < 1))·

(((1 < 2) · (2 < 2)) ⊃ (1 < 2))·

(((2 < 1) · (1 < 1)) ⊃ (2 < 1))·

(((2 < 1) · (1 < 2)) ⊃ (2 < 2))·

(((2 < 2) · (2 < 1)) ⊃ (2 < 1))·

(((2 < 2) · (2 < 2)) ⊃ (2 < 2))·

Now this formula is false under the interpretation of ‘ < ’ as less than or greater than (cf. the sixth conjunct). Hence it is inconsistent with (a) and (b). This example has been written out at length to make plausible the impossibility of satisfying (a), (b), and (c) in a finite domain: Formula (a) guarantees that an individual doesn’t have the relation to itself, formula (b) that for any individual there always is another individual with that relation to it, and formula (c) rules out the possibility of a “circle” (as above in our “less than or greater than” interpretation).

Now if we conjoin (a), (b), and (c) in a single formula, we get

d) ((∀x)(x < x) · (∀x)(∀y)(x < y).

(∀x)(∀y)(∀z)(((x < y) · (y < z)) ⊃ (x < z))).

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