Monthly Archives: August 2011

Symbolic Logic 5E: 3.5

“For each of the following arguments, construct both a formal proof of validity and an indirect proof and compare their length”


Formal Proof

  1. A ∨(B ∧ C)
  2. A→C …Therefore, C
  3. ¬A→(B ∧ C) (1,IMP)
  4. ¬C→¬A (2,TRANS)
  5. ¬C→(B ∧ C) (4,3,HS)
  6. C ∨ (B ∧ C) (5,IMP)
  7. (C ∨ B) ∧ (C ∧ C) (6,DIST)
  8. C ∧ C (7,SIMP)
  9. C (8,TAUT)

Indirect Proof

  1. A ∨(B ∧ C)
  2. A→C …Therefore, C
  3. ¬C (IP)
  4. ¬A (3,2,MT)
  5. B ∧ C (4,1,DS)
  6. C (5,SIMP)
  7. C ∧ ¬C (6,3,CONJ)


Formal Proof


Indirect Proof

  1. (D ∨ E)→(F→G)
  2. (¬G ∨ H)→(D ∧ F) …Therefore, G
  3. ¬G (IP)
  4. ¬G ∨ H (3,ADD)
  5. D ∧ F (4,2,MP)
  6. D (5,SIMP)
  7. D ∨ E (6,ADD)
  8. F→G (8,1,MP)
  9. F (5,SIMP)
  10. G (9,8,MP)
  11. G ∧ ¬G (10,3,CONJ)

3. In back of book.


Formal Proof


Indirect Proof

  1. (M ∨ N)→(O ∧ P)
  2. (O ∨ Q)→(¬R ∧ S)
  3. (R ∨ T)→(M ∧ U) …Therefore, ¬R
  4. R (IP)
  5. R ∨ T (4,ADD)
  6. M ∧ U (5,3,MP)
  7. M (6,SIMP)
  8. M ∨ N (7,ADD)
  9. O ∧ P (8,1,MP)
  10. O (9,SIMP)
  11. O ∨ Q (10,ADD)
  12. ¬R ∧ S (11,2,MP)
  13. ¬R (12,SIMP)
  14. ¬R ∧ R (13,4,CONJ)

5. In back of book.

∧ = And
∨ = Or
→ = If…then
¬  = Not (negation)
≡   = Logical Equivalence


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Symbolic Logic 5E: 3.4

“Give conditional proofs of validity for Exercises 21*, 22, 23, 24, and 25 on pages 47-48.”

21. In back of book.


  1. (T→E) ∧ (A→L) …Therefore, (T ∧ A)→(E ∧ L)
  2. T ∧ A (CP)
  3. T (2,SIMP)
  4. T→E (1,SIMP)
  5. E (3,4,MP)
  6. A→L (1,SIMP)
  7. A (2,SIMP)
  8. L (7,6,MP)
  9. E ∧ L (5,8,CONJ)
  10. (T ∧ A)→(E ∧ L) (2-9,CP)


  1. E→S
  2. E→(S→N)
  3. S→(N→F) …Therefore, E→F
  4. E (CP)
  5. S (4,1,MP)
  6. S→N (2,4,MP)
  7. N (5,6,MP)
  8. N→F (5,3,MP)
  9. F (7,8,MP)
  10. E→F (4-9,CP)


  1. A→(B ∨ C)
  2. E→(C ∨ P)
  3. ¬C …Therefore, ¬(B ∨ P)→¬(A ∨ E)
  4. ¬(B ∨ P) (CP)
  5. ¬B ∧ ¬P (4,DeM)
  6. ¬B (5,SIMP)
  7. ¬B ∧ ¬C (6,3,CONJ)
  8. ¬(B ∨ C) (7,DeM)
  9. ¬A (8,1,MT)
  10. ¬P (5,SIMP)
  11. ¬P ∧ ¬C (3,9,CONJ)
  12. ¬(E ∨ P) (11,DeM)
  13. ¬E (12,2,MT)
  14. ¬A ∧ ¬E (9,13,CONJ)
  15. ¬(A ∨ E) (14,DeM)
  16. ¬(B ∨ P)→¬(A ∨ E) (4-15,CP)

