Complete Question:
Translate this valid argument into symbolic form, then prove it using rules of inference/identities. "If the teller or the cashier had pushed the alarm button, the vault would have locked automatically, and the police would have arrived within three minutes. Had the police arrived in three minutes, the robbers' car would have been overtaken. However, the robbers' car was not overtaken. Therefore the teller did not push the alarm button. [ Let T="the teller pushed the alarm button", C="the cashier pushed the alarm button", V="the vault locked automatically", P="the police arrived within three minutes", 0="the robbers' car was overtaken" ]
Answer:
1) (T v C) => (V Λ P )
2) P => O
3) ⌐ O / ∴ ⌐ T
4) ⌐ P 2, 3, M.T.
5 ⌐ P v ⌐V 4, Add.
6 ⌐V v ⌐ P 5,Com.
7 ⌐ (T v C ) 1,6. M.T
8 ⌐ T Λ ⌐ C 7, De.M.
9∴ ⌐T 8, Simp.
Translate this sound argument into symbols, then use inference or identity rules to prove it.
"Had the teller or cashier pressed the alarm button, the vault would have automatically closed and the cops would have arrived in three minutes." The criminals' car would have been overtaken if the cops had arrived in three minutes.
The criminals' automobile, on the other hand, was not overtaken. As a result, the teller did not activate the alarm. [Let T="the teller pressed the panic button," C="the cashier pressed the panic button," V="the vault automatically locked," P="the cops came in three minutes," 0="the thieves' automobile was overtaken"]
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