SOLUTION: For any TFL sentences 𝛼, 𝛽, and 𝛾 such that 𝛼 is a contradiction, 𝛽 is a tautology, and 𝛾 is neither a contradiction nor a tautology, do the following entailment

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Question 1171518: For any TFL sentences 𝛼, 𝛽, and 𝛾 such that 𝛼 is a contradiction, 𝛽 is a tautology, and 𝛾 is neither a contradiction nor a tautology, do the following entailments hold:
3.2.1. 𝛼 β†’ 𝛼 ⊨ 𝛽 β†’ (𝛽 β†’ 𝛼)
3.2.2. Β¬(𝛽 β†’ 𝛾) ⊨ 𝛾 ↔ 𝛼
Thank you!
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
𝛼 β†’ 𝛼 ⊨ 𝛽 β†’ (𝛽 β†’ 𝛼)

Substitute FALSITY for 𝛼, TRUTH for 𝛽, 
𝛼 β†’ 𝛼 ⊨ 𝛽 β†’ (𝛽 β†’ 𝛼)
FALSITY β†’ FALSITY ⊨ TRUTH β†’ (TRUTH β†’ FALSITY)
            TRUTH ⊨ TRUTH β†’ FALSITY
            TRUTH ⊨ FALSITY


Does not hold.   



3.2.2. Β¬(𝛽 β†’ 𝛾) ⊨ 𝛾 ↔ 𝛼

Substitute FALSITY for 𝛼, TRUTH for 𝛽, and leave 𝛾 as it is,

    Β¬(TRUTH β†’ 𝛾) ⊨ 𝛾 ↔ FALSITY
  Β¬(Β¬TRUTH or 𝛾) ⊨ 𝛾 ↔ FALSITY
 Β¬(FALSITY or 𝛾) ⊨ 𝛾 ↔ FALSITY
 (Β¬FALSITY & ¬𝛾) ⊨ 𝛾 ↔ FALSITY
    (TRUTH & ¬𝛾) ⊨ 𝛾 ↔ FALSITY
              ¬𝛾 ⊨ 𝛾 ↔ FALSITY

Since the left side becomes ¬𝛾, then 𝛾 would have to be a falsity and
would imply falsity. 

Yes, it does hold.

Edwin
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