Validity, Truth, and Soundness

Validity

Truth Premises, Conclusions, and Support

Arguments

Conclusions or Main Claims

Premises

Support

Assumptions, Suppositions, Axioms

Evidence

Authority

Explanations and Anecdotes Introduction to Inference identifiers

The language in which an argument is presented often contains words or phrase to help identify its parts, especially its premises and conclusion. These words and phrases are identifiers of the function played in the argument. Unfortunately, identifiers are only as precise as the persons using them, and both the individual making an argument and the one evaluating it are liable to make mistakes by inexact or sloppy use of identifiers. Since the purpose of an argument is to communicate an idea clearly, the careful use and interpretation of identifiers is an important skill for critical thinking.

The following are some of the most common premise and conclusion identifiers:

These are only a few of the words and phrases commonly used to identify premises and conclusions. In addition, keep in mind that:

1. some of these words can also appear within the context of an argument, but without indicating an inference. "So," for example, has several meanings, only one of which is a synonym for "therefore."
2. sloppy usage may produce confusing identifiers. A common answer to the question, "What would you think if the sky suddenly clouded up and turned very dark," is "I would assume it was going to rain." Yet "it is going to rain" here is a conclusion, not an assumption or premise.
3. "if" and "then" are often used to identify premises and conclusions, respectively. However, "if" and "then" are also used to introduce the two halves of a conditional premise. In either usage, "then" is sometimes omitted; and it has other meanings, as well.
4. an identifier may not immediately precede or follow the word or phrase whose function it is indicating. For example, in the sentence "Thus, whenever the sun rises, the rooster crows," there are two claims: a premise, "the sun rises," and a conclusion identified by "thus" (but not immediately following it), "the rooster crows."
5. in cases where there are no identifiers, the most frequent order is conclusion first, followed by one or both premises. If both premises are given, they are often conjoined with "and" or "but." For example, "I like Mozart. I like most classical composers, and Mozart was a classical composer."

Hierarchy of Support

Facts and Opinions

Burden of Proof

Soundess

The first rule in evaluating any argument is never bother to disagree with a conclusion, because if you find nothing wrong with its form (or how the argument is made) and nothing wrong with its content (or the assumptions on which the argument is based), then you must accept its conclusion. As a result, to challenge an argument, you must challenge either its form or its content, not its conclusion directly. Because we can always evaluate the form of an argument, but not always its content, the process of analyzing an argument usually begins with its form.

Validity. When the form of an argument is acceptable, that is, when its premises and conclusion are in the proper relationship, we say that the argument is valid. A valid argument, then, is one that is in an acceptable form; and invalid argument is one in an unacceptable form. Rules for determining the validity of an argument are given in the sections on inductive and deductive reasoning. If an argument is found to be invalid, all judgment of its must be suspended because, to be acceptable, an argument must be valid. The conclusion of an invalid argument is not necessarily wrong; because of the invalidity, there is simply no way to evaluate that argument.

Truth. If, however, the form of an argument is found to be valid, then the content of its premises must be evaluated, to determine if they are true or false. A true premise is one that you believe has or can be verified, or is self-evident, in the case of a verifiable statement, or has or can be justified, or is self-evident, in the case of an evaluative or advocatory statement. The verification or justification usually comes in the form of support, such as evidence, expert opinion, and supporting arguments.

As a general rule, in judging premises and their support, you should accept as verifiable or justifiable all claims that follow these three rules:

1. They are not in conflict with what you know or understand to be true.
2. They do not require you to believe or accept other unsupported elements that are in conflict with what you know or understand to be true.
3. They bear the proper burden of proof.

Burden of Proof refers to the sense you have, in any dispute, of how much each side needs to prove in order to win your agreement. Sometimes, this burden of proof is an established rule: in the United States, for example, the criminal court system operates on the rule that a person is innocent until proven guilty, which means that the prosecution carries all of the burden of proof; if the defendant is not proven guilty, then he or she should not be convicted of a crime, even if the defense cannot or does not prove him or her innocent of that crime.

