Invalidating Validity-1

POWELL:
I promised to give arguments indicating that so-called "valid deductive arguments" aren't really valid in the way usually claimed. Here I go. First, let me quote what the various logical terms mean. I will assume that Copi and Cohen "Introduction to Logic" 11th edition (copyright 2002) represents the consensus view of introductory logic teachers on these issues.

COPI & COHEN Glossary / Index:
Argument:
Any group of propositions of which one is claimed to follow from the others, which are regarded as providing support or grounds for the truth of that one, <snipped page numbers>

Deduction:
One of the two major types of argument traditionally distinguished, the other being induction. A deductive argument claims to provide conclusive grounds for its conclusion; if it does so it is valid, if it does not it is invalid, <snipped page numbers>

Induction:
One of the two major types of argument traditionally distinguished, the other being deduction. An inductive argument claims that its premisses give only some degree of probability, but not certainty, to its conclusion, <snipped page numbers>

Valid:
A deductive argument whose premisses, if they were all true, would provide conclusive grounds for the truth of its conclusion, is said to be valid. Validity is a formal characteristic; it applies only to arguments, as distinguished from truth, which applies to propositions, <snipped page number>

Modus Ponens (M.P.):
One of the nine elementary valid argument forms; a rule of inference according to which, if the truth of the hypothetical premiss is assumed, and the truth of the antecedent of that premiss is also assumed, we may conclude that the consequent of that premiss is true. Symbolized as: p =) q , p, therefore q, <snipped page numbers>

Modus Tollen (M.T.):
One of the nine elementary valid argument forms; a rule of inference according to which, if the truth of the hypothetical premiss is assumed, and the falsity of the consequent of that premiss is also assumed, we may conclude that the antecedent of that premiss is false. Symbolized as p =) q, ~q, therefore ~p, <snipped page numbers>

COPI & COHEN (pg. 42-43):

1.7 Deduction and Validity

Every argument makes the claim that its premisses provide grounds for the truth of its conclusion. Indeed, that claim is the mark of an argument. But there are two major classes of arguments: deductive and inductive. These two classes differ fundamentally in the way in which their conclusions are supported by their premisses. In this section we give a brief account of deduction.

A deductive argument makes the claim that its conclusion is supported by its premisses conclusively. In contrast, an inductive argument does not make such a claim. If, in interpreting a passage, we judge that such a claim is being made, we treat the argument as deductive; if we judge that such a claim is not being made, we treat it as inductive. Since every argument either makes this claim of conclusiveness or does not, every argument is either deductive or inductive.

When an argument makes the claim that its premisses (if true) provide irrefutable grounds for the truth of its conclusion, that claim will be either correct or not correct. If it is correct, that argument is valid. If it is not correct (that is, if the premisses when true fail to establish the conclusion irrefutably), that argument is invalid.

For logicians, therefore, the term validity is applicable only to deductive arguments. To say that a deductive argument is valid is to say that it is not possible for its conclusion to be false if its premisses are true. Thus we define "validity" as follows: A deductive argument is valid when, if its premisses are true, its conclusion must be true.

Every deductive argument makes the claim that its premisses guarantee the truth of its conclusion, but not all deductive arguments live up to that claim. Deductive arguments that fail to do so are invalid.

COPI & COHEN (pg. 43-44)

1.8 Induction and Probability

Inductive arguments do not claim that their premisses, even if true, support their conclusions with certainty. They make a weaker but nonetheless important claim that their premisses support their conclusions with probability, which always falls short of certainty. What was said above about validity and invalidity therefore does not apply to inductive arguments: Inductive arguments are neither valid nor invalid. ^47

- - - - - begin footnote

Note 47: In everyday speech the terms "valid" and "invalid" have taken on much wider and looser meanings. One hears it said, for example, that a fine motion picture "makes a valid statement," or that some emotional response to an act or event is a "valid reaction," and so forth. English is beautifully rich. But as logicians we use the terms valid and invalid far more narrowly; they indicate nothing more than the success, or lack of success, of a deductive argument in making its claim that if the premisses are true its conclusion must be true.

- - - - - - end footnote

We can still evaluate them [inductive arguments, added by Powell] of course. Indeed, the appraisal of inductive arguments is one of the leading tasks of scientists in every sphere. The premisses of an inductive argument provide some support for its conclusion, and the higher the level of probability the premisses confer on the conclusion, the greater the merit of the argument. In general, we say that inductive arguments may be "better" or "worse," "weaker" or "stronger," and so on. But even when the premisses are all true and provide very strong support for the conclusion, in an inductive argument the conclusion is never certain.

<snipped comments about where different techniques are treated in the text>

The difference between inductive and deductive arguments is deep. Because an inductive argument can yield no more than some degree of probability for its conclusion, it is always possible that additional information will strengthen or weaken it. Newly discovered facts may cause us to change our estimate of the probabilities, and thus may lead us to judge the argument to be better (or worse) than we thought it was. In the world of inductive argument - - - even when the conclusion is thought to be very highly probable - - - all the evidence is never in. It is this possibility of new data, perhaps conflicting with what was believed earlier, that keeps us from asserting that any inductive conclusion is absolutely certain.

Deductive arguments, on the other hand, cannot gradually become better or worse. They either succeed or do not succeed in exhibiting a compelling relation between the premisses and conclusion. The fundamental difference between deduction and induction is revealed by this contrast. If a deductive argument is valid, no additional premisses could possibly add to the strength of that argument. For example, if all humans are mortal, and if Socrates is human, we may conclude without reservation that Socrates is mortal - - - and that conclusion will follow from those premisses no matter what else may be true in the world, and no matter what other information may be discovered or added. If we come to learn that Socrates is ugly, or that angels are immortal, or that cows give milk, neither those findings nor any other findings can have any impact on the validity of the original argument.

