MHB Truth Table Precedence: Evaluating Implication Rules

  • Thread starter Thread starter lyd123
  • Start date Start date
  • Tags Tags
    Table Truth table
lyd123
Messages
11
Reaction score
0
Hello!
The question is attached.

I know that " $\implies $ " (implies) has precedence from right to left. But because " l- " appears after
P$\implies ($Q $\implies$ R ), in my truth table do I evaluate:(P$\implies ($Q $\implies$ R ) ) $\implies$ ((P$\implies$Q ) $\implies$ R ) )
or

P$\implies ($Q $\implies$ R ) $\implies$ (P$\implies$Q ) $\implies$ R )

Thank you for any help. :)
 

Attachments

  • 2018-12-22 (2).png
    2018-12-22 (2).png
    1.7 KB · Views: 128
Physics news on Phys.org
The turnstile separates formulas but is not a logical connective itself. Therefore $$A\vdash B$$ is equivalent to the fact that $$A\to B$$ is a tautology. This formula has $A$ and $B$ as subformulas joined by $\to$, but it cannot have a subformula that consists of a strict subformula of $A$ and $B$, for example. So it's wrong to consider $P\to(Q\to R)\to(P\to Q)\to R$, which is $P\to((Q\to R)\to((P\to Q)\to R))$ because it has a subformula $(Q\to R)\to((P\to Q)\to R)$, which consists of a part of $A$ and the whole $B$.

lyd123 said:
I know that " $\implies $ " (implies) has precedence from right to left.
I also like this convention, but I've seen textbooks that consider $\to$ to be left-associative, so one has to be careful.
 
I think I understand now.. so I should use (P⟹(Q ⟹ R ) ) ⟹ ((P⟹Q ) ⟹ R ) ),
which would give me the attached truth table.
View attachment 8719

So it is not a tautology. Is this correct?
 

Attachments

  • 2018-12-22 (3).png
    2018-12-22 (3).png
    3.2 KB · Views: 138
Yes, it is correct. The converse implication is a tautology. This follows from the fact that $P\to Q\to R$ is equivalent to $PQ\to R$ (I omitted conjunction) and $PQ$ implies $P\to Q$.

There is a typo in column R, second last row.
 
Just a quick question from a novice.

Do [math]\implies[/math] and [math]\rightarrow[/math] mean the same thing? I note that the OP and Evgeny.Makarov are using two different symbols.

-Dan
 
topsquark said:
Do [math]\implies[/math] and [math]\rightarrow[/math] mean the same thing? I note that the OP and Evgeny.Makarov are using two different symbols.
This completely depends on the textbook or other source. Implication can be denoted by $\rightarrow$, $\to$ and $\supset$, and in addition arrows can be short of long. Some authors use different notations for metalevel (a contraction for "if... then..." in English) and object level (a part of the formal language we study) implications. I used a single arrow because it occurs in the attached image in post #1, which I assume comes from the instructor, and because it is shorter in LaTeX ([m]\to[/m] vs [m]\Rightarrow[/m] or [m]\Longrightarrow[/m]).
 
Hi all, I've been a roulette player for more than 10 years (although I took time off here and there) and it's only now that I'm trying to understand the physics of the game. Basically my strategy in roulette is to divide the wheel roughly into two halves (let's call them A and B). My theory is that in roulette there will invariably be variance. In other words, if A comes up 5 times in a row, B will be due to come up soon. However I have been proven wrong many times, and I have seen some...
Thread 'Detail of Diagonalization Lemma'
The following is more or less taken from page 6 of C. Smorynski's "Self-Reference and Modal Logic". (Springer, 1985) (I couldn't get raised brackets to indicate codification (Gödel numbering), so I use a box. The overline is assigning a name. The detail I would like clarification on is in the second step in the last line, where we have an m-overlined, and we substitute the expression for m. Are we saying that the name of a coded term is the same as the coded term? Thanks in advance.

Similar threads

Replies
6
Views
3K
Replies
12
Views
2K
Replies
3
Views
1K
Replies
12
Views
7K
Replies
2
Views
1K
Replies
45
Views
4K
Replies
1
Views
1K
Replies
4
Views
3K
Back
Top