Generally, by initiating a claim one takes on a greater degree of the burden of proof than the same position would warrant otherwise. If, for example, Warren said, "California became a state in 1850," he would be expected to offer more proof for his position than if someone else said "California became a state in 1851," and Warren disagreed. In an easily verifiable case like that, the burden of proof is almost even, so the person making the claim is usually expected to support it first.

In most arguments, however, it is usually the side that supports altering or rejecting the status quo--the current beliefs, practices, and information--which has most of the burden of proof. The more controversial the matter, generally speaking, the more evenly is the burden of proof shared by all sides; and the more extreme or unusual one side of an argument is, the greater its burden of proof. In such extreme cases, initiating the claim is normally insufficient to offset the burden of proof. Thus, if Aziza says, "I don't believe in ghosts," we might be willing to accept her claim without any support, even though she has initiated it, because the burden of such an argument would be carried overwhelmingly by the side that supports a belief in ghosts.

Intentionally shifting the burden of proof, in order to avoid offering support for one's premises, is a logical fallacy.

Consider the following arguments:

1.       I can prove there is life on Mars. Samples of Martian rocks show evidence of the kind of chemical reaction that can only involve a living organism.

2.       I can prove there is life on Mars. Spectroscopic analysis through the Hubble telescope has revealed a purplish area on the Martian surface, and according to Mozyritzski's Second Law, that purplish area must be associated with living organisms.

3.       I can prove there is life on Mars. A spaceship filled with Martians abducted me last night.

4.       Prove there is life on Mars? Can you prove there isn't?

The fourth one is the easiest to deal with: at the minimum, a claim of life on Mars carries some of the burden of proof, and therefore has to be substantiated. The fact an opponent cannot disprove the claim is insufficient for the claim to be accepted; it must be proved. The third argument makes the same claim and does support it, but the support (that the speaker was abducted by Martians) requires you to believe something else that is itself unsupported and even more unusual. The second argument is similar to the third, although it may be easier to accept Mozyritzski's Second Law (whatever that is) than Martian abductors; we can reject Martian abductors without further consideration, but to accept or reject an argument based on Mozyritzski's Second Law, we first need to find out what it is, whether it applies in this case, and how accepted it is generally. The first argument was, in fact, made by scientists in 1996, and it is certainly the most creditable of the four examples here. That "chemical reaction" may be no more verifiable than Mozyritzski's Second Law, but it is more accessible. (In fact, other scientists soon disupted the claim.) So, as presented above and without further support, those four arguments appear in descending order of their acceptibility. Yet even the claim, "There's no life on Mars," would carry some of the burden of proof, if for no other reason than someone initiated it.

Soundness. Finally, if an argument is valid and its premises are true, it is termed a sound argument, and its conclusion must be accepted. In many cases, however, there is insufficient reason to find the premises of a valid argument totally true; the more complex the argument, the less likely that it will be considered undeniably sound. In such cases, we often talk of the "relative soundness" of an argument by describing it as strong or weak. A strong argument is valid in form, and with premises and support that make a compelling case for its acceptance. A weak argument is also valid in form, but its premises and support do not compell their acceptance.

The Rules of Validity

An argument must meet all of the following conditions to be valid. Failing to meet one or more conditions shows an argument to be invalid.

1. The middle term must be distributed at least once.
2. If a term is distributed in the conclusion, then it must be distributed in its premise.
3. If one of the premises is negative, then the conclusion must be negative, and if the conclusion is negative, then one of the premises must be negative.
4. There must not be two negative premises.

Immediate Inference

Immediate inference provides tools useful for determining whether a proposition is true, false or indeterminate after a number of manipulations have been performed on it and given its initial truth or falsity. Immediate inference is somewhat similar to an argument with one premise and one conclusion. Distribution is not a kind of immediate inference, but will prove useful in understanding immediate inference.