In the case of every valid deductive argument, the conclusion that follows with certainty from its premisses follows from an enlarged set of premisses with the same certainty, regardless of the nature of the additional premisses. If an argument is valid, nothing in the world can make it more valid; if a conclusion is validly inferred from some set of premisses, nothing can be added to that set to make that conclusion follow more strictly, or more logically, or more validly.

But this is not true of inductive arguments, in which the relationship claimed between premisses and conclusion is much less strict and different in kind.

<snipped some examples>

A deductive argument is one whose conclusion is claimed to follow from its premisses with absolute necessity, this necessity not being a matter of degree and not depending in any way on whatever else may be the case. In sharp contrast, an inductive argument is one whose conclusion is claimed to follow from its premisses only with probability, this probability being a matter of degree and dependent upon what else may be the case.

Inductive arguments do not always acknowledge explicitly that their conclusions are inferred only with some degree of probability. On the other hand, the mere presence of the word "probability" within an argument is no sure indication that the argument is inductive. This is so because there are some strictly deductive arguments about probabilities themselves *

- - - - - - begin footnote

Note *: If, for example, we learn that the probability of three successive heads in three tosses of a coin is 1/8, we may infer deductively that the probability of getting at least one tail in three tosses of the coin is 7/8.

- - - - - - end footnote

POWELL:
Now, I'll add a few dictionary definitions.

Merriam-Webster On-Line Dictionary:
Main Entry: de·duc·tion
Pronunciation: di-'d&k-sh&n, dE-
Function: noun
Date: 15th century
1 a : an act of taking away <deduction of legitimate business expenses>
b : something that is or may be subtracted <deductions from his taxable
income>
2 a : the deriving of a conclusion by reasoning; specifically : inference in which the conclusion about particulars follows necessarily from general or universal premises -- compare INDUCTION
b : a conclusion reached by logical deduction

Main Entry: in·duc·tion
Pronunciation: in-'d&k-sh&n
Function: noun
Date: 14th century
1 a : the act or process of inducting (as into office)
b : an initial experience : INITIATION c : the formality by which a civilian is inducted into military service
2 a (1) : inference of a generalized conclusion from particular instances -- compare DEDUCTION 2a
(2) : a conclusion arrived at by induction
b : mathematical demonstration of the validity of a law concerning all the positive integers by proving that it holds for the integer 1 and that if it holds for an arbitrarily chosen positive
integer k it must hold for the integer k+1 -- called also mathematical induction
3 : a preface, prologue, or introductory scene especially of an early English play
4 a : the act of bringing forward or adducing (as facts or particulars)
b : the act of causing or bringing on or about
c : the process by which an electrical conductor becomes electrified when near a charged body, by which a magnetizable body becomes magnetized when in a magnetic field or in the magnetic flux set up by a magnetomotive force, or by which an electromotive force is produced in a circuit by varying the magnetic field linked with the circuit
d : the inspiration of the fuel-air charge from the carburetor into the combustion chamber of an internal combustion engine e : the sum of the processes by which the fate of embryonic cells is determined and morphogenetic differentiation brought about

POWELL:
In the next posts of the same title I will give my arguments.

John Powell

Invalidating Validity-2
Dialetheism Argument

POWELL:
The first argument I will present is based on the possible existence of true contradictions and the possible failure of the Law of Non-Contradiction.

http://plato.stanford.edu/entries/dialetheism/

PLATO STANFORD:
Stanford Encyclopedia of Philosophy

Dialetheism

A dialetheia is a true contradiction, a statement, A, such that both it and its negation, ~A, are true. Hence, dialeth(e)ism is the view that there are true contradictions. Dialetheism opposes the so-called Law of Non-Contradiction (LNC) (sometimes also called the Law of Contradiction): for any A, it is impossible for both A and ~A to be true. Since Aristotle's defence [sic] of the LNC, the Law has been orthodoxy in Western philosophy. Nonetheless, there are some dialetheists in the history of Western Philosophy. Moreover, since the development of paraconsistent logic in the second half of this century, dialetheism has now become a live issue once more.

POWELL:
The fact that some philosophers are seriously considering the possibility that A and ~A could both be true will serve as my first argument that so-called "valid deductive arguments" are not valid in the way usually claimed.

Let's use Modus Ponens as our first example.

1) If p then q
2) p
3) therefore, q

Now, let p = A and q = ~A (not A)

4) If A then ~A
5) A
6) therefore, ~A

Modus Ponens indicates this inference should be valid, since the argument form is supposed to be valid. However, there are problems. If premiss 2, A, is true then the conclusion should also be A, but by the Law of Non-Contradiction, ~A can't then be true. The LNC, however, was thrown out the window with this contradictory substitution. Anyway, if you allow for dialetheism to be true then you can get weird results.

Now, consider Modus Tollens:

7) If p then q
8) ~q
9) therefore, ~p

As before, let p = A and q = ~A

This results in

10) If A then ~A
11) ~ (~A)
12) therefore, ~A

If ~(~A) is the same as A then Modus Tollens looks identical to Modus Ponens with this substitution.

The point is that if dialetheism is true then neither M.P. nor M.T. are known to be a valid inferences because you can no longer rely on the conclusion being true or noncontradictory because if the premisses are true then there could be contradictions.

Let's consider some "real life" examples so one doesn't falsely assume this is just word games. I'll mention 3 from science and then a couple of more common ones from the natural language.

Example 1: Is the classical Law of Conservation of Matter true? Is the classical Law of Conservation of Energy true? Or, is E = mc^2 true? Today, we'd say that E=mc^2 is true, but when Einstein first proposed it there was an apparent contradiction.

