The following argument is said to be invalid in my book.
All dogs are animals. All cats are animals. Hence, all dogs are cats.
While this clearly sounds invalid, if we try to prove it, does this really comes out invalid, I mean it seems to be valid?
Here's my proof:
1. (x)(Dx ⊃ Ax) 2. (x)(Cx ⊃ Ax) [ Therefore, (x)(Dx ⊃ Cx) ] 3. Proof by contradiction, assuming the opposite of conclusion: ~(x)(Dx ⊃ Cx) 4. Therefore, (∃x)~(Dx ⊃ Cx) {from 3} 5. Therefore, ~(Da ⊃ Ca) {from 4; Drop existential} 6. Therefore, Da {from 5; Simplifying} 7. Therefore, ~Ca {from 5; Simplifying} 8. Therefore, (Da ⊃ Aa) {from 1; Drop universal} 9. Therefore, (Ca ⊃ Aa) {from 2; Drop universal} 10. Therefore, Aa {from 6 and 8; Inference rule} 11. Therefore, ~Aa {from 7 and 9; Inference rule} 12. Therefore, (x)(Dx ⊃ Cx) {from 3; 10 contradicts 11}