Distribution

A term is distributed if its proposition makes a claim about each and every member of that category. With the A proposition "All men are pigs." for example, a claim is being made about each and every man. But the same is not true for the predicate term: pigs. Other pigs, in addition to those referred to, may exist. All A propositions are the same in this regard: the subject term is distributed and the predicate term is not distributed. Similarly, all other categorical propositions are consistent about which terms they do or do not distribute. E propositions distribute both the subject and the predicate terms. I propositions do not distribute either term. O propositions are a little contrary to intuition. Obviously O propositions do not  refer to all members of the subject class. But they are said to refer to all members of the predicate class. The idea is that, from the few of the subject class referred to, the entire predicate class is excluded. For example, in "Some cows are not clean." the each and every member of the clean things is excluded from the subgroup of cows referred to by "some cows".

Two Rules:

• The subject of any Universal proposition is distributed, but the subject of any Particular proposition is not.
• The predicate of any Negative proposition is distributed, but the predicate of any Affirmative proposition is not.

Conversion

 Conversion is the manipulation which simply switches the subject and predicate terms.

A Propositions

The converse of "All cars are automobiles." would be "All automobiles are cars." From the truth of the former one cannot tell whether the latter is true or false so it is indeterminate. Someone might object that the latter is clearly false, since trucks are automobiles. But one knows that from outside information rather than from logical inference. To see that one cannot rely on the converse of a true A proposition being false consider: "All bachelors are unmarried males." The converse is "All unmarried males are bachelors." which is happens to be true. Similarly, if one starts with a false A proposition such as "All crows are pink." the converse "All pink things are crows." is indeterminate. Again this particular example might seem to produce a false, but consider that "All mammals are cows." which is false converts into "All cows are mammals." which is true. In sum, conversion is not legitimate for A propositions.

E Propositions

The converse of the true proposition "No bachelors are married." is the also true statement "No married people are bachelors." The converse of the false "No crows are birds." is the false "No birds are crows." This result is not a feature of the examples we happened to choose or our having outside information. If all of one group is excluded from another, then obviously all of the other group must equally be excluded from the one. Conversion is legitimate for E propositions.

I Propositions

The converse of the true "Some crows are albinos." is the true "Some albinos are crows." and the converse of the false "Some cows are purple." is the equally false "Some purple things are cows." As with the E proposition, this is always the case. If some members of one kind of thing are members of another kind then, since the two kinds must have members in common in order for this to be true, some members of the other kind must be members of the first kind. And in order for an I proposition to be false,  no members can be common between the two kinds, which will equally make the converse of the original I proposition false. Conversion is legitimate for I propositions.

O Propositions

The converse of the true "Some cows are not brown." is the indeterminate "Some brown things are not cows." As in the case of the A proposition, if you think the converse is true, consider the true "Some cows are not pregnant holsteins." which becomes "Some pregnant holsteins are not cows." which is false. The converse of the false "Some cows are not female." is the indeterminate "Some females are not cows." Again if this converse looks true, consider "Some holstein cows are not female holsteins" which becomes the false "Some female holsteins are not holstein cows." (Assuming that cow properly refers to only female bovines.)

Note on Distribution

The fact that E and I are symmetrical in distributing or not distributing their terms is significant here. In the case of the unsymmetrical A propositions where only the subject is distributed, converting results in a claim about every member of the former predicate class. But the basis of the new claim is supposed to be the original claim which did not make a claim about every member of the predicate class. Thus, about that class we end up claiming more than we were given as true (or false). Nothing justifies the stronger claim about that class.

Partial Conversion (Conversion by Limitation)

A propositions may be legitimately partially converted. Consider the true "All ducks are birds." The subaltern will also be true: "Some ducks are birds." which can be legitimately converted into "Some birds are ducks." Partial conversion, then involves taking the subaltern first and then converting. Partial conversion only is legitimate for true A propositions.

