# Question regarding proof of ❋3.47 in Principia Mathematica by Whitehead and Russell

❋3.03 in the the last step seems unnecessary. Can someone explain to me why 3.03 is listed? The last step can be written out in full like this:

``````                  ⊦: p .⊃. q ⊃ r                  (1)

[(1).❋2.83.❋1.11] ⊦: p .⊃. r ⊃ s :⊃: p .⊃. q ⊃ s  (2)

⊦: p .⊃. r ⊃ s                  (3)

[(2).(3).❋1.11] ⊦: p .⊃. q ⊃ s  ⊃⊦. Prop
``````

Notice that ❋3.48 didn't use ❋3.03

• This is not a question. Can you please explain why this belongs here, and what kind of answer you're expecting? – iphigenie Jan 4 '14 at 12:03
• I concur with @iphigenie. – DBK Jan 4 '14 at 16:04

I think that *3.03 is not really needed neither in the proof of *3.47 nor in that of 3.48 [see Alfred North Whitehead & Bertrand Russell, Principia Mathematica to 56 (2nd ed - 1927), page 114].

I think that the derivations of *3.47 and *3.48 are a little bit "sloppy". The appeal to *3.03, i.e.

Given two asserted elementary propositional functions "⊢ φ(p)" and "⊢ ψ(p)" whose arguments are elementary propositions, we have ⊢ φ(p) . ψ(p).

where the "." (dot) is the conjunction, is used in *3.47 to license the step from :

⊢ [(p ⊃ r) . (q ⊃ s)] ⊃ (p.q ⊃ q.r) --- (1)

and

⊢ [(p ⊃ r) . (q ⊃ s)] ⊃ (q.r ⊃ r.s) --- (2)

to :

⊢ [(p ⊃ r) . (q ⊃ s)] ⊃ (p.q ⊃ r.s) --- form (1) and (2) by *3.03 and 2.83.

But *3.03 license the step from ⊢(1) and ⊢(2) to ⊢(1).(2), while *2.83 needs ⊢(1) ⊃ [ (2) ⊃ ...].

I think that it is enough - with suitable substitutions in *2.83 - to use *1.11 twice [both in *3.47 and *3.48] to get the result.

Consider *3.47 ; with the following substitution into *2.83 :

[(p ⊃ r) . (q ⊃ s)] / p --- p.q / q --- q.r / r --- s.r / s

we will have (assuming that I've restored the parentheses in the right way ...) :

⊢ ( [(p ⊃ r) . (q ⊃ s)] ⊃ (p.q ⊃ q.r) ) ⊃ [ ( [(p ⊃ r) . (q ⊃ s)] ⊃ (q.r ⊃ r.s) ) ⊃ ( [(p ⊃ r) . (q ⊃ s)] ⊃ (p.q ⊃ r.s) ) ].

The last formula is :

⊢ (1) ⊃ [ (2) ⊃ *3.47 ].

Thus, I believe that *1.11 suffices.

• Brilliant! Thanks, @Mauro, for confirming that 3.03 is not necessary. – George Chen Mar 25 '14 at 18:10
• nevertheless, I would refrain from using such words as "modus ponens" simply because PM didn't use these words. Modus ponens in many modern textbooks fail to distinguish between "if ... then ..." and "therefore." We've discussed in another thread that this distinction is important to get out of Lewis Caroll's puzzle. – George Chen Mar 25 '14 at 18:15