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No. The way this is worded all x are y but not all y are necessarily x.
.
Example:
All Gorillas are Apes.
Some Apes are Chimpanzees.
Some Gorillas are Chimpanzees. (Not True)
.
All Dogs are Mammals.
Some Mammals are Cats.
Some Dogs are Cats. (Not True)
.
Analysis:
All A are B.
Some B are C.
Therefore, Some A are C.
.
∀x(Ax → Bx)
∃x(Bx ∧ Cx)
∴ ∃x(Ax ∧ Cx)
Truth Tree
..............................|Bn ET(n)
..............................|Cn ET(n) premise
............................./.\
.........................../....\
.......UT prem ~An / ......\
........................../\........\
......................../..\.........\
UT neg con ~An /.....\ ~Cn \ Bn UT prem
.............invalid ↑..... ↑...... /\
...................................../...\
.............................~An /......\ ~Cn UT negative conclusion
...................................↑.......x Invalid
Ignore the periods. The spaces were being deleted so I included periods to make the truth tree readable.
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∙ 11y ago
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-8
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