Obversion

Obversion is a manipulation involving two changes: the quality is changed (either from affirmative to negative or negative to affirmative) and the compliment of the predicate class is substituted for the predicate term. The compliment of a class is the class of everything not in the original class. The class of non-dogs is the compliment of the class of dogs. Equally the class of dogs is the compliment of the class of non-dogs. We will take the prefix "non" to be trustworthy in picking out the compliment of a class. Other prefixes are not trustworthy. The class of immature things is not the compliment of the class of mature things, because some things are neither. Obversion is always legitimate.

A Propositions

The obverse of "All believers are heaven bound." is "No believers are non-heaven-bound." If the proposition is true so is the obverse and if the proposition is false so is the obverse. The obverse of an A proposition is always an E.

E Propositions

The obverse of "No reptiles are birds." is "All reptiles are non-birds." If the proposition is true so is the obverse and if the proposition is false so is the obverse. The obverse of an E proposition is always an A.

I Propositions

The obverse of "Some cars are Fords." is "Some cars are not non-Fords." If the proposition is true so is the obverse and if the proposition is false so is the obverse. Notice that the word "not" and the prefix "non" perform different roles logically. The "not" is part of the copula and determines the quality of the proposition. The "non" is part of the predicate term and helps identify the predicate class. The obverse of an I proposition is always an O.

O propositions

The obverse of "Some cars are not Fords." is "Some cars are non-Fords." If the proposition is true so is the obverse and if the proposition is false so is the obverse. Notice that the word "not" and the prefix "non" perform different roles logically. The "not" is part of the copula and determines the quality of the proposition. The "non" is part of the predicate term and helps identify the predicate class. The obverse of an O proposition is always an I.

Contraposition

Contraposition is also a manipulation involving two changes: both terms are replaced by their compliments, and the terms are switched (as in conversion). Contraposition is legitimate for A propositions and O, but not for E and I. One may use the other forms of immediate inference to derive the contrapositive over three steps: first obvert, second convert and third obvert again. Performing these steps will confirm that contraposition is legitimate for A and O, but not legitimate for E and I. The reason is the middle step of conversion that is not always legitimate.

Summary

In the following table 'S' stands for the Subject term in the original proposition, and 'P' for the Predicate term in the original proposition. Except where marked Not Legitimate if the original proposition is true, the resulting proposition is true, and if the original proposition is false, then the resulting proposition is false. (Note that Partial Conversion is not included in this table.)
 

For more information see Garth Kemerling's Web Page on Immediate Inference.

http://www.candleinthedark.com/syllogisms.html

Once the task of forming a categorical syllogism in standard form is accomplished, validity of the form can be ascertained according to the following table

Figure 1: AAA, EAE, AII, EIO
Figure 2: EAE, AEE, EIO, AOO
Figure 3: IAI, AII, OAL, EIO
Figure 4: AEE, IAI, EIO

In the traditional interpretation of categorical propositions from the square of opposition, more valid forms exist. I present here only the modern forms, which are far more stringent.

Ok, lets try deconstructing an argument. Lets imagine you are offered the following, rather mind numbing argument, that you wish to destroy on simple grounds of validity...

Original argument:

No Republicans are Democrats, so no Republicans are big spenders, since all big spenders are Democrats.

The first thing to do is to figure out the major Premise - the point of the argument. This is the predicate of the conclusion.

The word "so" is a tip off that "no Republicans are big spenders" contains the conclusion. It labels the preceding statement as a support for it, and the last statement is invalidated as well by the word "since", which marks it as a premise as well.

P = big spenders (the predicate of the conclusion)
S = Republicans (the subject of the conclusion)
m = Democrats - occurs in both premises.

In standard form, this argument becomes: Categorical proposition type:
All big spenders (P) are Democrats (m) A
No Republicans (S) are Democrats (m) E
Therefore, no Republicans (S) are big spenders (P) E

This form is figure number 2 , specifically AEE-2

A quick check on the list tells us that , unfortunately, this argument is unconditionally valid. (Although it should still prove a simple matter to dispute the first premise informally...)

