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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 $P$ and $Q$, and $P$ enters into some true proposition, and the substitution of $Q$ for $P$ wherever it appears results in a new proposition that is likewise true, and if this can be done for every proposition, then $P$ and $Q$ 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 $P$ and $Q$ 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I was chasing for a historical reference on this “principle of substitution of equals” (or what do people call it?)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.

• 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

• Richard Cartwright, Identity and Substitutivity, p. 119-133 of: Milton Munitz (ed.) Identity and Individuation, New York University Press (1971) [pdf]
• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeSep 23rd 2022

• 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.

• 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) [archive.org]

and included a screenshot into the entry (here)

• 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.)

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

I have:

• made salva veritate a redirect to this entry

• 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

(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)

• 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. $A = B \;\Leftrightarrow\; A = B$). What’s a really early record of somebody making the symmetry of the equality-relation explicit?