Example 2. Is light a particle or a wave? Are things like electrons particles or waves?

Example 3. Are physical quantities quantized (come in discrete bundles) or not? Quantum Mechanics assumes yes. General Relativity assumes no. Which is it?

Example 4: A "couple" (2 or 3?), a few (3 to 4?), and many (more than 4?).

Example 5: "Like" and "love." Women will often pressure a man to say he "loves" her, when the man may want to say he only "likes her," perhaps "a lot." Some of these men perhaps want to avoid having to defend themselves when they don't "show your love" to her satisfaction.

Example 6: "Dislike" and "hate." It's more common to hear something like "I dislike him so much that I hate him." However, it's linguistically possible to have someone meaningfully say: "I don't dislike him. I hate him."

Example 7: "Hungry" and "Starving." This is similar to the dislike / hate example. Let's look at this one a little more in detail.

These two terms are sometimes used synonymously and sometimes they are not, even by the same person. When they aren't treated as synonyms then "hungry" usually means something like "feeling hunger pains" and "starving" means "very hungry" or perhaps even "hunger so bad that damage to organs or death of the complete organism is imminent." The idea is that on a "hunger" spectrum, the word "hungry" goes from barely feeling hunger pains up to where damage to organs or even death is imminent, and "starving" covers the more extreme forms of hunger.

For discussion sake, let's assume that hunger level 0 is neutral, neither hungry nor full. Hunger levels up to 50 are "hungry" and above 50 are "starving" with level 100 death by starvation. Negative levels of hunger are feeling satisfied to feeling full.

Now, one could think of "starving" as a kind of hunger that's from level 50 to 100. Therefore, "starving" is a kind of "hungry". On the other hand, one could think of "starving" as distinct from "hungry" since "hungry" only goes up to hunger level 50. In other words, "starving" is not "hungry."

Given that introduction, consider the following syllogism.

13) If Jack is starving then Jack is hungry.
14) Jack is starving.
15) therefore, Jack is hungry.

This argument may be valid if "starving" means "very hungry." In other words, if starving isn't just normal hunger, but strong hunger. This argument could be saying semantically that if Jack is "very hungry" then Jack is "hungry."

However, if "starving" and "hungry" are considered as exclusive of each other, in which "starving" is level 50-100 while "hungry" is level 0-50, then things could be different.

A person hearing the argument above might initially accept premiss 13 as true, but by the time he gets to the conclusion, the same person could have switched to the other distinction between hungry and starving, so the conclusion wouldn't necessarily follow.

The logician might strongly assert that IF premisses 13 and 14 are true then 15 MUST be true, but that's not necessarily the case since the moments that the truth values of premisses 13 and 14 are considered are not necessarily the same moment that the truth value of the conclusion 15 is considered. One must assign truth values to 13, 14, and 15 simultaneously to have a chance of the argument being valid. There is a relativistic problem with this procedure that I'll treat in a later argument.

Now, getting back to the problem of allowing contradictions.

What is the solution to make these arguments really valid? One way might be to just disallow propositions which violate the LNC. Based on what I've read on Internet sites, I think this is how most logicians handle the problem. Probably Copi and Cohen do the same thing, but I'll have to get to that part. However, doing this would mean that so-called valid deductive arguments could only be valid for a restricted range of possible substitutions. That would make so-called valid deductive arguments merely statistical again.

An example of such an excluded substitution would be a decisional conditional like:

16) If the President wins re-election then I will eat my hat.

Even if the President were to win re-election that would not necessarily mean that I would eat my hat. This is a promise, not a logically binding relationship. A truth value cannot be reliably assigned to the conditional until the President wins re-election and I eat my hat or the last possible moment transpires that I could possibly fulfill my promise, but failed to.

Including modal terms like "necessarily," "probably," and "possibly" in the conditionals could also cause problems.

A similar, but more cumbersome solution is to add a premiss which explicitly disallows contradictory substitutions into the logical forms. Let's apply it to M.P.

17) LNC is true and Dialetheism is false (or something to that effect)
18) If p then q
19) p
20) therefore, q

This revised M.P. might be valid, but you can't be sure that any given substitution is sound because dialetheism might be true and the LNC might be false. It is not necessarily the case that dialetheism is false and LNC is true. Who knows what future ideas philosophers will come up with?

Consider again the words of Copi and Cohen:

COPI and COHEN::
If an argument is valid, nothing in the world can make it more valid; if a conclusion is validly inferred from some set of premisses, nothing can be added to that set to make that conclusion follow more strictly, or more logically, or more validly.

POWELL
It appears that no one can know whether the so-called deductively valid arguments, such as M.P. and M.T., really are valid or not because we don't know if dialetheism will turn out to be true.

The point is that neither the classically valid Modus Ponens or Modus Tollens with only two premisses are certain of being valid. In order to be MORE assured that the inferences are valid one must add a LNC premiss. The inferences of the original M.P. and M.T. are suspect because of the possibility that dialetheism could be true and LNC could be false. The revised Modus Ponens + LNC assumption (and M.T. + LNC) might be valid, but that's still questionable. At least the revised M.P. appears to be MORE valid than the original Modus Ponens. However, this "matter of degree" defeats it as well since deductively valid arguments aren't supposed to be matters of degree, but all or nothing. Consider,

COPI and COHEN:
A deductive argument is one whose conclusion is claimed to follow from its premisses with absolute necessity, this necessity not being a matter of degree and not depending in any way on whatever else may be the case.

In sharp contrast, an inductive argument is one whose conclusion is claimed to follow from its premisses only with probability, this probability being a matter of degree and dependent upon what else may be the case.

