# Schubert, Schenker, and propositional calculus…oh, my.

The complexity of Schubert’s lied Der Musensohn lies in its simple harmonic syntax within the deep-middleground Schicht.  There the oscillation between G major and B major creates an ostensible problem for fleshing out an orthodox Urlinie: The B major prolongations do not act as either third dividers or applied dominants to the submediant; thus, they cannot be used as evidence for such voice-leading paradigms (i.e., neither D major nor E minor arrives after the B major prolongation).  Well, how do the  B major prolongations fit into the large-scale voice-leading infrastructure—if it is possible, into an orthodox Schenkerian background?

I’ve written a paper I hope to publish that reveals a very tenable and (hermeneutically) satisfying solution to this problem, but I’ve always been bothered by the above logical argument concerning the role of B major.  For example, is it logically valid—formally speaking—to say “if B major is a third-divider, then D major follows”?  The short answer is yes: If the antecedent is false, the entire conditional is always true.  But I’ve assumed B major isn’t a third-divider from the beginning, so I’ve established a situation where my conditional is guaranteed to be true.  No good.  What about the negation of the consequent (i.e., modus tollens)?  This tells us that because D major does not materialize, B major is not a third-divider.  But, again, this is derived from the conditional as designed.  Circumventing these issues involves a more complicated proof that does not derive the identity of B major with such immediacy:

R = D major arrives (after B major), S = E minor arrives (after B major), P = B major is a third-divider, Q = B major is an applied dominant

Assumptions and proof:

¬R, ¬S, [((P → R) ˅ (Q → S)) ˄ ¬((P → R) ˄ (Q → S))]  ˫  ¬P ˄ ¬Q

__________________________

**1. ((P → R) ˅ (Q → S)) ˄ ¬((P → R) ˄ (Q → S))
A
**2. ¬R
A
**3. ¬S
A
**4. (P → R) ˅ (Q → S)
1 (˄E)
**5. ¬((P → R) ˄ (Q → S))
1 (˄E)
**6. | P → R
H
**7. | ¬R
2 RE
**8. | ¬P
6, 7 MT
**9. (P → R) → ¬P
6, 8 (→I)
**10. P → ¬(P → R)
9 (TRANS)
**11. | P
H
**12. | ¬(P → R)
10, 11 (→E)
**13. | ¬(¬P ˅ R)
12 MI
**14. | P ˄ ¬R
13 DM
**15. P → (P ˄ ¬R)
11, 14 (→I)
**16. | P
H
**17. | P ˄ ¬R
15, 16 (→E)
**18. | ¬R
17 (˄E)
**19. P → ¬R
16, 18 (→I)
**20. ¬P ˅ ¬R
19 MI
**21. | ¬ ¬P
H
**22. | P
21 DN
**23. | ¬R
20, 22 DS
**24. ¬ ¬P → R
21, 23 (→I)
**25. P → R
24 DN
**26. ¬P
9, 25 (→E)
**27. | Q → S
H
**28. | ¬S
3, RE
**29. | ¬Q
27, 28 MT
**30. (Q → S) → ¬Q
27, 29 (→I)
**31. Q → ¬(S → Q)
30 TRANS
**32. | Q
H
**33. | ¬(S → Q)
31, 32 (→E)
**34. | ¬(¬S ˅ Q)
33 MI
**35. | S ˄ ¬Q
34 DM
**36. Q → (S ˄ ¬Q)
32, 35 (→I)
**37. | Q
H
**38. | S ˄ ¬Q
36, 37 (→E)
**39. | ¬S
38 (˄E)
**40. Q → ¬S
37, 39 (→I)
**41. ¬Q ˅ ¬S
40 MI
**42. | ¬ ¬Q
H
**43. | Q
42 DN
**44. | ¬S
41, 43 DS
**45. ¬ ¬Q → S
42, 44 (→I)
**46. Q → S
45 DN
**47. ¬Q
30, 46 (→E)
**48. ¬P ˄ ¬Q
26, 47 (˄E)

Q.E.D.