### The Logical Dentist

The human brain doesn’t always think logically. College students’ performance on logic problems is not a pretty sight. Steven Pinker (How The Mind Works, 1997) discusses the following student logic test: There are some archeologists, biologists, and chess players in a room. None of the archeologists are biologists. All of the biologists are chess players. What, if anything follows? A majority of students conclude that none of the archeologists are chess players, which is not valid. None of them conclude that some of the chess players are not archeologists, which is valid. In fact, one fifth claim that the premises allow no valid inferences.

Patrick Shaw (Logic and its Limits, 1997) defines a logical argument as “one which is sound; a logical person is one who habitually uses sound arguments.” Sound arguments are essential for logical decision-making. Sound arguments are progressively built, brick by brick, by assembling a string of premises that lead to a reasonable concludion. The conclusions are increasingly valuable if they stand up to observation over time. The logical argument building process often sounds like… If ‘X’ is true, and ‘Y’ is also true, therefore ‘Z’ must then be true. A sound argument can be very simple, such as:

Many animals build nests according to a pattern, which varies little within the species. In some instances, the offspring have had no opportunity to learn from their parents. There must, therefore, be at least some innate tendency controlling the activity. (An argument from Boring, Langfeld, and Weld, Foundations of Psychology, 1948.)

Premise 1: Many animals build nests according to a pattern, which varies little within the species.

Premise 2: In some instances the offspring have not been taught by their parents to build the characteristic nest.

Conclusion: There is at least some innate tendency controlling the activity.

Another example of s simple logical argument:

All fish are cold-blooded, and no whales are cold-blooded; so whales are not fish.

Premise 1: All fish are cold-blooded.

Premise 2: Whales are warm-blooded.

Conclusion: Whales are not fish.

Of course, not all logical arguments are so simple, and the task of assessing arguments– the validity of the premises and conclusions both– is a challenging one that is influenced not only by our capacity for logical thought, but also our beliefs and personal experience. Simply because an argument is formatted as ‘if… then… therefore’ does not make it a sound argument. Consider a slightly more vague argument:

Premise 1: All vitamins are nutritious.

Premise 2: Some nutritious things are not cheap.

Conclusion: Some vitamins are expensive.

Most people will hesitate to agree with this conclusion and even if it is accepted it is of marginal value due to the vagueness of both the premises and the conclusion. If any of the premises are not true, then the conclusion will likely not be true– but the argument may remain sound from a purely logical point of view. As Shaw points out: “It must be stressed that to ask whether a conclusion follows is not the same as asking whether that conclusion is true. From the point of view of logic, truth is not of immediate account. A conclusion follows from the premises in this sense: if one grants the premises then one must, to be consistent, also accept the conclusion. If the premises are true, then the conclusion must be true. Which is not to say that the premises and conclusion are true: whether or not they are is a different problem.” For example:

Premise 1: All students are teapots.

Premise 2: Our dog is a student.

Conclusion: Our dog is a teapot.

Obviously both premises are false and so is the conclusion. Yet the argument is logically sound with the conclusion properly built upon the premises. Anyone who accepts these premises would be logically committed to accept the conclusion. The lack of concern with truth can seem strange at first, but limiting logical arguments to only the realm of known truths and current beliefs would limit the boundary of useful conclusions that might be examined.

Mathematics is filled with logical arguments. You may remember from your school days such problems as this…

Bill is eight years older than John, and in two years time he will be twice as old as John. How old is Bill?

Conclusion sequence:

From premise 2, it follows that x / 2 = 2y + 4

Subtracting two from each side we get x = 2y + 2

Since y + 8 and 2y + 2 both equal x, it follows that they equal each other : 2y + 2 = y + 8

Subtracting y + 2 from each side we get y = 6

John’s age is 6 and Bill is 8 years older; therefore Bill is 14.

Mathematical logic demonstrates how a series of very trivial steps can eventually lead to an answer that is a considerable distance from the original problem. In the history of science we often observe how scientists made many observations about the Earth, assembled them into premises, and were then able to make useful conclusions.

For example, because coal seams have been found in Antartica (observation/premise), the climate there was once warmer than it is now (sub-conclusion/new premise), therefore either the geographical location of the continents has shifted (possible conclusion to test further) or the whole earth was once warmer than it is now (alterative conclusion to test). The eventual theory of plate tectonics, certainly a beautifully logical argument widely accepted today, arose from building a series of useful premises based on field observations and testing alternative conclusions.

Logic and Truth… to be continued…

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