# Deductive Reasoning proof logic

I have a question for a deductive reasoning proof i made for the following question: (i needed to prove that these three premises can come to the conclusion of : ~R <--> ~T Im new to this so im not completely sure that this is correct.

Premises 1-3

1. R--> (S-->T)
2. S
3. ~T

End of Premises --------

1. Sub proof Assumption S
2. Sub proof T
3. (end of sub proof) S-->T . -->Intro, 4-5
4. R . --> Elim 1,6
5. T -->Elim 2,6
6. Sub proof Assumption R
7. Sub proof T
8. Sub proof ~T
9. Sub proof ⊥
10. (end of sub proof) ~R . ~Intro, 9-11
11. Sub proof ~R
12. Sub proof ~T
13. New Sub proof ~T
14. New sub proof ~R
15. ~R <--> ~T <--> Intro 13-14, 15-16
• Your use of $\to$-elim in step 7 is wrong. You cannot derive $R$ from $R \to (S \to T)$ and $S \to T$. Dec 6, 2019 at 11:48

We have to derive $$\lnot R \to \lnot T$$ and vice-versa; then conclude with $$\leftrightarrow$$-intro.

The first part is straightforward, From 3rd premise : $$\lnot T$$, using $$\to$$-intro we get immediately:

4) $$\lnot R \to \lnot T$$.

For the second part :

5) $$R$$ --- temporary assumed for a sub-proof

6) $$T$$ --- from 5) and 2nd premise from 1st one

7) $$\bot$$ --- contradicition of 6) with 3rd premise

8) $$\lnot R$$ --- from 5) and 8) by $$\lnot$$-intro, discharging temporary assumption.

9) $$\lnot T \to \lnot R$$ --- from 8) by $$\to$$-intro.

10) $$\lnot R \leftrightarrow \lnot T$$ --- from 4) and 9) by $$\leftrightarrow$$-intro.

• Ok I see now how you got to step 4 but I dont understand much after that in the second part. Dec 6, 2019 at 11:53
• Oh I didnt know it was possible to go from premise 1 and 3 directly to the ¬𝑅→¬𝑇. in line 4 Dec 6, 2019 at 12:04
• @Michaellaswhich - NO; from premise 3) only. You can use $\to$-intro because you can assume $\lnot R$ and re-iterate $\lnot T$. Then apply $\to$-intro to get $\lnot R \to \lnot T$, discharging $\lnot R$. It can be abbreviated in this way : $\to$-intro allows to derive $A \to B$ from $B$, with $A$ whatever. Dec 6, 2019 at 12:09