Category Archives: Mathematical Philosophy

What the Tortoise said to Achilles shows the nature of implications.

The reason Tortoise couldn’t stop hypothesizing is that an implication never asserts its constituent propositions. Given: A. P is true. B. if P is true then Q is true, (P implies Q). B only asserts the implication.  Even if both … Continue reading

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Symbols in Principia Mathematica

“(∃x)φx” — “There exists an x for which φx is true,” or “there exists an x satisfying φx” page 16. 1st ed. “(x)φx” — “the x which satisfies .” Page 31. 1st ed. “E!(x)φx”  — “the x satisfying φx exists”. Page … Continue reading

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It is meaningless to say a set is or is not a member of itself.

A defining function can be used to determine a set whose members are arguments that satisfy . Let denote the set determined by . To say is or is not a member of itself is to use as an argument … Continue reading

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An example of vicious-circle fallacy in proving identity

Define ” is identical with ” as meaning “whatever is true of is true of ,” i.e. ” always implies .”            (1) Dem. If is identical with and ” is identical with , then is identical with a. Although the … Continue reading

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Dem. of ❋5.14 from 1st ed. of Whitehead & Russell’s PM written out in full

❋5.14 ⊦:. p ⊃ q .v. q ⊃ r Dem. ⊦. [Simp] ⊦: p . ~q .⊃. ~q         (1) ⊦. [❋2.21] ⊦: ~q .⊃. q ⊃ r         (2) ⊦. [(1).(2).Syll] ⊃⊦: p.~q .⊃. q ⊃ r : [Transp] … Continue reading

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Dem. of ❋4.86 from 1st ed. of Whitehead & Russell’s PM written out in full

Courtesy of Albert Steppi who provided answer from stackexchange.com. ❋4.86 ⊦:. p ≡ q .⊃: p ≡ r .≡. q ≡ r Dem. ⊦. [❋4.22]   ⊃⊦: p ≡ q . q ≡ r .⊃. p ≡ r        (1) ⊦. [❋4.22]  … Continue reading

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Dem. of ❋4.77 from 1st ed. of Whitehead & Russell’s PM written out in full

❋4.77 ⊦:. q ⊃ p . r ⊃ p .≡: q v r .⊃. p   Dem. ⊦. [ ❋3.44]   ⊦:. q ⊃ p . r ⊃ p .⊃: q v r .⊃. p         (1) ⊦. [Add]   ⊦. r ⊃ … Continue reading

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