Concerning the equivalence

Concerning the equivalence

The boolean operators $\land , \lor$ ,and $\neg$ are easily expressed in Western languages,e.g. "and ", "or" and "not" respectively. Also the implication $\Rightarrow$ can be accommodated, usually by "if... then". The equivai- valence $\equiv$ is rendered by "iP and only if", but this an artefact of mathematical prose: people are definitely uncomfortable with it, as is shown by the fact that in pronouncing $A \equiv B$ ,verbal constructs like "holds" or "is true" and quotes are often inserted: "A" holds if and only if "B" holds. This linguistic clumsiness was for one of my mathematical colleagues proof that the equivalence did not come naturally and reason to warn against its use in reasoning.

We consider variables ranging from 0 to 1 ;

Let $X \wedge Y$ denote $X_{\max}Y$ , and let $X \vee Y$ denote $X_{\min}Y$ . All sorts of familiar relations hold true, such as

$$ \begin{array}{l} X \wedge \mathrm {t r u e} = X \quad X \wedge \mathrm {f a l s e} = \mathrm {f a l s e} \ X \vee \mathrm {f a l s e} = X \quad X \vee \mathrm {t r u e} = \mathrm {t r u e} \ X \wedge X = X \quad X \vee X = X \ \end{array} $$

$$ \begin{array}{l} X \wedge Y = Y \wedge X \quad X \vee Y = Y \vee X \ X \wedge (Y \wedge Z) = (X \wedge Y) \wedge Z \ X \vee (Y \vee Z) = (X \vee Y) \vee Z \ X \vee (Y \wedge Z) = (X \vee Y) \wedge (X \vee Z) \ X \wedge (Y \vee Z) = (X \wedge Y) \vee (X \wedge Z) \ \end{array} $$

Now the negation. Let ${7x}$ denote $1 - x$ . Again all sorts of familiar formulae may be used:

$$ \begin{array}{l} (1 7 7 X) = X \ \eta (X \wedge Y) = \eta X \vee \eta Y \quad \eta (X \vee Y) = \eta X \wedge \eta Y \ \end{array} $$

We have lost, however, $\neg X \vee X =$ true ; as this relation is known as "Terhium non dahur", this loss should not surprise us: it is the obvious price to pay for continuous logic.

Let $X \equiv Y$ denote the absolute value of the difference between $X$ and $Y$ . Again all sorts of familiar formulae hold true:

$$ \begin{array}{l} \because X \equiv Y = X \equiv Y \quad X \equiv Y = Y \equiv X \ X \equiv Y = X \land Y \equiv X \lor Y \ X \wedge Y \equiv X = X \vee Y \equiv Y \ X \equiv Y = (X \wedge Y \equiv X) \wedge (X \wedge Y \equiv Y) \ \end{array} $$

In the last formula but 1 we recognize the two expressions for the implication $X \Rightarrow Y$ ,in the last one the equivalence as " if and only if".

IF we take $X \Rightarrow Y$ as short for ${X\Lambda Y} \equiv X$ or for

$$ \begin{array}{l} X \Rightarrow Y = ? Y \Rightarrow ? X \ (X \Rightarrow Y) \land (X \Rightarrow Z) = X \Rightarrow Y \land Z \ (X \Rightarrow Y) \vee (X \Rightarrow Z) = X \Rightarrow Y \vee Z \ (X \Rightarrow Y) \land (Z \Rightarrow Y) = X \lor Z \Rightarrow Y \ (X \Rightarrow Y) \vee (Z \Rightarrow Y) = X \wedge Z \Rightarrow Y \ \end{array} $$

a wealth of formulae which might explain the popularity

For the equivalence, however, we lose a number of

$\left( {X \equiv Y}\right) \equiv Z\;$ and $\;X \equiv \left( {Y \equiv Z}\right) \;$ in general difference: equivalence has lost its associativity; $\left( {X \equiv Y}\right) \vee Z\;$ and $\;{XvZ} \equiv {YvZ}\;$ in general different. $\tau \left( {X \equiv Y}\right)$ and ${\eta X} \equiv \gamma$ in general differ: no $\left( {X \equiv Y}\right) \land \left( {Y \equiv Z}\right) \Rightarrow \left( {X \equiv Y}\right) \land \left( {Y \equiv Z}\right) \land \left( {Z \equiv X}\right)$ in general different from true : equivalence is no longer transitive.

With the exception of the loss of the "Tertium non darur" - which seems to be suspect anyhow - all formulae that lose their validity in the transition From binary to continuous logic involve the equivalence! And that is - see our introduction - precisely the logical connective our natural languages hardly cater for.

The moral is two-fold. Firstly we conclude that,

in all probability, "natural logic" is not as binary at

In fact, quite a number of formulae still valid surprised me and I must confess that at this stage I could only verify them by a painful case analysis. (Another way of stating this is by pointing out that this note contains a whole bunch of surprising little theorems of possible interest.)

Nuenen, 19 August 1985

Prof. dr. Edsger W. Dyrkstra Department of Computer Sciences The University of Texas at Austin Austin, TX 78712-1188 United States of America