# Truth Table Precedence: Evaluating Implication Rules

• MHB
• lyd123
In summary, the question is about the precedence of the logical connective "implies" ($\implies$) in a truth table. The answer is that it has precedence from right to left, meaning that in a formula like (P$\implies$(Q $\implies$ R)), the subformula (Q $\implies$ R) is evaluated first and then the entire formula is evaluated. This is different from the formula (P$\implies$Q) $\implies$ R, where the subformula (P$\implies$Q) is evaluated first and then the entire formula is evaluated. The turnstile symbol ($\vdash$) does not have a logical connective itself and is equivalent to the formula A
lyd123
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. :)

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• 2018-12-22 (2).png
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The turnstile separates formulas but is not a logical connective itself. Therefore $$\displaystyle A\vdash B$$ is equivalent to the fact that $$\displaystyle A\to B$$ 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
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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 $$\displaystyle \implies$$ and $$\displaystyle \rightarrow$$ mean the same thing? I note that the OP and Evgeny.Makarov are using two different symbols.

-Dan

topsquark said:
Do $$\displaystyle \implies$$ and $$\displaystyle \rightarrow$$ 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]).

## 1. What is a truth table and how is it used to evaluate implication rules?

A truth table is a mathematical tool used to determine the truth values of a compound statement based on the truth values of its components. In the context of evaluating implication rules, a truth table is used to determine the validity of a conditional statement by analyzing the truth values of the statements that make up the conditional.

## 2. What is the order of precedence for evaluating implication rules?

The order of precedence for evaluating implication rules follows the acronym PEMDAS: parentheses, exponents, multiplication and division, addition and subtraction. In other words, parentheses must be evaluated first, followed by any exponents, then multiplication and division, and finally addition and subtraction.

## 3. How do I know which operation to perform first when evaluating implication rules?

When evaluating implication rules, it is important to follow the order of precedence (PEMDAS) and evaluate the operations from left to right. This means that any operations within parentheses should be evaluated first, followed by any exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right.

## 4. Can the order of operations be changed when evaluating implication rules?

No, the order of operations cannot be changed when evaluating implication rules. Following the correct order of precedence (PEMDAS) is necessary in order to accurately evaluate the truth values of a conditional statement and determine its validity.

## 5. How can I check if my evaluation of an implication rule is correct?

To check if your evaluation of an implication rule is correct, you can use a truth table to compare the truth values of the individual statements to the truth value of the entire conditional statement. If the truth values match, your evaluation is correct. It is also important to double check your order of operations to ensure that you followed the correct order of precedence (PEMDAS).

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