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Symbolic Logic 5E: 3.3

“Prove the validity of each of the following arguments by the method of assigning truth values”

1. In back of book.

2. E=T ; F=T ; G=F ; H=F ; I=F

3. J=T ; K=F ; L=F ; M=F ; N=F

4. O=F ; P=T ; Q=T ; R=T ; S=T

5. In back of book.

6. X=F ; Y=T ; Z=F

7. A=F ; B=F ; C=T ; D=T ;  E=T ; F=T ; G=T ; H=F

8. I=T ; J=T ; K=F ; L=F; M=T ; N=F ; M=F

9. P=T ; Q=T ; R=F ; S=T ; T=T ; U=T ; V=F

10. In back of book.

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Symbolic Logic 5E: 3.2, III, Second Half

Note: I cheated and used conditional proof (CP) for problems 27-29. I don’t think this is such an egregious offense, as CP is introduced a mere four pages later. The point of doing these problems pre-CP is to see how helpful CP is. So save yourself some time and trust me: CP makes solving proofs much, much easier!


  1. A→(B→C)
  2. C→(D ∧ E) …Therefore, A→(B→D)
  3. ¬A ∨ (B→C) (1,IMP)
  4. ¬A ∨ (¬B ∨ C) (3,IMP)
  5. (¬A ∨ ¬B) ∧ (¬A ∨ C) (4,DIST)
  6. ¬A ∨ ¬B (5,SIMP)
  7. ¬(A ∧ B) (6,DeM)
  8. ¬(A ∧ B) ∨ D (7,ADD)
  9. (A ∧ B)→D (8,IMP)
  10. A→(B→D) (9,EXP)


  1. E→F
  2. G→F …Therefore, (E ∨ G)→F
  3. ¬E ∨ F (1,IMP)
  4. ¬G ∨ F (2,IMP)
  5. (¬E ∨ F) ∧ (¬G ∨ F) (3,4,CONJ)
  6. F ∨ (¬E ∧ ¬G) (5,DIST)
  7. (¬E ∧ ¬G) ∨ F (6,COMM)
  8. ¬(¬E ∧ ¬G)→F (7,IMP)
  9. (E ∨ G)→F (8,DeM)


  1. [(H ∧ I)→J] ∧ [¬K→(I ∧ ¬J)] …Therefore, H→K
  2. (H ∧ I)→J (1,SIMP)
  3. ¬(H ∧ I) ∨ J (2,IMP)
  4. (¬H ∨ ¬I) ∨ J (3,DeM)
  5. ¬H ∨(¬I ∨ J) (4,ASSOC)
  6. H→(¬I ∨ J) (5,IMP)
  7. ¬K→(I ∧ ¬J) (1,SIMP)
  8. ¬(I ∧ ¬J)→K (7,TRANS)
  9. (¬I ∨ J)→K (8,DeM)
  10. H→K (6,9,HS)


  1. [L ∧ (M ∨ N)]→(M ∧ N) …Therefore, L→(M→N)
  2. L→[(M ∨ N)→(M ∧ N)] (1,EXP)
  3. L→[¬(M ∨ N) ∨ (M ∧ N)] (2,IMP)
  4. L→[(¬M ∧ ¬N) ∨ (M ∧ N)] (3,DeM)
  5. L→(M≡N) (4,EQUIV)
  6. L→[(M→N) ∧ (N→M)] (5,EQUIV)
  7. ¬L ∨ [(M→N) ∧ (N→M)] (6,IMP)
  8. [¬L ∨ (M→N)] ∧ [¬L ∨ (N→M)] (7,DIST)
  9. ¬L ∨ (M→N) (8,SIMP)
  10. L→(M→N) (9,IMP)