POWELL:
It appears to me that so-called valid deductive arguments are subject to matters of degree. Perhaps arguments like M.P. and M.T. have worked flawlessly for most logicians for thousands of years, but new knowledge reveals that they actually need additional explicit premisses or implicit restrictions to be more sure that they are valid. This makes them appear more like what Copi and Cohen call inductive arguments, those for which the inference is less than 100% certain.

Therefore, due to the dialetheism argument, so-called deductively valid arguments are really just statistical arguments with nearly 100% certainty that the inference is correct if the premisses are true.

John Powell

Invalidating Validity-3
Ambiguity of Language Argument

POWELL:
In the previous argument, I tried to show that so-called valid deductive arguments (specifically Modus Ponens and Modus Tollens) aren't really valid as advertised because the possible failure of the Law of Non-Contradiction or the truth of Dieletheism (the possibility that A and not A could both be true) makes knowledge of the inference uncertain. In this second argument, I will use the ambiguity of language to argue that deductively valid arguments aren't really valid as advertised.

Let's use the classical deductively valid argument

1) All men are mortal
2) Socrates is a man
3) therefore, Socrates is mortal

How do you know the Socrates in premiss 2 is the same as the Socrates in the conclusion 3? You don't. Maybe the Socrates in premiss 2 is the philosopher, but the one in the conclusion 3 is an immortal angel also named Socrates. You could have two true premisses, but the conclusion is not true. Therefore, this classically valid argument is not really valid as advertised.

The natural language does not insist that every time you use a word it has exactly the same meaning. Numbers are probably least subject to this variability.

Consider some other examples I am critical of.

http://www.unc.edu/~theis/phil20/arguments.html

PHIL 20:
Example 1:
Premise 1 St. Paul is a city in Minnesota.
Premise 2 Last week I was in St. Paul.
------------
Conclusion Therefore, last week I was in Minnesota.

This is a good argument. It is valid and since the premises are true, it is also sound.

POWELL
I don't think this is valid because there could easily be more than one place named St. Paul in the world.

PHIL 20:
Example 2:
Premise 1 I am the star player on Carolina's basketball team
Premise 2 Everyone on Carolina's basketball team is over 7 feet tall.
------------
Conclusion Therefore, I am over 7 feet tall

This is a valid argument, but since the premises are false, it is not sound.

POWELL:
I don't believe this is even a valid argument because the "Carolina basketball team" mentioned in both premiss 1 and 2 could be different ones. Both North and South Carolina could easily have more than one basketball team.

Let me make up an argument.

4) All the players on Carolina's basketball team are black men.
5) Jane is on Carolina's basketball team
6) therefore, Jane is a black man

I'm confident that the author of PHIL 20 would consider this to be a valid deductive argument. In other words, that if premiss 4 and 5 are true then conclusion 6 must be true, cannot be false. However, PHIL 20 would probably think that it's an unsound argument since a person named Jane is unlikely to be on a basketball team composed entirely of black men.

Now, imagine an alien, Xz206yl, is listening in on a conversation between Jack and Jill as they watch basketball on T.V.

Jack asks, "Are all the players on Carolina's basketball team black men?"

Jill truthfully says,"Yes."

Xz206yl translates those words into its own language.

There is heard a click, but basketball is still being presented on the T.V. Unknown to Xz206yl, Jack just changed the channel to a game being played by the Women's Carolina Basketball team.

"Hey isn't that Jill playing on the Carolina basketball team?" Jack asks.

"Yes," Jill truthfully says.

Xz206yl translates these words.

Is Xz206yl justified in concluding that it must absolutely be true that Jane is a black man if the statements were truthfully spoken and the translation was correct? No. Given the limited information, we shouldn't criticize Xz206yl for concluding that Jill is probably a black man, but Xz206yl would be unjustified in concluding that it must be an absolutely correct inference even though the argument follows a so-called valid deductive form.

The inference that Jane is a black man does not follow from this possible dialogue.

Let's consider another example, more dialogue.

Eve: "I'm starving. Can we hurry up and order?"

Adam: "Sure"

Jack, a poor man walks up. "I bet you're hungry. Could you people help out a starving poor man? Aren't you starving?"

Eve: "No, but I'm really hungry."

The point here is that in normal conversation Eve could both claim to be starving and, yet, deny that she's starving. Language allows that kind of freedom. Because of that you can't be sure that the words used in so-called deductive arguments always refer to exactly the same thing in all premisses and in the conclusion.

Different people have different meanings for the same words and even the same person changes their understanding of the meanings of the same words as they gain more experience. Words like "man" and "mortal" will have a slightly different meaning to every single person. Furthermore, what you "see" as "green" may be different than what my visual system perceives.

Useful communication between human beings and of a person with his own mind is possible because humans are reasonably similar in physical characteristics and experiences and, so, there is reasonably close overlap in understanding and the meanings of words tend to change slowly with time compared with the life span of the communicators. The concepts for numbers, like 1, 2, etc. may be the least subjective or changeable, however even here there is some variation. To the Hebrews, for example, the number "7" may have meant more things than just this many: I I I I I I I.

Logicians try to invent logical languages that satisfy their logical rules, their axioms and their theorems. Natural languages like English do not necessarily follow those rules.

Because of the ambiguity of language, so-called deductively valid arguments really aren't valid as advertised. The words in various parts of the argument could mean different things to different people and even to the same person at different times.

What is the solution to overcome this ambiguity of language problem and to make the so-called valid deductive argument valid as advertised? I don't think there is a solution. You can't force the definitions of words to be understood identically by everyone or even by the same person from moment to moment. There is no way to be absolutely certain that any two people have exactly the same understanding for the same word. In fact, it is more sensible to expect that two people or the same person at different times will have a slightly different understanding for what any certain word means.

Therefore, due to the ambiguity of language argument, so-called valid deductive arguments are just statistical arguments in which the conclusion is nearly 100% certain if the premisses are true.

