On Logic

[img]http://imgs.xkcd.com/store/imgs/formal_logic_store_1.png[/img]

Formal logic is a great tool for use in mathematical proofs, scientific papers, and even forum debates. Even better, it is relatively simple to understand. There are a few main elements:

[b]Conditional[/b]: [i]p -> q[/i]
Conditionals are statements written in if-then format. They are statements structured as "If [hypothesis], then [conclusion]." An example of this is "If the sky is blue, then it is not raining." In that conditional, the hypothesis- the cause- was "the sky is blue", and the conclusion- the effect- was "it is not raining". Note that both the hypothesis and the conclusion are statements by themselves, with a subject and a predicate. These can also be mathematical statements, as in "If (x+1)7=28, then x=3."

[b]Inverse[/b]: [i]not p -> not q[/i]
The inverse of a conditional is when you negate both the hypothesis and the conclusion. The inverse of "If the sky is blue, then it is not raining." is "If the sky is not blue, then it is raining." (Note that the double negative in the conclusion was cancelled.)

[b]Converse[/b]: [i]q -> p[/i]
The converse of a conditional is when you switch the hypothesis and conclusion statements. The converse of "If the sky is blue, it is not raining." is "If it is not raining, the sky is blue." The converse is not always true; it can be helpful for analyzing a statement.

[b]Contrapositive[/b]: [i]not q -> not p[/i]
Essentially, the contrapositive is the inverse of the converse of a conditional statement. The contrapositive of "If the sky is blue, it is not raining." is "If it is raining, the sky is not blue." Logically, the contrapositive agrees with the conditional.

[b]Definition[/b]: [i]p IFF q[/i]
If both the conditional and converse forms of a statement is true, it can be written as a definition. Definitions are if-and-only-if, or IFF, statements, meaning that they are formatted in a "[hypothesis] if and only if [conclusion]" structure. If you look at our mathematical conditional, "If (x+1)7=28, then x=3," you can see that the converse, "If x=3, then (x+1)7=28." is also true. Thus, "(x+1)7=28 if and only if x=3."

[b]Law of Detachment[/b]
The Law of Detachment is one of the laws of formal logic. Basically, it states that, if the hypothesis of a conditional is true, then the conclusion is logically true. If the statement "If the sky is blue, it is not raining," is true, and "the sky is blue" is true, then "it is not raining" must also be true.

[b]Law of Syllogism[/b]
The Law of Syllogism states that if you have two conditionals, and the conclusion of one is the hypothesis of the other (as in "p -> q" and "q -> r"), then the statements can be combined ("p -> r"). Suppose that you have "If the sky is not blue, it is raining," and "If it is raining, the grass is wet." Using the Law of Syllogism, you can conclude that "If the sky is not blue, the grass is wet." Not the best example, but it functions.

[b]Law of Identity[/b]
The Law of Identity essentially states that A is A or, more generally, that anything is itself. This is a sort of obvious law: 3=3; a pig is a pig. Existence necessitates identity; to lack identity is to lack existence. As Aristotle said, [quote]Now 'why a thing is itself' is a meaningless inquiry (for -- to give meaning to the question 'why' -- the fact or the existence of the thing must already be evident'-e.g. that the moon is eclipsed-but the fact that a thing is itself is the single reason and the single cause to be given in answer to all such questions as why the man is man, or the musician musical', unless one were to answer 'because each thing is inseparable from itself, and its being one just meant this' this, however, is common to all things and is a short and easy way with the question).[/quote]

However, self-evidence is not a sign of unimportance. The Law of Identity makes explicit that reality has a definite nature. Since it exists in a particular way, it has characteristics. Since reality has an identity, it is knowable. This is the basis for all of science.

[b]Law of the Excluded Middle[/b]
The Law of the Excluded Middle states that anything is either A or not A. If P is a proposition, it can be understood to mean "P or not-P." For example, if the statement is "Socrates is mortal," the Law of the Excluded Middle holds that "Socrates is either moral or not mortal" is logically true. There is a similar rule, the Principle of Bivalence, which states that every proposition is either true or false. This is different from the Law of the Excluded Middle, however, as the Law is also applicable to logic where a statement is true, false, or indeterminate. To see this, consider that 'P' means 'it is true that P' and 'not-P' means 'it is not true that P', from which it does not immediately follow that 'p is false'.

