Directory:Jon Awbrey/MathJax Problems

Computation and inference as semiosis

Equational reasoning, as distinguished from implicational reasoning, is well-evolved in mathematics today but grievously short-schrifted in contemporary logic textbooks. Consequently, it may be advisable for me to draw out and place in relief some of the more distinctive characters of equational inference that may have passed beneath the notice of a casual reading of these notes.

By way of a very preliminary orientation, let us consider the distinction between information reducing inferences and information preserving inferences. It is prudent to make make our first acquaintance with this distinction in the medium of some concrete and simple examples.

•	Example 1. Modus Ponens
		Information Reducing Inference
			\(\begin{array}{l} ~ p \Rightarrow q \\ ~ p \\ \overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~} \\ ~ q \end{array}\)
		Information Preserving Inference
			\(\begin{array}{l} ~ p \Rightarrow q \\ ~ p \\ \overline{\underline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~}} \\ ~ p ~ q \end{array}\)

Let us examine these two types of inference in a little more detail. A rule of inference is stated in the followed form:

\(\begin{array}{l} ~ \textit{Expression 1} \\ ~ \textit{Expression 2} \\ \overline{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~} \\ ~ \textit{Expression 3} \end{array}\)

The expressions above the line are called premisses and the expression below the line is called a conclusion. If the rule of inference is simple enough, the proof-theoretic turnstile symbol \({}^{\backprime\backprime} \vdash {}^{\prime\prime}\!\) may be used to write the rule on a single line, as follows:

\(\textit{Premiss 1}, \textit{Premiss 2} ~\vdash~ \textit{Conclusion}.\!\)

Either way, one reads such a rule of inference in the following manner:

From   \({\textit{Expression 1}}\!\)   and   \({\textit{Expression 2}}\!\)   infer   \({\textit{Expression 3}}.\!\)

Looking to Example 1, the rule of inference known as modus ponens says the following: From the premiss \(p \Rightarrow q\!\) and the premiss \(p\!\) one may logically infer the conclusion \(q.\!\)

Modus ponens is an illative or implicational rule. Passage through its turnstile incurs the toll of some information loss, and thus from a fact of \(q\!\) alone one cannot infer with any degree of certainty that \(p \Rightarrow q\!\) and \(p\!\) are the reasons why \(q\!\) happens to be true.

Further considerations along these lines may lead us to appreciate the difference between implicational rules of inference and equational rules of inference, the latter indicated by an equational line of inference or a 2-way turnstile \({}^{\backprime\backprime} \Vdash {}^{\prime\prime}.\!\)