John Powell

Invalidating Validity-4
Nonconservation of Identity Argument

POWELL:
In the first argument to show that so-called deductively valid arguments aren't valid as advertised, I used Dialetheism. In the second argument I used the ambiguity of language. In this argument I will use the nonconservation of identity.

Again, let's use the classical deductively valid argument

1) All men are mortal
2) Socrates is a man
3) therefore, Socrates is mortal

Because of the nature of our physical existence, the atoms which comprise our body keep changing over time. You are not the same being you were a moment before. At each moment you will have a slightly different assortment of particles in different energy states. Because human beings are very similar to what they were a second before, they have a concept of conservation of identity, but it's only approximately true on short time scales.

You imagine, for example, that the apple you saw a second ago is the same one you see now, but that's only approximately true. If you watch the apple long enough you won't want to eat it because it will change, it will decompose. Likewise, if you watch Socrates long enough he will die and decompose. Human beings and apples are very much like clouds which form, change, but eventually dissipate to make something else.

Therefore, even if the Socrates in premiss 2 is the same "person," as we say, as the Socrates in the conclusion (we're not talking about Socrates the philosopher and Socrates the angel anymore), it still can't really be exactly the same entity because of this problem. The Socrates from one moment to the next is so similar in attributes that you can have very high (but not absolute) confidence that a typical time interval is irrelevant.

Therefore, due to the nonconservation of identity argument, so-called deductively valid arguments are really just statistical arguments with nearly 100% certainty that the inference is correct if the premisses are true.

The solution to make these arguments valid? What you must do is to consider the truth value of the premisses and the conclusion at the exact same instant so there is no time for a change of state. Since people can usually only think about one thing at the same time, this would probably require more than one observer to accomplish. The validity of the inference could only be determined after their notes were compared and it was clear they had considered the truth value
simultaneously.

That will bring up perhaps my most complicated argument, the problem of simultaneity.

John Powell

Invalidating Validity-5
Simultaneity or Special Relativity Argument

POWELL:
Again, let's use the classical deductively valid argument

1) All men are mortal
2) Socrates is a man
3) Therefore, Socrates is mortal

In the ambiguity of language problem, we've considered that "man" might mean different things in different parts of the argument or the man Socrates might not be the Angel Socrates. In the Nonconservation of identity problem we've considered the fact that nothing material is ever the same if a time interval is involved. The new problem has to do with simultaneity. In order to know that the argument is sound you must test the truth value of all the premisses and the conclusion at the exact same instant. Otherwise, absolute certainty is destroyed. Things change over time.

For example, consider that premiss 1 is proposed 2000 some odd years ago, premiss 2 applies both then, now, and into the future, but conclusion 3 applies to the year 3001 A.D. on the day that Jesus Christ finally gets around to "quickly" coming back to Earth and turning men like Socrates into immortal angel men. The argument may have appeared to be valid 2000 years ago and continued to appear to be valid up to 3001 A.D.. However, once Socrates, the man, is resurrected and immortalized then the argument is unsound, because premiss 1 at that moment becomes false.

The point is that if you consider the premisses at a different time than the conclusion, then it's not valid, the conclusion is no longer certain of being true even if the premises were true when you measured them.

Consider another example, one I made up.

4) All of Jack's brothers are in the house.
5) Bill is one of Jack's brothers.
6) Therefore, Bill is in the house.

Logic teachers, I believe, would consider this to be a valid argument. They would consider that if premiss 4 and 5 are true then conclusion 6 must also be true. However, if Bill is on the way out the door there could be a problem. The logician can check premiss 4 as true and premiss 5 as true, but by the time he considers the conclusion 6, Bill may have left the house and premiss 4 is no longer true. Thus, the logician might go back and erase the "truth" check of premiss 4 and so the conclusion doesn't have to be true.

Now, what do you do during the second that Bill is in the doorway in which part of his body is in the house and part out? You could define him as "in the house" when more of his body is on the house side of some defined plane than outside that plane. Fine. This just helps to show that the logician may not be able to consider the truth value of 4 and then 6 before Bill has moved from being inside to outside the house or vice versa. Premiss 5 changes too slowly in truth value here to need to worry about it unless Bill is near death.

In other words, if you check mark the premisses as true, you can't be sure that the conclusion will be true when you get around to considering it. You have to check mark them all simultaneously.

Some logicians might try to avoid the check marking exercise as needless. However, what do they mean that some proposition is true that at other times could be false if it's not based on a truth-determination that occurs at some location in space at some moment of time?

So, again, one MUST consider the truth values of the premisses and conclusion simultaneously or close enough to simultaneous according to the time scale at which the truth values might change.

The problem, however, is that simultaneous measurements in one reference frame will not be simultaneous in a reference frame moving relative to it. This is something Einstein explained in his Special Theory of Relativity. Before I apply the principle to dethroning so-called valid deductive arguments, let me explain the simultaneity problem.

Imagine Jack is on the ground, while Jill is on a train car standing exactly at the center of the car. As measured by Jack, at the instant that Jill passes Jack, a bolt of lightning strikes both the front and the rear of the train car Jill is on leaving scorch marks on both the tracks and on the train car. Jack measures the time for the light flashes to reach him. They arrive at the same instant. He measures the distance between the marks on the tracks and finds that he was exactly at the center. Therefore, Jack will judge that the two flashes struck at exactly the same instant of time some fraction of a second before he saw the flashes.

Jill, however, is moving relative to Jack. If Jack measures the flashes to be simultaneous then Jill will receive the light flash from the front of her train car before she receives the flash from behind. Yet, she will measure herself to be exactly at the center of the train car. She, therefore, will conclude that the flash that happened at the front of the car occurred before the flash at the rear, since the signal reached her first, yet she was at the center.

