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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2022

    A late-night edit for entertainment – I’ll try to polish it up tomorrow when I am more awake:

    I have added a translated original quote from Leibniz, as given by Gries & Schneider 1998:

    Two terms are the same (eadem) if one can be substituted for the otherwithout altering the truth of any statement (salva veritate). If we have PP and QQ, and PP enters into some true proposition, and the substitution of QQ for PP wherever it appears results in a new proposition that is likewise true, and if this can be done for every proposition, then PP and QQ are said to be the same;

    Interestingly – and this is what I was searching for – Leibniz ends this paragraph with stating the converse:

    conversely, if PP and QQ are the same, they can be substituted for one another.

    I was chasing for a historical reference on this “principle of substitution of equals” (or what do people call it?) since this is the logical seed of path induction.

    I’d like to find a more canonical reference. But not tonight.

    diff, v13, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 23rd 2022
    • (edited Sep 23rd 2022)

    Sounds like the indiscernibility of identicals. Mike has some remarks here.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 23rd 2022

    Yes, or “principle of substitution” or “substitutivity” seems to be used a lot. I am still looking for a good reference on what Leibniz actually said about this.

    By the way, regarding that comment you point to:

    Strictly speaking, it is “transport” which is “indiscernibility of identicals”, while path induction is the specialization of that to identifications-of-identifications.

    This is why people find the J-rule un-intuitive: The J-rule is really just the intuitively clear Leibniz principle, but specialized to the (evident but) unfamiliar case of identifications-of-identifications!

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 23rd 2022

    added pointer to:

    • Richard Cartwright, Identity and Substitutivity, p. 119-133 of: Milton Munitz (ed.) Identity and Individuation, New York University Press (1971) [pdf]

    diff, v15, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 23rd 2022

    have extracted this page:

    • Clarence I. Lewis, Appendix (p. 373) of: A Survey of Symbolic Logic, University of California (1918) [pdf]

    Interestingly, Leibniz went on to state the “first law of thought” (aka refl). Will add the pointer there, too.

    diff, v16, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 23rd 2022
    • (edited Sep 23rd 2022)

    have dug out the original Latin version, as reproduced in

    • K. Gerhard (ed.), Section XIX, p. 228 in: Die philosophischen Schriften von Gottfried Wilhelm Leibniz, Siebenter Band, Weidmannsche Buchhandlung (1890) []

    and included a screenshot into the entry (here)

    diff, v16, current

    • CommentRowNumber7.
    • CommentAuthorGuest
    • CommentTimeSep 23rd 2022

    wouldn’t the topological version of identity of indiscernibles simply be the identity of indiscernibles after applying the shape modality in a cohesive homomtopy type theory?

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 24th 2022

    Maybe the question in #7 is referring to the section “In topology”? (Notice that this was added in revision 8 by Daniel Luckhardt, March 2017. By the way, I like the idea of that paragraph, but it leaves some room for clarification and maybe examples.)

    The standard shape modality does not capture the point-set-identification/distinction in this paragraph: under shape, all points in one connected component become “identified”.

    Instead, if one wants to bring a modality into Daniel’s paragraph, it would be that of “T0-reflection” (Kolmogorov quotient).

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 24th 2022

    Okay, I have added (here) a paragraph with more clarification on this example of “topological discernibility”.

    (But I am not invested into this subsection, just adding this for completeness. One could probably say much more here in relation to the topological model of intuitionistic logic.)

    diff, v21, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 25th 2022
    • (edited Sep 25th 2022)

    I have:

    • made salva veritate a redirect to this entry

    • added pointer to

      • W. V. O. Quine, §3 of Two Dogmas of Empiricism and Three grades of modal involvement, as reprinted in: Roger F. Gibson (ed.) Quintessence – Basic readings from the philosophy of W. V. Quine, The Belknap Press of Harvard University Press (2004) [ISBN:9780674027558]

      for prominent use of this (Leibniz’s original) term for the substitution principle

    • added pointer also

      (where I found that Quine reference from – interestingly, this and the other Wikipedia entry on Identity of indiscernibles do not currently talk to each other)

    diff, v22, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 25th 2022

    Further on ancient history:

    It looks like Leibniz’s ~1700 text fails to state the symmetry of his coincidentia (i.e. A=BA=BA = B \;\Leftrightarrow\; A = B). What’s a really early record of somebody making the symmetry of the equality-relation explicit?

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