Some simple propositions have a surface resemblance to compound propositions, but not all is as it appears. Do all propositions that appear to be compound need to be made simpler?
I show in sections 2-3 they do not by showing that doing so results in logically valid but semantically unsound, and practically misleading results. The takeaway is that 28,754 / 24,950 is not about 1.076 simply because you profit by 1.076% when you sell what cost you 24,950 for 26,852 rupees, and by 11.52% when you sell it for 28,754 rupees. Restricting or extending your logic can’t, and shouldn’t, change the way the world is and how much profit you’re entitled to at the market price at the time of sale on account of your commitment to one rule of inference or another.
Classical logic permits inferences from one argument to other new arguments. It also proffers such inference rules for producing new arguments from old ones as guarantee the new ones are sound if the old ones are.
Consider a. below.
a. If you’re to my left, then I’m to your right.
Say you were standing to my left you’d see that I really was, and also see that you were standing to my left. So, intuitively, a.’s premises are true, and a. is a valid assertion. Your intuitions coincide nicely with the classical inference rule for the conditional. These state:
P → Q ├ Q v ~P
Think of P as the fact that ‘you’re to my left’ and Q as the fact that ‘I’m to your right’. You’d think Q couldn’t be true if P weren’t. The classical inference rule for the conditional agrees. It formalizes that idea in its most general sense as follows:
It is true that “If you’re to my left, then I’m to your right”├ It is true that “Either I’m to your right or you’re not to my left.”
Why care about conditionals?
It is fortuitous that the classical conditional discussed above is also called the material conditional. In finance, and economics, we say some fact or the other is material information when its possession can influence the behaviour and decisions of market participants, and consequently influence market outcomes.
Consider the case vignette, suggestive of many similar situations in financial practice, below.
Say you’d acquired 10 grams of gold at 24,950 rupees last year and this year the market price had risen to 28754 rupees. You could either sell your gold for the market price today to make a whopping 3804 rupees, which is a profit of over 11.52%. Alternatively, you could hold on to the gold hoping the prices will rise further next year. May be the bigger profit you could make from higher gold prices would come in handy to pay for the wedding of your daughter’s wedding next year.
Say you thought you ought to sit tight with your gold, till the prices were higher than 11.52% in addition to your outlay of 24,950 rupees. But come next year, the price of gold has dropped to 26852 rupees. Now you could sell for 1902 rupees of underwhelming profit—a little over 1.076% profit on your initial outlay. Alternatively, you could wait for prices to rise.
Assessing which course of action is feasible for you given your goals and resources calls for conditional reasoning.
When are Conditionals Material?
Classical logic tells you:
p. = If it’s true that you sell what cost you 24,950 rupees for 28,754 rupees, then, it’s true that your profit is over 11.52%
q. = If it’s true that you sell what cost you 24,950 rupees for 26,852 rupees, then, it’s true that your profit is over 1.076%.
r. = If it’s true that you sell what cost you 24,950 rupees for 28,754 rupees, then, it’s true that your profit is over 1.076%, or If it’s true that you sell what cost you 24,950 rupees for 26,852 rupees, then, it’s true that your profit is over 11.52%.
r. is straightforwardly incorrect though it bears a surface resemblance to the classical inference rule of the form (A → B) & (C → D) ├ (A → D) v (C → B). The key thing to note here is that the resemblance is superficial; although p. and q. seem to contain connective occurrences [‘→’, ‘&’] they are actually simple propositions.
Note that the conditional occurring in each of p. and q. is not material in the economic sense. For it to be material in the economic sense the world would have to be otherwise than it is. Specifically, it would have to be a world where 28,754 / 24,950 is about 1.076, and where 26,852 / 24,950 is about 11.52. Our world is nothing like that. If it were you could sell in the second year and end up with a tidy profit of 3804. But, actually, you can only make that much if you sell in the first year; and, if you sell in the second year you will end up with only 1902 rupees in profit.
Why do some logicians bite this bullet, or shoot classical logicians who don’t with it? In logic p. and q. are called compound propositions. These contain simpler, or what are called atomic, propositions which don’t have connective occurrences. In p. and q. the connective ‘→’ standing for ‘If…then’ occurs, so they are compound rather than atomic. Logicians who think all compound propositions need to be made simpler, by removing connective occurrences, for analytical purposes would obviously think they ought to do so because p. and q. are compound.
But what analytical purpose is served by splitting p. and q. into atomic propositions? I dare say none. For all intents and purposes for p. to be true just is for both the implicit antecedent and the implicit consequent to be true. If it were not then you would among other things profit over 11.52% even if you did not sell what cost you 24,950 rupees for 28,754 rupees because P → Q ├ Q v ~P. Similarly, if q. were divided up artificially into its antecedent and consequent, then, it would appear you’d profit by over 1.076% even if you didn’t sell what cost you 24,950 rupees for 26,852 rupees. Would that it were so!
Some simple propositions have a surface resemblance to compound propositions, but not all is as it appears. Logic only promises the deduction of truth from truth; to ensure you end up with true conditional inferences simpliciter, you need to begin reasoning with true antecedents and true consequents. As we saw 28,754 / 24,950 is not about 1.076 simply because you profit by 1.076% when you sell what cost you 24,950 for 26,852 rupees, and by 11.52% when you sell it for 28,754 rupees. To sell what cost you 24,950 rupees for 28,754 just is what it is for your profit to be over 11.52%. Likewise, to sell what cost you 24,950 rupees for 26,852 rupees just is what it is for your profit to be over 1.076%. Changing your logic can’t, and shouldn’t change the way the world is and how much profit you’re entitled to on account of the market price on the day of your sale.
Goodman, V., & Stampfli, J. (2009). The Mathematics of Finance: Modeling and Hedging. Providence, RI: American Mathematical Society.
Priest, G. (2013). An Introduction to Non-Classical Logic: From If to Is. New York, NY: Cambridge University Press.