Another case of someone who intuited the modal fallacy early on?

plato.stanford.edu/entries/moore/#2

So, on the face of it, this thesis has here been inferred from Leibniz’ Law. Moore observes, however, that the step from (1) to (2) is invalid; it confuses the necessity of a connection with the necessity of the consequent. In ordinary language this distinction is not clearly marked, although it is easy to draw it with a suitable formal language.

Moore’s argument here is a sophisticated piece of informal modal logic; but whether it really gets to the heart of the motivation for Bradley’s Absolute idealism can be doubted. My own view is that Bradley’s dialectic rests on a different thesis about the inadequacy of thought as a representation of reality, and thus that one has to dig rather deeper into Bradley’s idealist metaphysics both to extract the grounds for his monism and to exhibit what is wrong with it.

 

Interesting. GE Moore is in good company, along with Leibniz.

A modal fallacy in linguistics

I’m writing this piece as i have gotten rather tired of explaining this point over and over. Writing an article about it saves me time.

The form of reasoning goes something like this:

1. This person uses some other spelling than the standard one for a word.
Therefore, 2. This person does not know how to spell the word.

It shud be relatively easy to see that this does not follow. Obviously, if one is familiar with spelling reform ideas, then that makes for easy counter-examples. But even people who have not thought/read about spelling reforms shud be somewhat familiar with familiar use of non-standard spellings in their native language. For instance, when speed is important, people may use alternative spellings becus they are shorter. For instance, using y for yes. Altho someone might see this as using abbreviations and not non-standard spellings. It can be rather difficult to distinguish between the two. Is becus an abbreviation that doesn’t count? How about ‘cus?

Another realm of counter-examples is when people deliberately ‘misspell’ a word for some other purpose, e.g. humor (making a pun), or to signal dialect (writing aks instead of ask), or murika instead of America thereby noting that the word is pronounced like that by many americans. Clearly, many people who do these things are aware of the standard spelling.

As an inductive inference?

In the above, i assumed that the argument was deductive, as in, the conclusion was supposed to follow with necessity by the reasoner. However, one might think of it as a probabilistic inference. Does it fare much better this way? Sort of. Certainly, sometimes people do try to write the standard spelling of some word, but for some reason write something else than that. This can be for many reasons: hitting the wrong key on the keyboard, being distracted at the moment of typing (typically results in one typing the word that was said to one) or just genuinely making a mistake (genuinely making a mistake! :D) becus one was wrong about how the word is spelled in the standard spelling.

Generally, tho, there are some patterns that one can use to make better guesses at whether people really did make a mistake becus they didn’t know the correct spelling, or something else is the case. There are lists of commonly misspelled words (example), if a person used one of the common but nonstandard spellings, then it increases the probability that it was a mistake. How does one spot typos? Usually, the character that is part of the nonstandard pattern is located near the intended one. This produces things like I luke you instead of I like you. Finally, the function words of a language are rarely misspelled becus they occur in high frequency. For that exact reason they also have the poorest spellings. If someone uses nonstandard spellings for such words, then that increases the probability that it is on purpose. Examples of this are words like could should would you I which might be written cud shud wud u i for various reasons.

Under which conditions wud the inference actually work deductively?

Perhaps if one added some extra premises then the inference wud work. Any candidates? Yes. Adding something like: Every person is always trying to use the standard spelling for every word. Implausible? Very! And it isn’t even enuf. There remains the possibilities of being distracted and hitting the wrong keys, and perhaps some other things i haven’t thought of.

Is it really a modal fallacy?

Yes.

Modal logic is that branch of logic which studies logical relations involving modalities. Modalities are ways, so to speak, in which propositions can be true or false. The most commonly studied modalities are necessity and possibility, which are modalities because some propositions are necessarily true/false and others are possibly true/false. (source)

Name for the fallacy?

Can’t think of anything good. It shud be short and relevant. Things like the anti language reformist fallacy is not short but at least it’s descriptive.