The classical solution would be to claim that Jill is really moving and Jack is not, and to assume that information of things can travel infinitely fast. However, motion is all relative to the observer and the fastest information can travel is at the speed of light. There is no known experiment that can show that it is really Jack who is stationary and Jill who is moving. The constant and limited speed of light helps to produce this problem. In fact, an observer moving with the Sun might measure both Jack and Jill to be moving in the same direction, but at different speeds. You can't claim that the flashes really were simultaneous just because that's what Jack obtained. All observers in non-accelerating reference frames have equal claim to such things.

The conclusion Einstein came up with is that what is simultaneous to Jack won't be simultaneous to someone moving relative to Jack (such as Jill).

Simultaneity is not a constant of the universe, it's NOT invariant to relative motion.

Now, how does that apply to determining the validity of deductive arguments?

The problem with determining the truth value of the premisses and conclusion of an argument is that it will only be valid for people stationary with respect to the truth testers. People who are moving will judge that the measurements were not done simultaneously and, therefore, the validity claim is not justified. This only becomes a serious problem when the truth values can change rapidly and people are moving near the speed of light, so one can say in practice that the validity of such arguments is approximately correct. However, that destroys it completely, because it's all or nothing for so-called valid arguments.

Therefore, due to the relativistic problem of simultaneity, so-called deductively valid arguments are really just statistical arguments in which the conclusions are virtually (but not exactly) 100% probable of being true if the premisses are true.

John Powell

Invalidating Validity-6
Modal Argument

POWELL:
Since posting these arguments to the II Errancy Annex, I have come up with an additional argument, the modal argument. I will place it before what used to be my final argument.

It is argued by modal logicians that the following conditional would be incorrect:

1) If p then necessarily q

Modal logicians argue that the problem is that it is not necessarily the case that q be true by itself. In some possible worlds or situations, q might be false. Therefore, a possible resulting conclusion "therefore, necessarily q" would be false. At most it would necessarily be the case that the conditional was true. So, modal logicians argue, the conditional should be worded in the following way:

2) Necessarily (if p then q)

A problem, however, is whether q would be certain to be true if this were used in a M.P. type argument. If q isn't certain or necessarily true in this situation then the argument fails to satisfy what is meant by a valid deductive argument. One cannot conclude with absolute confidence that q is true even if premise 1 and 2 were true because q is not certain to be true because it's not necessarily true. The conclusion q could be false.

It is my current opinion that what logicians really mean by Modus Ponens is probably something more like the following:

3) If (in the possible world w, p) then (in w, necessarily q).
4) In the possible world w, p.
5) therefore, in w, necessarily q.

Perhaps because this is too cumbersome for the translation of arguments from a natural language, the "world w" and "necessarily" parts are omitted. I think this should be clarified in introductory logic texts.

My argument here then, is that because introductory logic texts, like Copi & Cohen do not make this modal / possible world correction clear early on, sufficiently explicit, the conventional wording of the M.P. argument does not satisfy the necessary conditions to be valid. Because of this, one could not be certain that the conclusion would be true even if the premises were true. One can only justifiably claim that the conclusion is very likely to be true if the premises are true.

John Powell

Uh, John, you need to put the book down and go outside and take a little walk.

Invalidating Validity-7
Measurement Error Argument

POWELL:
Although I'm giving this as my last argument, it was actually the first one I used for myself. Based on my scientific training, I intuitively felt there had to be a problem with so-called deductively valid arguments. How can anyone be perfectly justified in claiming to know for absolutely certain much of anything? Any measurement has some error associated with it. How can you be perfectly justified in feeling sure that the conclusion will be true even if the premisses were true? Logicians appeared to be saying you are perfectly justified in feeling absolutely certain, but what I had learned about science did not support that position. Things once thought by virtually everyone to be absolutely certain like the Earth is the center of the universe, Euclid's geometry, our concepts of space and time, Classical mechanics, etc., have been shown to be less than certain or plainly wrong.

Let me ask a simple question that attempts to cut right to the point between "certain" valid deductive arguments and "probably true" inductive arguments.

What's the difference between an inference that is 100% certain and one that is only
99.999999999999999999999999999999999999999999% certain (add as many 9's as you want)?

To Copi and Cohen, 100% certain would be deductively valid, while the other would be inductively strong or something like that.

You might call this tiny distinction "a quibble," but that's all you need to dethrone deductively valid arguments as they are advertised in introductory logic texts.

There is a conceptual difference between 100% and virtually 100%, I'll admit, but there is probably no measurement that can distinguish them. When one speaks of "certainty," as in a deductively valid argument, it's only "close to absolute certainty."

There is not a huge difference between what Copi and Cohen term "deductive" and "inductive" since all such arguments are statistical. The question is one of degree. So-called valid deductive arguments are really just those with virtually 100% certainty, while the so-called inductive argument are less certain than that.

Therefore, due to the measurement error problem, so-called deductively valid arguments are really just statistical arguments with nearly 100% certainty that the inference is correct if the premisses are true.

If the difference between these kinds of arguments is merely whether the certainty is virtually 100% probable or less than that, then the basis for distinguishing them as "deductive" and "inductive" is weakened considerably. They might as well all be called "statistical arguments." Actually, they might as well just be called "arguments" because it's not helpful to characterize some arguments as statistical if there are none which are not. It would be like speaking of male men if there are no other kind.

The terms "deductive" and "inductive" could be used, instead, to refer to "going from the general to the specific" and "going from the specific to the general" respectively, as scientists and mathematicians and dictionaries and even some logicians use those terms.

I propose the following term changes.

Deductive argument:
An argument in which a conclusion about a specific case is inferred from a general principle.

