# Determine Consistency Of System Specifications

I am looking at an example problem in my text:

"Determine whether these system specifications are consistent:

'The diagnostic message is stored in the buffer or it is re-transmitted.'

'The diagnostic message is not stored in the buffer.'

'If the diagnostic message is stored in the buffer, then it is re-transmitted.' "

The commentary the author gives on this problem isn't very helpful. Naturally, I am not quite sure how to approach this problem. Are the first two sentences suppose to be conditions, and we have to determine if the last sentence is consistent with the two conditions? Shouldn't rules of inference be used to determine the consistency of the statements? If so, that would seem quite odd, seeing as the textbook doesn't introduce that topic until a few sections later.

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I think you are supposed to take all three statements as hypotheses and see whether or not an inconsistency can be derived from them. – Zhen Lin Dec 11 '12 at 14:48
@ZhenLin So, you are saying it would be incorrect to think of the first two sentence as conditions for the third sentence? – Mack Dec 12 '12 at 18:06

Let $b$ stand for the diagnostic message is stored in the buffer, and let $r$ stand for the diagnostic message is re-transmitted. The three statements can then be abbreviated $b\lor r$, $\lnot b$, and $b\to r$. The question is then whether there is an assignment of truth values to $b$ and $r$ that makes all three statements true. If there is not, they’re mutually inconsistent; if there is, they’re consistent, though they may be satisfied for only one possible assignment of truth values to $b$ and $r$. Indeed, that turns out to be the case here:
$$\begin{array}{cc} b&r&\lnot b&b\lor r&b\to r\\ \hline \text{T}&\text{T}&\text{F}&\text{T}&\text{T}\\ \text{T}&\text{F}&\text{F}&\text{T}&\text{F}\\ \color{red}{\text{F}}&\color{red}{\text{T}}&\color{red}{\text{T}}&\color{red}{\text{T}}&\color{red}{\text{T}}\\ \text{F}&\text{F}&\text{T}&\text{F}&\text{T} \end{array}$$
As you can see, they are consistent, but they hold simultaneously only when $b$ is false and $r$ is true.
@EMACK: One can prove ‘algebraically’ that $$\lnot b\land(b\lor r)\land(b\to r)\leftrightarrow\lnot b\land r\;,$$ and since $\lnot b\land r$ is clearly consistent, this shows that $\lnot b\land(b\lor r)\land(b\to r)\leftrightarrow\lnot b\land r$ is consistent as well. – Brian M. Scott Dec 12 '12 at 20:00