Dennett (1995) and th modal falacy

“In the “weak form,” it is a sound, harmless, and on occasion useful application of elementary logic: if x is a necessary condition for the existence of y, and y exists, then x exists. If consciousness depends on complex physical structures, and complex structures depend on large molecules composed of elements heavier than hydrogen and helium, then, since we are conscious, the world must contain such elements.

But notice that there is a loose cannon on the deck in the previous sentence: the wandering “must.” I have followed the common practice in ordinary English of couching a claim of necessity in a technically incorrect way. As any student in logic class soon learns, what I really should have written is:

It must be the case that: if consciousness depends … then, since we are conscious, the world contains such elements.

The conclusion that can be validly drawn is only that the world does contain such elements, not that it had to contain such elements. It has to contain such elements for us to exist, we may grant, but it might not have contained such elements, and if that had been the case, we wouldn’t be here to be dismayed. It’s as simple as that.

Some attempts to define and defend a “strong form” of the Anthropic Principle strive to justify the late location of the “must” as not casual expression but a conclusion about the way the universe necessarily is. I admit that I find it hard to believe that so much confusion and controversy are actually generated by a simple mistake of logic, but the evidence is really quite strong that this is often the case, and not just in discussions of the Anthropic Principle. Consider the related confusions that surround Darwinian deduction in general. Darwin deduces that human beings must have evolved from a common ancestor of the chimpanzee, or that all life must have arisen from a single beginning, and some people, unaccountably, take these deductions as claims that human beings are somehow a necessary product of evolution, or that life is a necessary feature of our planet, but nothing of the kind follows from Darwin’s deductions properly construed. What must be the case is not that we are here, but that since we are here, we evolved from primates. Suppose John is a bachelor. Then he must be single, right? (That’s a truth of logic.) Poor John—he can never get married! The fallacy is obvious in this example, and it is worth keeping it in the back of your mind as a template to compare other arguments with.” (p. 165-166, Darwin’s Dangerous Idea)

Stranj that he does not just mention it by name or go into mor detail, or eevn sujest mor litratur on it.

John Nolt – Logics, chp. 11-12

I am taking an advanced logic class this semester. Som of the reading material has been posted in our internal system. I’ll post it here so that others may get good use of it as well. The text in question is John Nolt’s Logics chp. 11-12. I remade the pdfs so that they ar smaller and most of the text is copyable making for easyer quoting. Enjoy

John Nolt – Logics, chp 11-12

Edit – A comment to the stuff (danish)

Dær står i Nolt kap. 12, at:

”We have said so far that the accessibility relation for all forms of alethic

possibility is reflexive. For physical possibility, I have argued that it is transitive as

well. And for logical possibility it seems also to be symmetric. Thus the accessibil-

ity relation for logical possibility is apparently reflexive, transitive, and symmetric.

It can be proved, though we shall not do so here, that these three characteristics

together define the logic S5, which is characterized by Leibnizian semantic… That

is, making the accessibility relation reflexive, transitive, and symmetric has the

same effect on the logic as making each world possible relative to each.” p. 343

mæn min entuisjon sagde maj strakes, at dette var forkert, altså, at de er forkert at

R1. For any world, that world relates to itself. (Reflexsive)

R2. For any world w1 and for any world w2, if w1 relates to w2, then w2 relates to w1. (Symmetry)

R3. For any world w1, for any world w2, and for any world w3, if w1 relates to w2, and w2 relates to w3, then w1 relates to w3. (Transitive)

er ækvivalænt mæd

R4. For any world w1 and for any world w2, w1 relates to w2. (Omni-relevans)

Mæn de er ganske rægtigt, at ves man prøver at køre diverse beviser igænnem, så kan man godt bevise fx (◊A→☐◊A) vha. en modæl som er reflexsive, symmetrical og transitive (jaj valgte 1r1, 2r2).

Mæn stadig er dær någet galt. Di forskællige verdener er helt isolerede, modsat vad di er i givet R4. Jaj googlede de, og andre har osse bemærket de:

”Requiring the accessibility relation to be reflexive, transitive and symmetric is to require that it be an equivalence relation. This isn’t the same as saying that every world is accessible from every other. But it is to say that the class of worlds is split up into classes within which every world is accessible from every other; and there is no access between these classes. S5, the system that results, is in many ways the most intuitive of the modal systems, and is the closest to the naive ideas with which we started.”