20. In back of book.


  1. S→(T ∧U)
  2. (T ∨ U)→V …Therefore, S→V
  3. ¬S ∨ (T ∧U) (1,IMP)
  4. (¬S ∨ T) ∧ (¬S ∨ U) (3,DIST)
  5. ¬(T ∨ U) ∨ V (2,IMP)
  6. (¬T ∧ ¬U) ∨ V (5,DeM)
  7. (V ∨ ¬T) ∧ (V ∨ ¬U) (6,DIST)
  8. ¬S ∨ T (4,SIMP)
  9. S→T (8,IMP)
  10. V ∨ ¬T (7,SIMP)
  11. ¬T ∨ V(10,COMM)
  12. T→V (11,IMP)
  13. S→V (9,12,HS)


  1. ¬W ∨ [(X→Y) ∧ (Z→Y)]
  2. W ∧ (X ∨ V) …Therefore, Y
  3. W (2,SIMP)
  4. (X→Y) ∧ (Z→Y) (3,1,DS)
  5. X ∨ V (2,SIMP)
  6. Y ∨ Y (5,4,CD)
  7. Y (6,TAUT)


  1. (A ∨ B)→(C ∧ D)
  2. ¬A→(E→¬E)
  3. ¬C …Therefore, ¬E
  4. A ∨ (E→¬E) (2,IMP)
  5. A ∨ (¬E ∨ ¬E) (4,IMP)
  6. A ∨ ¬E (5,TAUT)
  7. ¬C ∨ ¬D (3,ADD)
  8. ¬(C ∧ D) (7,DeM)
  9. ¬(A ∨ B) (8,1,MT)
  10. ¬A ∧ ¬B (9,DeM)
  11. ¬A (10,IMP)
  12. ¬E (11,6,DS)


  1. (F→G) ∧ (H→I)
  2. F ∨ H
  3. (F→¬I) ∧ (H→G) …Therefore, G≡¬I
  4. G ∨ I (2,1,CD)
  5. I ∨ G (4,COMM)
  6.  ¬I→G (5,IMP)
  7. ¬G ∨ ¬I (2,3,CD)
  8. G→¬I (7,IMP)
  9. (G→¬I) ∧ (¬I→G) (8,6,CONJ)
  10. G≡¬I (9,EQUIV)

25. In back of book.


  1. Q ∨ (R ∧ S)
  2. (Q→T) ∧ (T→S) …Therefore, S
  3. Q→T (2,IMP)
  4. T→S (2,IMP)
  5. Q→S (3,4,HS)
  6. (Q ∨ R) ∧ (Q ∨ S) (1,DIST)
  7. Q ∨ S (6,SIMP)
  8. ¬S→¬Q (5,TRANS)
  9. ¬Q→S (7,IMP)
  10. ¬S→S (6,9,HS)
  11. S ∨ S (10,IMP)
  12. S (11,TAUT)


  1. (U→V) ∧ (W→X) …Therefore, (U ∨ W)→(V ∨ X)
  2. U→V (CP)
  3. V ∨ X (2,1,CD)
  4. (U ∨ W)→(V ∨ X) (2-3,CP)


  1. (Y→Z) ∧(A→B) …Therefore, (Y ∧ A)→(Z ∧ B)
  2. Y ∧ A (CP)
  3. Y→Z (1,SIMP)
  4. Y (2,SIMP)
  5. Z (4,3,MP)
  6. A→B (1,SIMP)
  7. A (2,SIMP)
  8. B (7,6,MP)
  9. Z ∧ B (5,8,CONJ)
  10. (Y ∧ A)→(Z ∧ B) (2-9,CP)


  1. (C→D) ∧ (E→F)
  2. G→(C ∨ E) …Therefore, G→(D ∨ F)
  3. G (CP)
  4. C ∨ E (3,2,MP)
  5. D ∨ F (4,1,CD)
  6. G→(D ∨ F) (3-5,CP)


