chatGPT, Jobs, and Hell

In our first quiz, I asked you to think of two different ways that one might logically interpret the sentence

If chatGPT learns logic, then I am out of a job and all hell breaks loose.

Students reported at least three different ways that we might reasonably interpret that sentence (using the statement letters

  1. \(C \longrightarrow (J \land H)\):

    If chatGPT learns logic, then (I am out of a job and all hell breaks loose).

  2. \((C \longrightarrow J) \land H\):

    (If chatGPT learns logic, then I am out of a job) and (all hell breaks loose).

  3. \((C \longrightarrow J) \land (J \longrightarrow H)\)

    (If chatGPT learns logic, then I am out of a job) and (if I am out of a job, then all hell breaks loose).

I want to compare the first and third, which might seem to be logically equivalent (but which are not). In fact, the 3rd implies the 1st, but not vice versa.

\[ (C \longrightarrow J) \land (J \longrightarrow H) \] Inference (hs):

\(\longrightarrow\)

\[ C \longrightarrow H \]

We can illustrate the use of a temporary hypothesis to demonstrate that

\[ (C \longrightarrow J) \land (J \longrightarrow H) \] Inference:

\(\longrightarrow\)

\[C \longrightarrow (J \land H)\]

Here we go:

  1. \(C \longrightarrow J\)
  2. \(J \longrightarrow H\)
  3. \(\hspace{1in} C\)
  4. \(\hspace{1in} J\)
  5. \(\hspace{1in} H\)
  6. \(\hspace{1in} J \land H\)
  7. \(C \longrightarrow (J \land H)\)

  1. hyp
  2. hyp
  3. temp. hyp
  4. 1,3,mp
  5. 2,4,mp
  6. conj
  7. temp. hyp discharged

Now let's negate the wff at left:

\[ [(C \longrightarrow J) \land (J \longrightarrow H)]' \] Equivalent to:

\(\leftrightarrow \)

\[ (C \longrightarrow J)' \lor (J \longrightarrow H)' \] \[ (C' \lor J)' \lor (J' \lor H)' \] \[ (C \land J') \lor (J \land H') \]

But notice that this is not the same as the negation of \(C \longrightarrow (J \land H)\):

\[ [C \longrightarrow (J \land H)]' \] Equivalent to:

\(\leftrightarrow\)

\[ [C' \lor (J \land H)]' \] \[ C \land (J \land H)' \] \[ C \land (J \land H)' \] \[ C \land (J' \lor H') \]
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