A Guide to Logic and Reasoning Pt. 2

A Guide to Logic and Reasoning

Introduction

         In the last piece we discussed the concepts of the Logical Absolutes, Reality, Occam's Razor, and Pragmatism. In this piece, we're not really going to introduce many new things, but rather demonstrate how to use these things to formulate arguments, evaluate propositions, and how to gather conclusions from your propositions. That said, let's get into the thick of it by starting with the most basic unit in logic, the Proposition.

Propositions

         A proposition in it's most basic sense is a statement, that's all. "It's a nice day today." is a proposition, as is "I've painted my house blue." These propositions aren't the only kind either, propositions don't even have to make sense. For example, "eijbnlaisjdbnfoqnwewerbg" is a valid proposition as well. These things we call propositions are what we use to formulate arguments and arrive at conclusions about our world. Think of it as a color wheel. If you mix blue and yellow you get green. In much the same way, if you put the proposition, "I don't like rainy days." with the proposition, "It's is a rainy day." you can arrive at the conclusion "I don't like this day." this kind of stacking propositions to arrive at conclusions can be summarized as follows.

A
B
therefore C

These are generally written with the following notation, where P stands for proposition, and C for conclusion.
P1 (Read as Proposition/Premise 1)
P2 (Read as Proposition/Premise 2)
Therefore C (Therefore, conclusion)

         This is the basic Logical Form called Syllogism. In any argument of Syllogism, two propositions must be linked by some attribute, (Rain in this case) to draw a relationship between the other elements in the propositions, (Myself, and the day it is). Now, Syllogism isn't the only form an argument can take but it is the most common and easiest to understand. Think of it as linking chains together to build a longer chain. The first proposition and the second proposition must have something in common with their statements, this is their link. Now, you can then add a third proposition into the argument and link it with something in either the first or second proposition. You can keep on doing this even across multiple arguments by using the conclusion of one argument as the proposition for another. Before we can start bending the rules, however, let's learn what the rules of the game are.

Is it Valid? Is it Sound?

         There's absolutely nothing holding you from putting random propositions together and seeing what you get, but as we discussed in the last piece logic, is a tool and it must be used carefully. This is where the issues of Soundness and Validity come into play. Consider the following famous syllogism.

P1 All men are mortal
P2 Socrates is a man
C Socrates is mortal

         A simple conclusion to reach. The first proposition defined a quality of all men. The second proposition defined an object as being a man, thus, we can conclude this object called Socrates has this quality. This is a classic example of something that is Valid and Sound. The Validity comes from the fact this syllogism is true in all circumstances. It doesn't matter what Socrates is, as long as he is in the group labelled "Man" this statement is true, and if he is not, then the second proposition was false to begin with. The Soundness comes from the argument itself, rather than the circumstances surrounding it. In order for something to be Sound it merely has to follow from these premises, regardless of the propositions' truthfulness or falsity. Thus, if something is valid, it follows that it must be sound seeing as validity applies to all circumstances while soundness just involves the propositions and conclusions at hand.

          So, you may ask, are there any situations wherein the syllogism is sound but not valid? To which I will say yes, there are. Consider the previous syllogism, and I'll mix some words around.

P1. All men are mortal
P2. Socrates is mortal
C. Socrates is a man.

          Now on the surface this makes sense. The P1 defines a quality of the group 'men' and P2 defines an object call 'Socrates' as having this quality. However, this is not how one is to properly Syllogism. The previous Syllogism works because 'Socrates' is an object defined as being apart of the group 'men'. In this fashion, there's no way he could not have this quality of 'mortal' since all objects in the group 'men' have it. This syllogism does things backwards by instead defining a group 'men' to have the quality of 'mortal' and then defining an object 'Socrates' to have this quality. By doing this, the argument assumes no other group has this quality of 'mortal' and falls flat when one finds out 'Socrates' is actually a dog in the group 'Animal.' Now, the argument is still sound, mind you, since the conclusion has the potential to be true given the premises. It isn't valid, however, since there exists at least one scenario where the conclusion can be wrong.

Argument Construction

         So, at this point I've given you the knowledge of what constitutes an argument, and what makes an argument good or bad. Now I'm going to teach you how to put an argument together. For this exercise I'll be building an argument against the Geocentric Conjecture. First, any argument needs premises, and for that, we'll need propositions to fill in those slots, preferably propositions that have something to do with the argument at hand. I'll pick a few. First, I'll take the proposition, "Geo-stationary Satellites orbit Earth at a distance from Earth." I'll next take the proposition, "Under the Geocentric Conjecture, Neptune orbits Earth at a distance from Earth different to that of Geo-stationary Satellites." The second to last proposition I need next, "Newton's Constant of Gravitation, and the formula that goes with it, calculates Earth to have two different masses in relationship to Geo-Stationary Satellites and in relationship to Neptune." And finally, "It is not possible for an object to have more than one measurement of mass in one moment." Now let's see how this stacks up.

