Proof in Fitch and counterexample in Tarski's World - From $A \to B$, infer $A \to (B \land C)$.

Question

Good official morning community,

Strengthening the Consequent: From A→B A→B , infer $A \to (B \land C)$.

I know that this proof is invalid and I want to make a counter example to prove that. How can I write $A \to (B \land C)$ and A→B A→B in Tarski's World. I tried writing sentences but it keeps telling me that it is of the wrong format.

what I have done so far is that I translated A→B A→B to tet(a) → → medium(a)

Would you please help me

You have to show that from $A \to B$, we cannot infer $A \to (B \land C)$. — Mauro ALLEGRANZA, Oct 11 '17 at 13:29
In the language of "solids" you can try with: $A= \text {Cube}(a) \land \text {Large}(a)$, $B= \text {Cube} (a)$, $C= \text {Cube} (a) \land \text {Small}(a)$. — Mauro ALLEGRANZA, Oct 11 '17 at 13:36
It is very simple: $C$ must contradict $A$ or $B$. Try with $C= \lnot Large(a)$ — Mauro ALLEGRANZA, Oct 11 '17 at 13:57
Maybe $Tet(a) \to Small(a)$ and $Tet(a) \to (Small(a) \land Large(a))$. — Mauro ALLEGRANZA, Oct 11 '17 at 14:16

score 2 · Accepted Answer · answered Oct 11 '17 at 17:19

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Create $1$ object in Tarski's World, make it a small cube, and label it $a$.

Then $Cube(a) \rightarrow Small(a)$ will evaluate to True, and $Cube(a) \rightarrow (Small(a) \land Tet(a))$ will evaluate to False.

answered Oct 11 '17 at 17:19

Bram28

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Thank you! That works! But why? I tried a very similar one but without including the tet(a) but it did not work with me. – Oct 11 '17 at 18:15
I actually feel kind of stupid now, I have been trying this for three consecutive days now. – Oct 11 '17 at 18:24
@markjohn It works because $Cube(a) \rightarrow Small(a)$ evaluates to $T \rightarrow T=T$, while $Cube(a) \rightarrow (Small(a) \land Tet(a))$ evaluates to $T \rightarrow (T \land F) = T \rightarrow F=F$. And I'd have to see exactly what you did in order to explain why your earlier attempts did not work . – Bram28 Oct 11 '17 at 19:14
Oh, I see! Thank you for explaining. I can see why I got them wrong, I was writing the second sentence as Cube(a)V(Small(a) → Cube(a), Which is totally wrong. Are you a teacher? – Oct 11 '17 at 19:21
@markjohn :) You can find me on page vii) – Bram28 Oct 11 '17 at 19:29
page vii -- which book ? – Oct 11 '17 at 19:47
@markjohn When doing Fitch, don't forget you now have that nice Add Support Steps feature :) – Bram28 Oct 11 '17 at 19:51
@markjohn I just replied to someone else struggling with Fitch proofs ... search for 'Fitch system for Logic Proofs' ... I go through the 'proper' thought process for setting up those proofs... maybe it helps you as well. And don't get discouraged!! – Bram28 Oct 11 '17 at 21:12
@markjohn The answer there may not have been accepted, but the answer was perfectly good. Fitch does not have MT (Modus Tollens), but the person that answered showed how to prove the pattern of ModusTollens at the end of the post. – Bram28 Oct 11 '17 at 23:06
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@markjohn That's a whole new question ... you better make that into a new post. And when you do so, you should show some of your own attempts, so we can see where you get stuck or confused. – Bram28 Oct 12 '17 at 00:23

Proof in Fitch and counterexample in Tarski's World - From $A \to B$, infer $A \to (B \land C)$.

1 Answers1