[b]Law of Noncontradiction[/b]
The Law of Noncontradiction states that nothing can be A and not-A, or, alternately, that a proposition cannot be both true and false. No one can believe that the same thing can, at the same time, be and not be. This law defines a counterexample as a means of disproving a statement, just as 2 is a counterexample of the statement "All prime numbers are odd." Some logicians, belonging to branches like dialetheism, a philosophy which holds that both a proposition and its negation can be true, contest this law, but, as this law defines a contradiction, any attempt to contradict it must rely on the acceptance that contradictions are false. As Avicenna said, [quote]Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned.[/quote]

An argument is a finite series of coherent propositions that support a conclusion. Knowing how to skillfully conduct an argument is key to battling the ignorance and "cancer" plaguing the Internet. Hans Eysenck was a prominent psychologist before his death in 1997. He assembled a list of rules or guidelines for successfully conducting a public argument based on personal experience and advise from his father, a nightclub entertainer. The rules are:

[quote]1. Never argue about something about which you are fundamentally ignorant.[/quote]
This is the number one rule of argument, as indicated by its high position. If you do not know what you are talking about, you cannot win a debate and will look like you are a fool. Eysenck notes that adherence to this rule alone would dramatically reduce the number of arguments in the world.

[quote]2. Do your homework, so that you really know everything there is to know about the topic in question.[/quote]
Essentially, this rule states to fully know your opponent's position. If you go into a debate in which you actually agree with your opponent, it can be quite confusing, misleading, and embarrassing. This ties in to the first rule: Know what you are talking about!

[quote]3. Keep what you have to say short, because if you go on for any length of time the audience will forget the points you are making.[/quote]
Be pithy. Verbosity is useful when conducting a lecture, not a debate.

[quote]4. Concentrate on the most important points, and don't go hunting after those that matter less.[/quote]

[quote]5. Having decided what are the most important points, force your opponent to answer these points, and don't let him [or her] escape by dragging in all sorts of irrelevant matters.[/quote]
Stay on topic, and make sure that your oppenent does, too!

The modern expression of logic and rationality over the Internet is skepticism. The Bayesian system of reevaluating one's own beliefs falls hand-in-hand with skepticism. Quite often, this leads to aggravation towards the inability of others to change their minds or acknowledge logic. Occassionally, skeptics are accussed by these others of being "trolls", an understandable confusion. It arises from the pandemic problem of pseudoskeptics.

The difference between true skepticism and pseudoskepticism is highlighted in the article [url=http://truthfall.com/pseudoscepticism/]Pseudoscepticism[/url] from the website [url=http://truthfall.com/]TruthFall[/url]. They correctly define true skepticism as objective inquiry and evidence-seeking that challenges all sides of a debate, including the skeptic's own beliefs. The goal is to seek and find the truth, no matter where the chase leads. A real skeptic can admit that he or she is wrong when faced with the evidence against his or her argument. Skeptics overcome bias and argue for the side with the most logical, rational, or evidential value in any conflict. Quoth Descartes on the duties assumed by being a skeptic:

[quote]The first was never to accept anything as true that I did not know to be evidently so: that is to say, carefully to avoid precipitancy and prejudice, and to include in my judgements nothing more than what presented itself so clearly and so distinctly to my mind that I might have no occasion to place it in doubt.[/quote]
Or to say, to avoid bias from influencing judgement and delve only in known facts.

[quote]The second, to divide each of the difficulties that I was examining into as many parts as might be possible and necessary in order best to solve it.[/quote]
To reduce the problem or argument into as many parts as possible: the great art of Reductionism. To best examine, define, or explain a quality of something, reduce it into parts that no longer contain that quality.

[quote]The third, to conduct my thoughts in an orderly way, beginning with the simplest objects and the easiest to know, in order to climb gradually, as by degrees, as far as the knowledge of the most complex, and even supposing some order among those subjects which do not precede each other naturally.[/quote]
To make connections; to build upon concepts in order to know, understand, and be able to explain each one before moving on to the next. Just as you broke the object with the quality down into parts which do not exhibit the quality, you now build it back up, bit by bit, examining how each connection creates or retains the quality.

[quote]And the last, everywhere to make such complete enumerations and such general reviews that I would be sure to have omitted nothing.[/quote]
To make a thorough examination of all available evidence for and against each and every explanation or argument before passing judgement and reaching a conclusion.

This is the process, the duties, of skepticism. The ordinary manifestation of logic and rationality, able to be applied to all fields. But then there are the pseudoskeptics. These people, posing as rational or logical individuals, have and argue for a pre-defined agenda. In the process, they immediately dismiss any opposite ideas or theories, no matter how much proof supports the other hypothesis. This is not an issue connected solely to skepticism; members of all beliefs or systems thereof can or will dismiss ideas opposing their ideology and shun those that express these ideas, all the while professing themselves to be "open-minded". However, it is most common for supporters of mainstream ideology to claim to be skeptics.

However, pseudoskepticism is [b]not[/b] skepticism. It should not be even confused or associated with skepticism. It soils the name and image of logic and rationality on the Internet. It is mere faith-based disbelief, and its practitioners should be [i]ashamed[/i] of themselves. The pseudoskeptics appear to move parallel to universal implementation of rationality and logic, but, in actuality, do more hurt than help. To restore the face of logic, rationality, Bayesianism, and skepticism, we must reveal the differences between the skeptics and their imitators in order to maintain our fight for the truth.

Seraphnb