12 July 2010

Mathematical Logic 1

Motivation by analogy.

Logic studies reasoning. Many people that are not familiar with logic are able to think (reasonably). For example, many children find the explanation “because I say so” less satisfying than “if you do not solve your homework then you will not receive a high grade, therefore you will receive a high grade only if you solve your homework.” A child may not agree that a high grade is a worthy goal, but the point is that the second explanation is somehow more satisfying. Some hidden mechanisms are at work, mechanisms that are transfered to children by training. Children learn how to reason mostly by imitation. I believe it is a very bad idea to teach a child how to reason by exposing him to logic, for that would be akin teaching a child how to walk by exposing him to the laws of mechanics, or teaching a child how to sing by exposing him to music theory, or teaching a child how to speak by exposing him to the rules of grammar. It is preposterous to try to teach a child how to speak by explaining the rules of grammar, because language is necessary in order to communicate the rules of grammar. Such circularity exists in the case of logic too. One cannot learn logic, unless one already knows how to reason.

Are then mechanics, music theory, grammar, and logic useless? Although mechanics does not help people to walk, it does help them to build cars; although grammar does not help people to speak, it does help them to communicate effectively. Mechanics is useful because cars are useful; grammar is useful because the ability to communicate is useful. Note that people built carts before understanding the laws of mechanics and they probably built engines before understanding the laws of thermodynamics. However, cars are better because we understand the laws of mechanics. Similarly, people may communicate without knowing grammar. However, people communicate better if they follow the rules of grammar.

Logic enriches the way people think. A logician immediately parses the sentence “if you do not solve your homework then you will not receive a high grade, therefore you will receive a high grade only if you solve your homework” as “(if (you do not solve your homework) then (you will not receive a high grade), therefore ((you will receive a high grade) only if (you solve your homework))”. Certain logical connectives (like ifthen, not, therefore, only if) are emphasized, because logic cares more about how statements are chained together into an argument, rather than what those statements mean. In fact, when being careful about what they say (or hear), logicians will likely translate on the fly that statement into a symbolic form that makes it easier to visualize the structure: “(¬ p ⇒ ¬ q) ⇒ (qp), where p stands for ‘you solve your homework’ and q stands for ‘you will receive a high grade’”. Here, logical connectives are represented by some standard symbols (¬ and ⇒) and the rest is hidden behind letters (p and q).

5 comments:

Michelangelo said...

I'm finding it hard to associate "(¬ p ⇒ ¬ q) ⇒ (q ⇒ p)" with a result. For me it's simply "q = p" (High grades are the result of the homework being done). Negating one will always negate the other. Anyway, you just sparked my interest in logic again.

rgrig said...

Michelangelo,
You may recall that p ⇒ q is ¬ p ∨ q (sometimes by definition!). Then the tautology from above follows from the commutativity of logical-or and from the fact that two negations cancel each-other.

rgrig said...

... but once you do logic for a while it just comes natural that ¬ p ⇒ ¬ q and q ⇒ p are the same, and you don't need to think of a proof anymore.

iPhone rose said...

I understood this one!

rgrig said...

Thanks, Snoopy. I'll try to write more like this one.

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