  1. (H→I) ∧ (J→K)
  2. H ∨ J
  3. (H→¬K) ∧ (J→¬I)
  4. (I ∧ ¬K)→L
  5. K→(I ∨ M) …Therefore, L ∨ M
  6. I ∨ K (1,2,CD)
  7. ¬K ∨ ¬I (2,3,CD)
  8. ¬I→K (6,IMP)
  9. K→¬I (7,IMP)
  10. ¬I→¬I (8,9,HS)
  11. I ∨ ¬I (10,IMP)
  12. (¬K ∧ I)→L (4,COM)
  13. ¬K→(I→L) (12,EXP)
  14. I→¬K (9,TRANS)
  15. I→(I→L) (14,13,HS)
  16. (I ∧ I)→L (15,EXP)
  17. I→L (16,TAUT)
  18. ¬I→(I ∨ M) (8,5,HS)
  19. I ∨ (I ∨ M) (18,IMP)
  20. (I ∨ I) ∨ M (19,ASSOC)
  21. I ∨ M (20,TAUT)
  22. ¬I→M (21,IMP)
  23. (I→L) ∧ (¬I→M) (17,22,CONJ)
  24. L ∨ M (23,11,CD)

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Article Summary: “The Supervenience of the Ethical on the Descriptive” by Frank Jackson


Jackson argues that ethical properties are merely descriptive properties – there are no properties above and beyond the descriptive. Jackson develops his thesis in light of the a priori truth that ethical properties supervene, or depend upon, descriptive properties. Then, Jackson handles two objections to his view.

Jackson begins by clarifying what ethical supervenience means, as opposed to psychological supervenience. In the case of psychological supervenience, physicalism (a metaphysical doctrine) holds that a full account of the physical world will entail a full account of the psychological world. The psychological depends upon the physical. But this is an intraworld, contingent supervenience thesis, rather than a global one. Another way of putting the point is that a full account of the psychological world does not entail a full account of the physical world: because the psychological is multiply realizable, there could be many different physical accounts which entail the psychological world. So the fact that the physical entails the mental in the way it does in our world is a contingent reality; it could have been otherwise.

Ethical supervenience, meanwhile, is a global supervenience thesis: it holds across all possible worlds. The descriptive entails the ethical in virtue of the meaning of these words; it cannot be otherwise. Suppose, for example, that two situations are exactly alike descriptively. Is it possible for us to condemn an actor in one situation and not the other? It seems not. The morality of the situation is determined by what occurs descriptively: an actor’s motivations, what the actor did, the outcome of the act, etc. It cannot be otherwise.

Now Jackson presents his argument: for any sentence ‘x’ about ethical nature, that sentence will be fully entailed by the world’s descriptive nature (this is a reiteration of the global supervenience thesis described above). Since the ethical depends upon the descriptive, it can be expressed fully in descriptive terms. So, any sentence ‘x’ about ethical nature ultimately reduces to merely descriptive terms, and ethical properties are merely descriptive properties.

Jackson then aims at clarifying his thesis: he says, for example, that it does not necessitate that an account of the ethical world is symmetrical with an account of the descriptive world. This is because a full account of the ethical world is consistent with infinitely many accounts of the descriptive world: one which includes electron E, for example, and another that excludes it.

Further, Jackson insists that his thesis does not imply that it is acceptable to do away with ethical vocabulary. If a given ethical property is an infinitely large descriptive disjunction, for example, the only way to express it will be to use an ethical term as shorthand. So too with ‘baldness’: baldness is really just referring to an infinitely large disjunction of possible hair distributions, but we point to a paradigmatic example of a bald person and use the word as shorthand out of necessity. But this does not mean ‘baldness’ is some extra feature of the world; so too, ethical terms, despite their necessity, do not imply extra features of the world.

Jackson then deals with two variations of an objection to the idea that ethical properties are merely descriptive properties. The general objection is that logically equivalent predicates may nonetheless point out distinct properties. In the case of ethical predicates, while they may be logically equivalent to descriptive sentences and necessarily co-extensive with them, it is still possible that they are pointing to distinct properties besides merely descriptive ones. The specific example in the first variation of the objection is the case of triangles: there seems to be a property of equiangularity and a property of equilaterality, despite their always occurring alongside one another.

