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

$C$ -- chatGPT learns logic
$J$ -- I'm out of a job
$H$ -- all hell breaks loose):

$C ⟶ (J \land H)$ :

If chatGPT learns logic, then (I am out of a job and all hell breaks loose).
$(C ⟶ J) \land H$ :

(If chatGPT learns logic, then I am out of a job) and (all hell breaks loose).
$(C ⟶ J) \land (J ⟶ H)$

(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 ⟶ J) \land (J ⟶ H)

Inference (hs):

$⟶$

C ⟶ H

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

(C ⟶ J) \land (J ⟶ H)

Inference:

$⟶$

C ⟶ (J \land H)

Here we go:

$C ⟶ J$
$J ⟶ H$
$C$
$J$
$H$
$J \land H$
$C ⟶ (J \land H)$

hyp
hyp
temp. hyp
1,3,mp
2,4,mp
conj
temp. hyp discharged

Now let's negate the wff at left:

[(C ⟶ J) \land (J ⟶ H)]^{'}

Equivalent to:

$\leftrightarrow$

(C ⟶ J)^{'} \lor (J ⟶ 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 $C ⟶ (J \land H)$ :

[C ⟶ (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^{'})

Website maintained by Andy Long. Comments appreciated.
Updated on 01/22/2023 23:49:19