# How does Schröder explain the apparent oddity of ❋5.11.12.13.14 in 1st ed of Whitehead and Russell's PM?

The footnote refers to Schröder's work. I'd appreciate if someone can explain Schroder's insights and spare me some hard reading.

• What exactly are you asking? It's not clear what you're unclear on. Jan 11 '14 at 23:38
• The foot note said Schroder explained the apparent oddity of the above propositions. I'd like to learn Schroder's insights but I don't want to delve into his volumes yet. Jan 11 '14 at 23:52
• Do you agree that there's an oddity? Why did the note interest you? Jan 14 '14 at 21:51
• @Kevin, That is a very good question. I don't have the calibre to recognize the oddity. But the whole book is about making a big deal out of a picayune. As a matter of fact, my knowledge of modern math logic is an interference and does not help me to capture the core essence of this book. Jan 14 '14 at 21:58

## 1 Answer

Schröder says in his book that this formula looks paradox on the first glance since there is a conclusion $r$ on the right side while $r$ does not occur on the left side of the formula. The oddity is that the premise on the left side knows nothing about the statement $r$.

More literally: Very paradoxical at first glance the universal pair of formulas of propositional calculus:

• $(a ⊆ b) ⊆ (c ⊆ a)$
• $(a ⊆ b) ⊆ (b ⊆ c)$

Paradox because the statement $c$, over which the right conclusion reported in the left premise does not occur, it can thus contain no information on that statement, while is nonetheless entitled to conclude from the statement left-hand to refer to the right hand "deductive". (http://www.deutschestextarchiv.de/book/view/schroeder_logik0201_1891?p=294)

• @George Chen - very interesting Marc's historical research. Of course, the "paradoxical" law $\vdash (p \supset q) \lor (q \supset r)$ is only one of (many) "unnatural" side-effects of truth-functional conditional. In classical logic, being $q$ true or false, one of the two disjuncts will surely be true (the LHS when $q$ is true, the RHS when $q$ is false); thus, the formula is a tautology. Apr 15 '14 at 19:43
• Refreshing point of view! So it's just another way of saying $p$ either implies everything or is implied by everything. *5.14 is virtually self-evident now. Sorry about the delayed reply. It usually takes me a couple days to see the point. Apr 19 '14 at 22:01
• "Schröder says in his book that this formula looks paradox on the first glance since there is a conclusion r on the right side while r does not occur on the left side of the formula. The oddity is that the premise on the left side knows nothing about the statement r." If that's an oddity, then the formulas "C Cpq CCqrCpr" and "C Cpq CCrpCrq" have that oddity also. Nov 13 '14 at 6:48