Jackson thinks that this case conflates how we single out properties from how we distinguish how many properties there are. While it is true that equiangularity and equilaterality are distinct, this is so because they are ways in which we single out a property. The property itself, understood as a feature of the world, is singular in nature. So, we are singling out one property in two different ways.

Jackson also attempts to diffuse a more elaborate variation of the above objection: suppose we construct a machine that can detect the equilaterality of a triangle, but not the equiangularity. The machine has a light which blinks when this feature of the triangle is detected. This would seems to demonstrate that these two features of a triangle are distinct properties, since one is causally efficacious (it causes the machine’s light to shine) while the other is not. Jackson thinks that the force of this example derives from the machine’s operating in a segmented fashion: it first measures the sides of the triangle, then blinks if they are equal. It never gets a chance to measure its angles. In any case, Jackson thinks the example pertains merely to whether angles and sides are equivalent, not whether being an equilateral triangle and being an equiangular triangle are distinct properties or not.

Jackson finishes his rebuttal by deploying Mackie’s ‘argument from queerness’ (see my summary of Mackie on the ‘metaethics’ page). He does not understand what an ethical property could possibly be if not a descriptive one: what could a person mean when they say that something is ‘right’ if not the merely descriptive facts of the thing being picked out? What else could possibly be picked out? To Jackson, it is better to suppose such non-natural properties do not exist than to suppose they exist in some strange, unknown way.

Jackson concludes by stating one important implication of his argument: all  moral realists must express ethical properties in descriptive terms. Since his thesis is modal and not metaphysical, it does not claim such properties exist. It merely says that if indeed they do exist (or if one is a moral realist), then they must be expressed as descriptive, natural features of the world, regardless of one’s specific metaphysical allegiances.

Logical Outline

Primary Argument: Ethical properties = Descriptive properties

1. Ethical nature depends upon descriptive nature (global supervenience thesis).
2. Descriptive nature can be expressed in full as a sentence with merely
descriptive terms.
3. Therefore, ethical nature can be expressed in full as a sentence with
merely descriptive terms. [1,2]
4. Therefore, when one expresses something about ethical nature, one is
merely expressing something about descriptive nature. [from 3]
5. Therefore, ethical properties are merely descriptive. [from 4]

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Symbolic Logic 5E: 3.2, III, First Half

“Construct a formal proof of validity for each of the following arguments”

1. In back of book.


  1. C …Therefore, D→C
  2. C  ∨ ¬D (1,ADD)
  3. ¬D  ∨ C (2,COMM)
  4. D→C (3,IMP)


  1. E→(F→G) …Therefore, F→(E→G)
  2. (E ∧ F)→G (1,EXP)
  3. (F ∧ E)→G (2,COMM)
  4. F→(E→G) (3,EXP)


  1. H→(I ∧ J) …Therefore, H→I
  2. ¬H ∨ (I ∧ J) (1,IMP)
  3. [(¬H ∨ I) ∧ (¬H ∨ J)] (2,DIST)
  4. ¬H ∨ I (3,SIMP)
  5. H→I (4,IMP)

5. In back of book.


  1. N→O …Therefore,  (N ∧ P)→O
  2. ¬N ∨ O (1,IMP)
  3. (¬N ∨O ) ∨ ¬P (2,ADD)
  4. ¬P ∨ (¬N ∨O ) (3,COMM)
  5. P→(¬N ∨O ) (4,IMP)
  6. P→(N→O) (5,IMP)
  7. (P ∧ N)→O (6,EXP)
  8. (N ∧ P)→O (7,COMM)


  1. (Q ∨R)→S …Therefore, Q→S
  2. ¬(Q ∨R) ∨ S (1,IMP)
  3. (¬Q ∧ ¬R) ∨ S (2,DeM)
  4. (S ∨ ¬Q) ∧ (S ∨ ¬R) (3,DIST)
  5. S ∨ ¬Q (4,SIMP)
  6. ¬Q ∨ S (5,COMM)
  7. Q→S (6,IMP)