Inductive argument:
An argument in which a conclusion about a general principle is inferred from one or more specific cases.

Valid argument:
An argument in which the statistical inference is virtually 100% certain. The conclusion would be virtually certain of being true if the premises were true. Note that very little, if anything, is absolutely certain.

Strong argument:
An argument in which the statistical inference is more than 50% probable, but less than virtually 100% certain. The conclusion would be probably true if the premises were true.

Weak argument:
An argument in which the statistical inference is less than 50% probable. The conclusion would be probably false if the premises were true.

Sound argument:
A valid argument in which the premises are all true. The conclusion is virtually certain of being true because the argument is valid and the premises are true.

Cogent argument:
A strong argument in which the premises are all true. The conclusion is probably true since the argument is strong and the premises are true.

Did I persuade anyone to accept that so-called deductively valid arguments aren't valid as advertised? Any comments?

If I failed to persuade you to doubt that so-called valid deductive arguments are the certain inferences they're claimed to be, maybe the fact that philosophers are debating these issues will motivate you to reconsider. Here's one example I came across when I was searching for details on modus ponens and modus tollens:

http://www.bu.edu/wcp/Papers/Logi/LogiDagl.htm

I didn't read the whole article or the references, but the mere fact that the issue is being debated by philosophers was enough to convince me that the strong claims of introductory logic texts and teachers concerning valid deductive arguments were probably overstated.

This is the last of the Invalidating Validity-x set of initial posts.

John Powell

Butters:
Uh, John, you need to put the book down and go outside and take a little walk.

POWELL:
Right.

I need these arguments posted so that from now on when I might claim that so-called valid deductive arguments are not the certain things they are claimed to be then any rebuttal to my assertion that does not directly rebut my specific arguments or does not point to a source that properly responds to my arguments then I can justifiably reply with:

"That's an argument by assertion, appeal to authority, and / or appeal to ignorance."

John Powell

John -

This is a subject I may be interested in discussing, but I (and presumably most others) simply don't have the time to wade through several full length posts. Could you maybe give a brief summary of one, some, or all of your arguments, or direct us to one in particular? Thanks.

psychopath

John -

This is a subject I may be interested in discussing, but I (and presumably most others) simply don't have the time to wade through several full length posts. Could you maybe give a brief summary of one, some, or all of your arguments, or direct us to one in particular? Thanks.

POWELL:
Certainly. I'll try.

I posted a series of 6 separate (but sometimes related) arguments that allege to demonstrate that so-called valid deductive arguments are not the "certain" things that introductory logic texts claim they are.

The arguments briefly are the following:

A1. Dialetheism argument.
Because dialetheism may be true, namely that true contraditions might exist, one cannot be absolutely certain that the conclusion of an argument of a so-called valid form such as M.P. is true even if the premises were true. One would have to add a premise "LNC is true" to be valid.

A2. Ambiguity of Language argument.
Because the meanings of words differ from person to person and even with the same person from moment to moment, one cannot be absolutely certain that the conclusion of a so-called valid deductive argument is true even if the premises were true. The meaning might change between the assigning of truth values to the premises and the conclusion. There appears to be no solution to this problem.

A3. Non-conservation of Identity argument.
Because physical things are only approximately the same from moment to moment, one cannot be absolutely certain that the conclusion of a so-called valid deductive argument that refers to them is true even if the premises were true, unless perhaps the truth values of premises and conclusion are determined simultaneously.

A4. Simultaneity argument.
Due to relativity, simultaneous events in one reference frame won't be simultaneous in another. In order to be certain that a time-variable argument is valid, the premises and conclusion must be assigned truth values simultaneously. However, if that is successful in one reference frame it will only be approximately successful in reference frames moving relative to it. Consequently, one cannot be absolutely certain that the conclusion will be true in your reference frame even if the premises are true in your reference frame.

A5. Modal argument.
Because arguments of the form

If p then necessarily q
p
therefore, necessarily q

are what are implied by M.P., but are deemed by modal logicians to be incorrect, one cannot be certain that q is true even if the premises were true. One cannot justifiably say "necessarily q" except perhaps if one uses the possible world w formulation.

A6. Measurement error argument.
Since there is no measurable difference between a conclusion that is 100% certain (deductive) and one that is 99.99999999...% certain (inductive) then to make the distinction is ill-advised. So many things have turned out to be wrong that were previously thought to be certain (such as geocentrism) that one cannot be absolutely certain that what logicians claim about valid deductive arguments is true.

Conclusions:

C1. So-called valid deductive arguments are really just statistical arguments such that the conclusions are very close to being certainly true if the premises were true, not certain of being true, as advertised by introductory logic texts.

C2. "Deductive" and "inductive" should not be distinguished as to the relative certainty of the conclusion as is done by modern introductory logic texts, but according to the historical distinctions (general to specific or vice versa) still used by mathematicians and scientists and some philosophers.

I hope this helps.

John Powell

A1. Dialetheism argument.
Because dialetheism may be true, namely that true contraditions might exist, one cannot be absolutely certain that the conclusion of an argument of a so-called valid form such as M.P. is true even if the premises were true. One would have to add a premise "LNC is true" to be valid.

In other words:

1. If dialetheism has any possibility of being true, conclusions drawn via M.P. are not 100% certain.

2. Dialetheism has a possibility of being true.

3. Conclusions drawn via M.P. are not 100% certain.

The thing is, you're using an argument of the modus ponens form (though you don't explicitly frame it as such) in an attempt to show the uncertainty of modus ponens. If your conclusion here happens to be true, then it follows that that conclusion itself may be false, and that conclusions drawn via M.P. ARE perhaps 100% certain, since it was arrived it through the use of M.P. So I'm not sure what this argument accomplishes.

