Every horse is an animal; therefore, every head of a horse is a head of an animal.
Here, I’ve changed a little both the example and the formalization with regard to what we
discussed in class (changing the formalization from the “strong” to the “weak” interpretation of
De Morgan’s sentence), for easier comparison with another example that we are going to discuss
later.
1. (x)(Hx  Ax)
2. (x)(y)[(Hx & Dyx)  (Ax & Dyx)]
3. (x)(y)[(Hx & Dyx) & (~Ax  ~Dyx)]
4. (y)[(Ha & Dya) & (~Aa  ~Dya)]
5. Ha
6. Dba
7. ~Aa | ~Dba
8. ~Aa | *
9. Ha  Aa
10. ~Ha | Aa
11. * | *
Premise
Conclusion
AP, QN2x, CN, DM, 2
x/a, EI, 3
y/b, EI, SIMP, 4
y/b, EI, SIMP, 4
y/b, EI, SIMP, 4
6, 7
x/a, UI, 1
IMP, 9
5, 11 | 8, 10
Every boy kissed some girl (or another); therefore, there is a girl every boy kissed (i.e.,
some girl was kissed by every boy).
Invalid (“quantifier-shift fallacy”), because: ‘Every boy kissed a girl’ is consistent with ‘there is
no girl every boy kissed’, i.e. for every boy there is a girl he did not kiss. ‘x’ ranges over boys, ‘y’
ranges over girls:
1.
2.
3.
4.
5.
6.
7.
8.
9.
(x)(y)(Kxy)
(y)(x)(Kxy)
(y)(x)~(Kxy)
(y)(Kay)
Kab
(x)~(Kxb)
~Kcb
(y)(Kcy)
Kcd
Premise
Conclusion
AP, QN2x,2
x/a, UI, 1
y/b, EI, 4
y/b, UI, 3
x/c, EI, 6
x/c, UI, 1
y/d, EI, 8
Refuting interpretation: Alex kissed Bella (5), but Corey did not kiss her (7), although he did
kiss Dora. So, in a four-member universe, with two girls and two boys, every boy kissed a girl,
but no girl was kissed by every boy. Note that the tree is not complete; in fact it would have an
infinite open branch (every UI would be followed by an EI, introducing a new name).
Every boy kissed the same girl (say, Alice); therefore, there is a girl every boy kissed.
For the formalization, note that it exploits the idea that the phrase ‘the same __’ marks out a
single thing of which the rest of the sentence holds. So, for instance, ‘every human activity
targets some ultimate end’ does not entail ‘there is an ultimate end targeted by every human
activity’, unless one can show that every human activity targets the same ultimate end (namely,
by showing that every human activity as such, ultimately targets human happiness, which is one
in kind, although not in number; see Nic. Eth.).
(x)(!y)(Kxy)  (x)(Ey)[Kxy & (z)(Kxz  y=z)], (every boy kissed one and the same single girl)
because (!y)(Ay)  (y)[Ay & (z)(Az  y=z)], where we can take 'A( )' to be 'Kx( )'.
1. (x)(y)[Kxy & (z)(Kxz  y=z)]
2. (y)(x)(Kxy)
3. (y)(x)~(Kxy)
4. (x)~(Kxa)
5. ~Kba
6. (y)[Kby & (z)(Kby  b=z)
7. Kbc
8. (z)(Kbc  b=z)
9. Kbc  b=c
10. ~Kbc
| b=c
*
| b=c
11.
| ~Kca
12. (Ey)[Kcy & (z)(Kcy  c=z)
13. Kcd
14. (z)(Kcd  c=z)
15. Kcd  c=a
16. ~Kcd
| c=a
*
| c=a
17.
| Kba
*
AP, QN2X, 2
y/a, UI, 3
x/b, EI, 4
x/b, UI, 1
y/c, EI, SIMP, 6
y/c, EI, SIMP, 6
z/c, UI, 8
IMP, 9
7,10
SI, 10,5
x/c, UI, 1
y/d, EI, SIMP, 12
y/d, EI, SIMP, 12
z/a, UI, 14
IMP, 15
16, 13
SI, 16, 7
17, 5