  1. T→¬(U→V) …Therefore, T→U
  2. T→¬(¬U ∨ V) (1,IMP)
  3. T→(U ∧ ¬V) (2,DeM)
  4. ¬T ∨ (U ∧ ¬V) (3,IMP)
  5. (¬T ∨ U) ∧ (¬T ∨ ¬V) (4,DIST)
  6. ¬T ∨ U (5,SIMP)
  7. T→U (6,IMP)


  1. W→(X ∧ ¬Y) …Therefore, W→(Y→X)
  2. ¬W ∨ (X ∧ ¬Y) (1,IMP)
  3. (¬W ∨ X) ∧ (¬W ∨ ¬Y) (2,DIST)
  4. ¬W ∨ X (3,SIMP)
  5. (¬W ∨ X) ∨ ¬Y (4,ADD)
  6. ¬W ∨ (X ∨ ¬Y) (5,ASSOC)
  7. W→(X ∨ ¬Y) (6,IMP)
  8. W→(¬Y ∨ X) (7,COMM)
  9. W→(Y→X) (8,IMP)

10. In back of book.


  1. E→F
  2. E→G …Therefore, E→(F ∧ G)
  3. ¬E ∨ F (1,IMP)
  4. ¬E ∨ G (2,IMP)
  5. [(¬E ∨ F) ∧ (¬E ∨ G)] (3,4,CONJ)
  6. ¬E ∨ (F ∧ G) (5,DIST)
  7. E→(F ∧ G) (6,IMP)


  1. H→(I ∨ J)
  2. ¬I …Therefore, H→J
  3. ¬H ∨ (I ∨ J) (1,IMP)
  4. (¬H ∨ I) ∧ (¬H ∨ J) (3,DIST)
  5. ¬H ∨ J (4,SIMP)
  6. H→J (5,IMP)


  1. (K ∨ L)→¬(M ∧ N)
  2. (¬M ∨ ¬N)→(O≡P)
  3. (O≡P)→(Q ∧ R) …Therefore, (L ∨ K)→(R ∧ Q)
  4. (K ∨ L)→(¬M ∨ ¬N) (1,DeM)
  5. (K ∨ L)→(O≡P) (4,2,HS)
  6. (K ∨ L)→(Q ∧ R) (5,3,HS)
  7. (L ∨ K)→(Q ∧ R) (6,IMP)
  8. (L ∨ K)→(R ∧ Q) (7,IMP)


  1. S→T
  2. S ∨ T …Therefore, T
  3. ¬T→¬S (1,TRANS)
  4. ¬S→T (2,IMP)
  5. ¬T→T (3,4,HS)
  6. T ∨ T (5,IMP)
  7. T (6,TAUT)

15. In back of book.

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Article Summary: “Morality as a System of Hypothetical Imperatives” by Philippa Foot


Foot argues that, contrary to commonly-held belief, moral judgments are not categorical imperatives, but rather are hypothetical imperatives like other judgments. Foot thinks this because she can see no basis for the claim that we always have a reason to obey moral rules. But if we do not always have a reason to obey, then it can be rational to ignore moral rules, and thus moral judgments cannot be categorical.

Foot discusses efforts to delimit the moral categorical imperative from other sorts, but believes that they all fail. The most basic theory about moral judgments is one about use: certainly, moral judgments are used categorically, as opposed to other sorts of judgments which tend to be hypothetical. But Foot points out that Kant (the originator of the claim that moral judgments are categorical) meant much more than this: he seems to think it is a metaphysical fact about moral rules that they are rationally binding, not merely that they are used as if they are rationally binding. Further, in certain circumstances, rules of etiquette are used categorically: when a man wants to disobey the rules of a club because he will never come back, nonetheless it is said he ought not to violate the club’s rules, despite his desires. So a theory of use cannot differentiate moral judgments from any other.

