Socrates & Plato

 Although truth is not strictly essential for pure logic, most arguments are designed to establish truth– to prove something. Proving something involves arguing validity; the arguer must strive to be consistent, to argue soundly towards the conclusion– and, of course, the argument must progressively build only on premises accepted as true by all involved. Shaw defines a proof as, “a valid argument starting from premises which are true and accepted as true by arguer and opponent, and proceeding to a conclusion which previously the opponent was unwilling to accept. It is showing the opponent that, given certain truths (which he accepts), he cannot consistently deny certain others.”

You may begin to sense that proofs are only possible and necessary because our mind is fallible, seeks shortcuts, is filled with innate preprogramming from an ancestral environment, and intellectually limited. If everyone had already considered all the possible consequences of every belief there would be nothing left to prove. In other words, there would be nothing to prove to a perfectly rational being– one that saw all the consequences of all their beliefs– because they could never be surprised. While we can’t hope to, and perhaps would not want to, become perfectly rational and logical, we can certainly benefit from learning to make more rational decisions in situations that demand logic– such as making investment decisions.

‘If, then, therefore’ arguments are often poorly positioned as proofs. Consider the following inference about a flashlight (torch):

 If the battery is low then the light is dim. The light is dim. Therefore the battery is low.

 This argument is invalid, because even if the premises were true there would be no guarantee of the truth of the conclusion. It may be true that if the battery is low then the light is dim, and true that the light is dim, but false that the battery is low. Perhaps another factor is causing the light to be dim, such as a poor connection, a faulty bulb, or a number of other reasons just as plausible as ‘the battery is low’. The argument may ultimately prove true– the battery being low may have actually caused the light to be dim– but this would be due to a lucky guess, not to the use of sound judgment and logic.

This is a very dangerous form of argument, although it is also a very common form of argument. Sherlock Holmes made this invalid argument style quite famous in the literary world– referring to it as brilliant deductive reasoning. Here is a passage from The Boscombe Valley Mystery:

It was about ten minutes before we regained our cab… Holmes still carrying with him the stone which he had picked up on the wood. ‘This may interest you, Lestrade,’ he remarked, holding it out. ‘The murder was done with it.’ ‘I see no marks,’ said Lestrade. ‘There are none,’ said Holmes. ‘How do you know then?’ said Lestrade. ‘The grass was growing under it. It had oly lain there a few days. There was no sign of a place whence it had been taken. It corresponds with the injuries,’ concluded Holmes.  

The argument is stated by Holmes as: if the stone were the murder weapon then it would not have lain there long and it would correspond with the injuries; it hadn’t lain there long and was consistent with the injuries; hence it is the murder weapon. Once again, we find that Sherlock Holmes, the great detective, has fallen into a logic trap with an invalid argument (but it certainly makes for fascinating reading).

As Shaw points out, Einstein’s theory of relativity could be considered an invalid argument on the same grounds. However, there is an important difference between the logic of Einstein and Sherlock Holmes. Einstein proposed his theory as the most plausible conclusion, after giving due scientific consideration to all competing theories, to explain observable premises, while Holmes has jumped to a conclusion from relatively brief observation without giving due consideration to other competing theories. Perhaps Holmes has found a rock thrown by a schoolboy in recent days, and that this particular rock is consistent with the injuries may also apply to dozens of other objects still in the area and many more objects that the murderer could have taken away from the crime scene. Einstein has moved from ‘If, then, therefore’ to ‘If, and ONLY if, then, therefore’. Sherlock Holmes has failed to validate his argument by considering a more exhaustive set of competing premises based on more in-depth observations.

Let’s consider some common uses of logic sequences heard in the financial sector.

If interest rates fall, then stock prices will rise. Interest rates have fallen this year; stock prices will be rising.  

 As common as this concept is among financial professionals– it resembles Sherlock Holmes invalid argument more than Einstein’s relativity argument. This argument fails the ‘If, and ONLY if, then’ test. While it may be true that lower interest rates are generally positive for business (lower borrowing rates and less competition from increasingly low return bonds and fixed deposits(, there is certainly no guarantee that falling interest rates will always result in rising stock prices (as history clearly shows). If stock prices do indeed rise, there is no guarantee that the rise was not caused by many other plausible explanations– such as increased earnings, a pickup in global economic activity, buy ratings from prominent analysts, or bubbly investor sentiment.

If the share price of s stock falls, then its dividend yield will rise. The price of stock XYZ has fallen, therefore its dividend yield will be higher.  

Once again, this is a valid but dangerous argument, which fails on two points. First, the premise is often false. When a share price falls the dividend yield will only rise if the dividend payout per share is held steady (or increased) by the board of directors. Indeed, a falling share price could be indicative of problems with the business that led to lower net earnings and poor cash flow, which could thus require a cut or elimination of the dividend in the near future. In this case, a falling share price would lead to a lower dividend, and perhaps even no dividend at all in the worst-case scenario. Secondly, this argument fails the ‘if, and ONLY if, then’ test because, even if the dividend is held constant as the share price falls– indeed resulting in a rising dividend yield– a rising dividend yield could just as easily result from an alternative cause, such as the board of directors deciding to increase the dividend payout. With a significant enough boost of the dividend payout one could certainly observe a rising dividend yield (a more healthy yield driven, presumably, by rising earnings and cash flows, which the directors believe are sustainable) even when the stock price rises or remains unchanged.

Sometimes invalid arguments appear in the negative form… to be continued…�
This article is extracted from the WallStraits publication, The Philosophical Investor, 2005.