As humans, we engage in arguments all the time. To many of us, an argument is simply a collection of sentences that tries to make a point, which is not a bad definition. But to be a little more concise:
An argument consists of premises and conclusions.
Premises are the collection of statements that the speaker regarding as true and does not need to provide support. (We can certainly always asking why those premises are true, but we will soon realize that keep pulling that thread will lead us into an endless chase. As I have discussed last week, in scientific fields, it is the norm to take certain statements to be true, not because we have proven them to be true, but rather we have yet to find empirical evidence to contradicting them).
Conclusions are the collection of statements that the speaker proves (or tries to prove) to be true, using the aforementioned proposed premises.
Before I go continue, I want to point out that the meaning of the word “true” is very nebulous in our daily language. But in mathematics, (handweavingly speaking,) the word true means one statement does not have a contradiction with another statement. If a statement is self-consistent (as most statements are), we can consider it as true (without the need of any justification) and use it as a premise. Notice that a statement can be both true and false at the same time. In such case, we will say that it is inconsistent. Just for fun, an example of a self-contradicting statement (or a paradox) would be:
This statement is false
From now on, when I use the word true or false, I will try to state what I mean. But as my discussion from On Science and Logic, It is a lot easier to prove a statement in a system (I will discuss more what a system is later) to be false, than to prove it to be true. If exists one contradiction statement in the system to the one we are examining, we may conclude that the statement in examination to be false, while to prove a statement to be true requires to demonstrate that the given statement is consistent with all the statements in the system.
Reader might also notice that truthhood can be a unearned status, we can say that something is true if we just want it to be true (but, of course we can also prove something to be true through formal logic).
Now, back to arguments. Here is an example of an argument (A):
All humans are mortal
I am a human
Therefore, I am mortal
The statement “All humans are mortal” and “I am a human” are the two premises that we took as true. “I am mortal” is the conclusion.
But, there is, actually a third premise in the argument above that we often overlook, that is the inference operator:
If all members of a group have property A, any member of the group has property A
The premise above may seem trivial, but without it, we can not infer the property of an individual from the property of the group.
Here is another example of an argument (B):
Some humans are mortal
I am a human
Therefore, I am mortal
We have noticed that both “Some humans are mortal” and “I am a human” are “true” “premises” (unless my parents have not told me something). Here, I am using the word “true” for the meaning of matching our empirical understanding of the universe. “I am mortal” is a “true“ conclusion. But this argument may sound fishy to some readers that’s because, by making this argument, we are adding a third premise for inference:
If some members of a group have property A, any member of the group has property A
Anyone who has learned about set theory in their math class would point out that the premise above is false. But actually, because it is a premise, and we accept the premise to be true. It can’t be false. In fact, if we list every statement we have used in argument (B):
Some humans are mortal
I am a human
If some members of a group have property A, any member of the group has property A
I am mortal
There is no contradiction in the system (I will elaborate more on its definition later) composed of the four statements. So what is going on?
Let’s take a look at the argument (C):
Some mammals are cats
I am a mammal
If some members of a group have property A, any member of the group has property A
Therefore, I am a cat
I have simply replaced the word “humans” with “mammals“, and “mortal” with “cat(s)“. So following the same structure, we can conclude that there is also no contradiction in the argument above. In fact, there is no contradiction in the argument above. But when we think that the conclusion “I am a cat” is False, what we really think is that the conclusion contradicts the empirical evidence. So, when we add empirical evidence to the collection of statements, we now have an inconsistent system:
Some mammals are cats
I am a mammal
If some members of a group have property A, any member of the group has property A
I am a cat
Empirically, I am not a cat
The examples above also show that it is easy to create a consistent system of statements with some empirical evidence (often, what psudo-science trying to show). But, it is really really hard to create a consistent system of statements with all empirical evidence (what scientific method tries to show).
So, instead of saying something to be true or false, maybe what we really want to say is if it is consistent or inconsistent.
Now I can give a formal definition to what a (formal) system is:
A formal system is the collection of the selected premises, and all the conclusions can be drawn from the premises.
It is important that a formal system must include all the conclusions. We can’t pick and choose what is in the system. For example, if I exclude the conclusion “I am a cat” from argument (C) which leads to the following collection of statements:
Some mammals are cats
I am a mammal
If some members of a group have property A, any member of the group has property A
Empirically, I am not a cat
There is no direct contradiction in this collection of statements, but the formal system is still inconsistent because we can draw a conclusion that is inconsistent with one of the premises.
One pair of consistent statements, even 1,000 pairs of consistent statements can not show that the formal system is consistent, we will need to show that all pairs of statements are consistent. But only one pair of inconsistent statements is needed to show that the formal system is inconsistent. We can’t pick and choose what conclusions to include the system to make it consistent.
So, when we disagree with someone’s argument, it can be one of the two reasons:
- Either we think their formal system of beliefs is inconsistent within itself.
- Or we think their formal system of beliefs is inconsistent within ours.
If we think someone’s formal system of beliefs is inconsistent within itself, we can construct a case using only statements from their system (without adding any premises of our own) to show such inconsistency. If the interlocutor cares about having consistent beliefs, he/she would be grateful for this counterexample.
If we think their formal system of beliefs is inconsistent within ours. Before we start criticizing others’ choice of premises, remember that premises are statements that we have taken as true, our choice of premises are just as valid (or invalid) as their choice. By choosing what lens we look at the problem through, we can see it in different colors. Maybe no perspective can give us a whole picture of every problem, but many of them can reveal different and important information. Remember, what we think is wrong in others’ premises, is exactly what they saw wrong in ours.
But first, let us examine if our own system of beliefs is consistent. In the next post, I will discuss Gödel’s incompleteness theorems and what happens if we do not have a consistent formal system of beliefs.
P.S. I want to use the postscript to talk a little about formal logic and informal logic. Formal logic is deductive reasoning that is defined in mathematics and has not contradicted by empirical evidence. “If all members of a group have property A, any member of the group has property A” is an example of formal logic, defined in set theories. Informal logic is reasoning that we use that sometimes will contradict by empirical evidence. “If some members of a group have property A, any member of the group has property A” is an example of informal logic. There are two commonly used informal logic inferences: inductive inference and abductive inference.
Inductive inference is generalization (not to be confused with mathematical induction). For example, if I observe one duck be white, and a second duck to be white, then I conclude that all ducks are white, which is inductive inference. Readers can very easily find cases that using inductive inference will lead to conclusions that contradict the empirical evidence.
Abductive inference is the process of elimination. When we have eliminated all the impossible, what remains, how improbable, must be the truth. But in reality, there is often an infinite number of possibilities, and we can rarely eliminate all the impossibles.
So, informal logic is inconsistent with empirical evidence. Therefore any system that includes both informal logic and empirical evidence is inconsistent. But informal logic is also extremely useful, scientists often use them to form up hypotheses, but then design experiment and use formal logic to test them. (Note, in this case, informal logic is not included in the system, only the proposed hypotheses, and we can test if it is consistent with empirical evidence).
RhaPsoDySea 
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