For Foot, the heart of the matter is whether moral rules are reason-giving: are we rationally compelled to obey, regardless of our desires? Foot proceeds to disputes the most significant attempts at justifying the reason-giving powers of moral rules. In the first place, such rules do not to Foot appear to be intrinsically reason-giving. That is to say, an action can be right and we can still have no reason to do it. An amoralist can rationally withdraw from the moral community and we will not be able to convict him of irrationality; we can say that he is vindictive, or evil, or something of the sort, but not irrational.

With the most common support for moral categorical imperatives removed, Foot proceeds to secondary reasons why we might always have a reason to obey moral rules. The first is the normativity of moral rules. But Foot rightly points out that many other sorts of judgments entail normativity; we do not, however, list them as categorical. Further, the notion that we simply ‘must’ adhere to rational precepts is, to Foot, likely an outcome of stringent teaching. So, we feel we must obey, but such feelings cannot serve as a support for a theory of categorical imperatives; regardless of how we feel, we might still be able to rationally disobey, after all. Finally, Foot dispenses with the idea that coercion (physical or psychological) might undergird the necessary obedience to moral rules. It is clear that Kant’s (and almost everyone else’s) conception of the categorical imperative is not one which people are coerced into; it should provide a reason to obey absent coercion.

So, Foot concludes that if moral judgments are not categorical, then they must be hypothetical. She then disputes the notion that, if this were so, and if everybody believed it, it would have a corrosive impact upon morality. Foot thinks that Kant himself believed this because he was a psychological hedonist with regard to all actions except those which adhered to moral rules: when we are not obeying moral rules, we are always promoting our own self-interest. Foot thinks that the fact that Kant believed this blinded Kant to the possibility of non-binding reasons for being moral.

People can, and do, have reasons for being moral despite moral rules not being binding. People love justice, liberty, charity, and other virtues. And since they desire these things, they will continue to promote them. So, people will still desire to promote the well-being of others, for example, despite the fact that they are not required to. Foot thinks that, in fact, dedication to a moral rule’s being voluntary might promote adherence: people will be more motivated to promote justice, say, if they feel they are volunteers banded together to promote the cause.

Logical Outline

Primary Argument: Moral judgments are hypothetical imperatives

1. Either moral judgments are categorical imperatives or otherwise they
are hypothetical imperatives.
2. Moral judgments are not categorical imperatives.

S1. In order for moral judgments to be categorical, there must
always be a reason to adhere to them.
S2. The use of moral judgments as categorical cannot be a reason
for adhering to them.
S3. Moral rules are not intrinsically reason-giving.
S4. The normative aspect of moral judgments cannot be a reason
for adhering to them.
S5. Coercion cannot be a reason for adhering to moral judgments.
S6. Our feelings cannot be reasons for adhering to moral judgments.
S7. Therefore, there are no necessary reasons for adhering to moral judgments. [S2-26]
S8. Therefore, moral judgments are not categorical. [S7,S1]

3. Therefore, moral judgments are hypothetical imperatives. [2,1]

Sub-argument: Morality would be preserved, even if hypothetical

1. If people would still desire to be just, charitable, etc. even if moral
judgments were hypothetical imperatives, then morality would continue
on as usual.
2. People would in fact continue to desire to be just, charitable, etc. even
if moral judgments were hypothetical imperatives.
3. Therefore, morality would continue on as usual, even if moral judg-
ments were mere hypothetical imperatives. [2,1]

Symbolic Notation

Primary Argument
1. (C ∨ H) ∧ ¬(C ∧ H)
2. ¬C…Therefore,H
3. C ∨ H (1, SIMP)
4. H (2, 3,DS)

Where C=Moral judgments are categorical imperatives and H=Moral
judgments are hypothetical imperatives.

1. D → M
2. D…Therefore,M
3. M (2, 1,MP)

Where D=People will continue to desire to be moral, even without
categorical imperatives, and M=Morality will continue on as usual.

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