## Thoughts on Boolean Laws of Thought

I first encountered symbolic

logic when I was a teenager. Unfortunately, I had great trouble following the ostensible explanations that I encountered, and I didn't recognize that my perplexity was not because the underlying subject were intrinsically difficult for me, but because the explanations that I'd found simply weren't very well written. Symbolic logic remained mysterious, and hence became intimidating. And it wasn't clear what would be its peculiar virtue over logic expressed in natural language, with which I was quite able, so I didn't focus on it. I was perhaps 16 years old before I picked-up any real understanding of any of it, and it wasn't until years after that before I became comfortable not simply with Boolean *expression* but with processing it as an *algebra*.

But, by the time that I was pursuing a master's degree, it was often how I generated my work in economics or in mathematics, and at the *core* of how I presented the vast majority of that work, unless I were directed otherwise. My notion of an ideal paper was and remains one with relatively little natural language.

Partly I have that notion because I like the idea that people who know mathematics shouldn't have to learn or apply much more than minimal English to read a technical paper. I have plenty of praise for English, but there are an awful lot of clever people who don't much know it.

Partly I have that notion because it is easier to *demonstrate* logical *rigor* by using symbolic logic. I want to emphasize that word demonstrate

because it is possible to be *just* as logically rigorous while expressing oneself in natural language. Natural language is just a *notation*; thinking that it is intrinsically less *rigorous* than one of the symbolic

notations is like thinking that Łukasiewicz Polish notation is less rigorous than infixing notation or vice versa. I'll admit that some people may be less inclined to various sorts of errors using one notation as opposed to another, but *which* notation will vary amongst these people. However, other people don't necessarily *see* that rigor when natural language is used, and those who are inclined to be *obstinate* are more likely to exploit the lack of *simplicity* in natural language.

But, while it may be more practicable to lay doubts to rest when an argument is presented in symbolic

form, that doesn't mean that it will be *easy* for readers to follow whatever argument is being presented. Conventional academic economists use a considerable amount of fairly high-level mathematics, but they tend to use natural language for the purely *logical* work.[1] And it seems that most of them are distinctly uncomfortable with extensive use of symbolic logic. It's fairly rare to find it heavily used in a paper. I've had baffled professors ask me to explain elementary logical transformations. And, at least once, a fellow graduate student didn't come to me for help, for fear that I'd immediately start writing symbolic logic on the chalk-board. (And perhaps I would have done so, if not asked otherwise.)

The stuff truly *isn't that hard*, at least when it comes to the sort of application that I make of it. There is a tool-kit of a relatively few *simple* rules, some of them *beautiful*, which are used for the lion's share of the work. And, mostly, I want to use this entry to high-light some of those tools, and some heuristics for their use.

First, though, I want to mention a rule that I *don't* use. This proposition, normally expressed in natural language as A is A

and called the Law of Identity

, is declared by various philosophers to be one of the three *Principles* of logic. But I have no g_dd_mn'd idea what to *do* with it. It's not that I would ever want to *violate* it; it's just that I literally don't see anything *use*ful to it. Ayn Rand and many of those for whom she is preceptrix treat it as an essential *insight*, but I think that it's just a dummy proposition, telling me that any thing can stand where that thing can stand.[2]

The remaining two Principles of logic are rather more appealing, but I seldom make use of them that is both direct and explicit.

There's the Law of Non-Contradiction, Now, I use proof by contradiction fairly often, but I don't *explicitly* haul-out the Law of Non-Contradiction when I do so. (Also, were all else equal, I'd avoid proof by contradiction, because I've often seen people flummoxed by it. But all else is rarely equal, and I don't generally delay work to find some more pleasing proof.)

Then there's the Law of the Excluded Middle, which I find myself using only even less directly than that of Non-Contradiction.

The tools that I use heavily and explicitly in symbolic logic are those that help me to find underlying parallels or symmetries that will allow me to reduce complex expressions to simpler expressions. Finding those parallels or symmetries is a matter of seeing how to turn conjunctions into disjunctions or vice versa, disjunctions into implications or vice versa, negations into affirmations or vice versa, so that expressions may be consolidated or analyzed and reconsolidated.

Of course a double negative is a positive, but people need to learn also to *turn that around*, when they *need a negative*, by converting positives into double negatives.

Redundant conjunctions or disjunctions can be simplified, but often simple propositions should be made redundantly to exhibit parallels in the structure containing them.

Implications can be converted into disjunctions with the protasis negated; *disjunctions can be turned into implications* with a negated term turned into a protasis.

Conjunction and disjunction each distribute over the other.

DeMorgan's Laws of Logic allow one to focus or spread negations, finding conjunctions where there were disjunctions and vice versa.

Universal quantifiers can be conceptualized as conjunctions or (given certain philosophical commitments) as a generalization thereöf.

Existential quantifiers can be conceptualized as disjunctions or (given certain philosophical commitments) as a generalization thereöf.

Above, I've written the inclusions for these quantifiers in the way that mathematicians and economists usually write them, but it's important to remember that an inclusion for a universal quantifier corresponds to an implication, and that for an existential quantifier corresponds to an conjunction, (In fact, I have come to prefer to write my inclusions in terms of implications and conjunctions.)

De Morgan's Laws apply to the quantifiers thus:

About the only other rules that I *consciously* use are the commutative properties of conjunction and of disjunction,

[1] There's an *idiotic* notion amongst a great many mainstream economists that the Austrian School tradition is somehow *less rigorous* simply because some of its most significant members eschew overt mathematics in favor of logical deduction expressed in natural language. But most of the mainstream is likewise not using symbolic

*logic*; neither is *necessarily* being less rigorous than otherwise. The *meaning* of variables with names such as

can be every bit as `q _{t}`

*muddled*as those called something such as

the quantity exchanged at this time. There are

*good*reasons to object to the rather

*wholesale*rejection of overt mathematics by many Austrian School economists, but

*rigor*is not amongst the

*good*reasons.

[2] (2010:09/01) I note that some authors use Law of Identity

to refer not to a principle that `A` can stand wherever `A` can stand for all `A`, but to a rule However, this is no more than a compressed expression of which is an alternate way of expressing which is a redundant expression of which is just different expression of which is just the Law of the Excluded Middle, rather than some further principle.