P1. Geo-Stationary Satellites orbit Earth

P2. Under the Geocentric Conjecture, Neptune orbits Earth at a distance different to that of Geo-Stationary Satellites.

P3. Newton's Constant of Gravitation, and the formula that goes with it, calculates Earth to have two different masses in relationship to Geo-Stationary Satellites and in relationship to Neptune.

P4. It is not possible for an object to have more than one measurement of mass in the same moment.

(From Premise four, we can conclude something already, written here as C1.)

C1. It's impossible for an object to have two mass measurements, regardless of when it occurs.

C2. Since it's impossible for Earth to have more than one mass measurement, and the Geocentric Conjecture requires this to be the case, the Geocentric Conjecture as is stands is False by contradiction.


          Here, you'll notice something special about this process. From my premises, I concluded first C1, and then used that as a new proposition to lead into C2. Also, concerning C2, there's something special about that conclusion. Whereas Syllogisms and logical forms attempt to use previous propositions to build new ones in order to gather information, arguments on the other hand have a specific proposition they are trying to reach. In the case of this argument I was looking for a proposition that negated the Geocentric Conjecture. Whenever you're making an argument for some kind of proof, or debate, you always must identify what goal you're wanting your argument to achieve. Once you know what your argument is supposed to do, you then look for those propositions that build your bridge of logic to get you to where you want to be. As a fun exercise, try to form an argument proving Earth orbits the Sun.

Differences in Reasoning

          Reasoning is, simply speaking, the relationship between your premises and conclusions, and how you establish them. There are two main types of reasoning, the one we've seen so far is called Deductive Reasoning. In this method of reasoning we are taking our propositions and establishing a conclusion based off of them. There's another kind of reason, however, known as Inductive Reasoning. With this kind of reasoning, you don't start with a series of propositions to arrive at a conclusion. Instead, the conclusion you reach won't be absolute. Instead, inductive reasoning only deals with degrees of certainty, but never fully guarantees the conclusion will always be true. Take a look at the following inductive argument, using inductive reasoning correctly.

P. The genetic code for dogs thus far dictates a dog has 4 legs
C. Therefore, Most dogs will have four legs.

         You can see how the conclusion doesn't guarantee all dogs you encounter will have four legs, it simply states that's the most likely case. Now let's take a look at another version of this argument, using inductive reasoning improperly.

P. The genetic code for dogs thus far dictates a dog to have four legs.
C. Therefore, Every dog has four legs

         Now, in this case, this argument is not true, using inductive reasoning deals with generalizations and thus will never give you a definitive answer for all cases. When you include a definitive conclusion you are either inducting improperly, or using deductive reasoning. In this case, there are several dogs that don't have four legs from birth defects, injuries, etc.

          So, you may be asking yourself the following question. "If Inductive reasoning can't give us absolute statements of Truth and Falsity, what good is it for?" To which I will say Inductive Reasoning has its uses. For example, Astrobiologists are scientists that postulate alien forms of life to help us search for where life in the universe might be. When it comes to alien life though, there's one saying NASA has above all else, "Follow the water." The argument is formulated asfollows.

P. All biological life we have encountered thus far requires some form of water to survive.
C. Therefore, most life in the universe depends on water to survive.

         Now, this conclusion has yet to be verified, but here is just one example of where real professionals use inductive reasoning in their line of study. It doesn't matter if the Conclusion is true or not, because truthiness isn't what it claims. All this argument claims is a probability of being correct, not a definite answer one way or the other.

Conclusion

         By this point in the piece you now know what a logical argument is, what it's meant to do, and how to construct one as well as how to tell if one is badly put together. In the next piece we'll be covering the various way to disprove certain premises and arguments, as well as the numerous fallacies that are often committed by debaters as well as how to expose them to your advantage. Until then, I'll leave the following controversial logical arguement for you to puzzle over, tell me in your review what your answer is to this argument. Simply tell me if it's Valid or Invalid, and if you answer Invalid, construct a counter argument to disprove it. Until next time!

P1. God is defined as being perfectly merciful

P2. God is defined as being perfectly Just/Fair

P3. If a soul is judged by God, the result must perfectly fit the extent of the soul's misdeeds in order to be perfectly just.

P4. If a soul is judged by God, the result must not contain any punishment or reprehension to be perfectly merciful.

P5. Complete forgiveness without punishment cannot fit any crime to the its perfect extent.

C1. The qualities for God defined by P1 and P2 are paradoxical to each other.

C2. this god, who is both perfectly merciful and perfectly just, cannot exist.