A6. Measurement error argument.
Since there is no measurable difference between a conclusion that is 100% certain (deductive) and one that is 99.99999999...% certain (inductive) then to make the distinction is ill-advised. So many things have turned out to be wrong that were previously thought to be certain (such as geocentrism) that one cannot be absolutely certain that what logicians claim about valid deductive arguments is true.

But if it is impossible to achieve absolute certainty due to human fallibility, error, etc., then there can be no certainty in your conclusion that deductive arguments are not 100% certain. It follows, then, that they may be.

It seems, I think, that similar objections could be raised to all of your arguments.

PSYCHOPATH:
The thing is, you're using an argument of the modus ponens form (though you don't explicitly frame it as such) in an attempt to show the uncertainty of modus ponens.

POWELL:
Perhaps I'm using an inductive argument, similar in form to the one you posted but with "probably" implied in the conclusion.

PSYCHOPATH:
If your conclusion here happens to be true, then it follows that that conclusion itself may be false, and that conclusions drawn via M.P. ARE perhaps 100% certain, since it was arrived it through the use of M.P. So I'm not sure what this argument accomplishes.

POWELL:
It should justify a rational person to doubt that so-called valid deductive arguments are the absolutely certain things advertised in introductory logic texts even if one can't phrase a sound deductive argument that proves this to be true.

PSYCHOPATH:
But if it is impossible to achieve absolute certainty due to human fallibility, error, etc., then there can be no certainty in your conclusion that deductive arguments are not 100% certain. It follows, then, that they may be.

POWELL:
Exactly. I might be wrong, but so might all the logicians who disagree with me, who claim that so-called valid deductive arguments are the certain things they are advertised to be. Don't you agree, Psychopath?

PSYCHOPATH:
It seems, I think, that similar objections could be raised to all of your arguments.

POWELL:
Perhaps, but the arguments are not identically the same.

You bring up good points, Psychopath. I admit that I might be wrong.

Q1) However, which is more likely?

A) That I'm wrong and that valid deductive arguments are the "perfect" things they are made out to be, that one can be ABSOLUTELY CERTAIN WITHOUT ANY DOUBT WHATSOEVER that the conclusions will be ABSOLUTELY NECESSARILY true, that they ABSOLUTELY CANNOT be false if the premises are true

or

B) That I'm right and so-called valid deductive arguments don't quite muster to that absolute level of perfection?

What do you think, psychopath?

There, I just used an inductive argument to tear deductive arguments down to size.

If you think valid deductive arguments do satisfy that absolute level of perfection, Psychopath, then my next two questions to you are

Q2) What is the sound deductive argument proving that position?

and, if you produce what you think is such an argument, then

Q3) How do you absolutely know that particular argument is sound, that the premises are absolutely true and the inference is valid in the absolute way advertised?

John Powell

Hello John!

Let's use Modus Ponens as our first example.

1) If p then q
2) p
3) therefore, q

Now, let p = A and q = ~A (not A)

4) If A then ~A
5) A
6) therefore, ~A

This is a fascinating point. Mathematics has in its arsenol a way to prove things indirectly called the Reducio ad Absurdum. The prove is of this form...

Prove: A
Assume:~A
~A-->B
But we know from an outside theorum or definition that ~B is true. Therefore, by MT we have ~~A, and conclude A. QED.

There is another form of this proof that is used often, but some do not consider valid. It looks like this...

Prove: A
Assume:~A
~A-->B
B-->~B
Therefore, A. QED?

The difference between the 2 forms is that ~B is derived independantly from the proof in example 1, and ~B is derived within the proof in example 2.

I would submit that B-->~B is never valid, and is of the form of many paradoxes like "Barber" and "Lier." So to try and discredit MT or MP by starting off with A-->~A is dubious. Also, it can be shown that A-->~A asserts ~A as being true. All of this violates LC, but not because MT is not valid, but because A-->~A is not valid.

Sincerely,

Brian

POWELL:
Brian, don't these "proofs" assume the Law of Non-Contradiction is true, that Dialetheism is false? If that axiom were false wouldn't these cease to be "proofs"?

Just because you consider rejection of the LNC and support of Dialetheism to be "dubius" does not necessarily infer "false," right?

BRIAN:
I would submit that B-->~B is never valid, . . .

POWELL:
Do you have a proof for this assertion (preferably what you think is a sound deductive argument) that doesn't assume LNC? What about the "real life" counter-examples I gave?

Can you prove LNC is true?

John Powell

John, I was just kidding you, As I can see that you have spent quite a long time thinking about this.

I must agree with you, even using logical arguments we cannot be 100% sure of anything being "true". This is a favorite argument of theologians. Is the law of contradiction always valid?
Are all events causal? etc.? I think it is important to show that we cannot KNOW this with100% certainty, as not to stop searching for a way to prove these ideas with 100% certainly. However, we cannot discount that adhering to these axioms is the only workable method we have for some sense of our world.

Take causality for example. Some like to claim that events are not casual. We cannot PROVE that they are, with 100% accuracy, but where does that leave us. As ALL events APPEAR to be casual, and that we can explain events and make future predictions by assuming causality, it only makes sense to ASSUME it is true, until it is PROVED otherwise, or until another method comes along that is more useful. Unfortunately for theists, the answer, God did it, also cannot be PROVEN, but beyond that, is not as useful, in fact it's not useful at all.

I believe the state of metaphysics and logic today are analogous to the discovery of Newtonian laws. These laws also only stated unproven axioms. They worked quite well (and still do) to make sense of our world. It should also be noted, that these laws were not "replaced" by a more complete understanding of physics, but were subsumed by them. So while the were not complete, they were "true" enough to be useful.

Almost forgot my favorite axiom,

One good observation is worth more than a centuray of bad philosophy.