Rules and Fallacies The first two rules have to do with the distribution of terms, and the last three have to do with the quality and quality of the propositions in the syllogisms. When any of these rules are broken, a corresponding formal fallacy is committed.

DISTRIBUTE To use (a term) so as to include all individuals or entities of a given class.

In an A proposition, the S is distributed, the P is not
In an E proposition, S and P are both distributed
In an I proposition, S and P are both undistributed
In an O proposition, S in undistributed, P is distributed

Now, because I find the concept of distribution complex, I want to delve into it further.

In the sentence "All S are P", S is extended completely to P, but we cannot assume that the reverse is true. Think of the famous example of rectangles and squares. We know all squares are rectangles, but we also know that all rectangles are not squares. So we see that to make the assumption that All S are P is equal to All P is S is a faulty one. So, in "All S are P", S is distributed, whereas P is undistributed. Got it?

In "No S is P" we can make the assumption of reversibility. Once we know that all pigs are not sheep, we can state that all sheep in turn, are not pigs. So in a universal negative statement, both terms are distributed. Easy, right?

In "Some S is P" we know by definition that we are not talking about all cases of S, so we know that neither of the terms is distributed.

In "Some S is not P" we do not say the same of all Ss, BUT we say of all Ps that they are not a certain S, so P IS distributed. Be careful with this one.

Finally the rule for conclusions - One premise should be negative (E or O), if the conclusion is negative. If the conclusion is affirmative (A or I), however, both premises should be affirmative.

If you go on to check out the Syllogistic Machine in the next section you'll see that the algorithm that evaluates the different syllogisms in the machine makes use of this property of premises and conclusion. Given the status of distribution of the terms in the conclusion, it makes certain demands on the status of distribution of the premises. If a term is distributed in the conclusion, it should also be distributed in the premise in which it occurs. Also the middle term should be distributed at least once. Violation of these rules are called "illicit premise" (major or minor) and "undistributed middle" respectively.

Here are the rules clearly laid out for you:

Recall:
In an A proposition, the S is distributed, the P is not
In an E proposition, S and P are both distributed
In an I proposition, S and P are both undistributed
In an O proposition, S in undistributed, P is distributed

Rule 1 - The middle term must be distributed in one of the premises, or the fallacy of the undistributed middle occurs:

All P are m
All S are m
All S are P Here, the m is undistributed in both premises (appears in same position) A common error.

Rule 2 - If a term in distributed in the conclusion, it must be distributed in the premise. If the rule is broken, the fallacy committed is either illicit major or illicit minor, depending on whether the Predicate or Subject is undistributed.

All m are P (A)
All S are m (A)
No S are P (E) Here P is distributed in the conclusion (E) but undistributed in the major Premise (A) this commits the fallacy illicit major

All P are m (A)
All m are S (A)
No S are P (E) Here S is distributed in the conclusion (A) but undistributed in the minor Premise (E) this commits the fallacy illicit minor

Rule 3 - Two negative premises are not allowed - otherwise, the fallacy of exclusive premises is committed.

No P are m
No S are m
No S are P

Rule 4 - Dig this: a negative premise requires a negative conclusion, and a negative conclusion requires one negative premise. Otherwise, man - bummer! --> drawing a negative conclusion from affirmative premises, or drawing an affirmative conclusion from a negative premise.

1) All P are m
All S are m
No S are P
Drawing a negative conclusion

2)No P are m
Some S are m
Some S are P
Drawing an affirmative conclusion...

Rule 5 - Are both premises universal? Then, the conclusion cannot be particular, otherwise, you break the existential fallacy!

All P are m
All S are m
Some S are P

Finally, in order to understand the Formal fallacy section, concerning argumentation form, an overview of propositional logic is required.

Please feel free now to move on to examine the Syllogistic Machine