Propositions, cross-world identity and truth values

1. There is a thing, x, such that it is a contingent proposition.

∃xCx

2. For any thing, x, if x is a contingent proposition then there is a possible world, w, where x is true, and there exists a possible world, w’, where x is false.

∃xCx→∃x∃w∃w’Pxaw∧Pxbw’

3. Thus, there is a thing, x, and there is a possible world, w, and there is a possible world, w’, such that x has the property true in world w and x has the property false in w’.

⊢ ∃x∃w∃w’Pxaw∧Pxbw’ [1, 2, MP]

4. For any thing, x, and for any possible world, w, if x has the property true or has the property false in w, then x exists in w.

∃x∃wPxaw→∃x∃wExw

5. Thus, there is a thing, x, and there is a possible world, w, and there is a possible world, w’, such that x exists in w and x exists in w’ and

⊢∃x∃w∃w’Exw∧Exw’∧Pxaw∧Pxbw’ [3, 4, MP]

6. For any thing, x, for any possible world, w, x has the property true iff x does not hasve the property false.

∀x∀w(Pxaw↔¬Pxbw)

7. For any thing, x, and for any thing, y, for any possible world, w, and for any possible world, w’, if x exists in w and y exists in w’, then (x and y are identical iff for any property, z, if x has it in w, then y has it in w’).

∀x∀y∀w∀w'(Exw∧Eyw’)→(x=y↔(∀zPxzw↔Pyzw’))

This set is inconsistent (5∧6→¬7)1. But it seems clear to me that we ought to give up 7.

Notes

1I thought about giving the proof tree but it would be awfully long. Until I have an automatic way of doing this, I won’t prove such complex argument beyond formalizing them.

Bachelors and essential properties

When explaining the distinction between essential and accidental properties bachelors are often used. The idea of essential properties is this:

The distinction between essential versus accidental properties has been characterized in various ways, but it is currently most commonly understood in modal terms along these lines: an essential property of an object is a property that it must have while an accidental property of an object is one that it happens to have but that it could lack. (SEP)

So, bachelors have the essential property of being unmarried. This is because a married bachelor is a contradiction. This is because the concept of being a bachelor implies the concept of being unmarried. Implication with concepts works just as one might intuitively think that it does; very similar to that of propositions (or whatever truth carrier that you like), but see Swartz & Bradley (1979) for a deeper explanation using possible worlds semantics.

So, since we’re good analytics, we shall invoke the power of formalization.

A. For any person, it is impossible that (the person is a bachelor and the person is married).

∀x¬◊(Bx∧Mx)

That’s all fine and dandy. But it seems then that essential properties are just a matter of semantics, of language, and has nothing to do with the bachelor at all. Based on some idea of essential properties and bachelors and so on, we might say that:

B. Bachelors can’t be married.

But then, how do we formalize this?

∀xBx→¬◊Mx

Going with some very literal interpretation we may get the above, but surely the proposition that that wff expresses is false. Let’s translate it to LAE1

For any person, if that person is a bachelor, then it is not the case that it is possibly the case that the person is married.

So, a bachelor can’t even marry. But this is wrong. Bachelors can and do marry. The only thing is that the concept of being a bachelor and that of being married cannot at the same time be instantiated in the same person. Nothing else. There is no contradiction in saying:

I was a bachelor for many years, but then I married Jane. Gosh she is so wonderful, blahblahblah…

We may reduce this tale to something like:

Bert the Bachelor was a bachelor at time t1, but then later he married and stopped being a bachelor.

Which we may formalize as something like:

Bbt1∧Mbt2∧t1≠t2

1Logically Aided English

Infallible knowledge, the modal fallacy and modal collapse

The much mentioned the modal fallacy is not a fallacy (that is, is a valid inference rule) if one accepts an exotic view about modalities and necessities that is logically implied by a particular understanding of infallible knowledge and a knower.

Infallible knowledge

Some people seem to think that some known things are false and thus the need for a term like infallible knowledge, for that kind of knowledge that cannot be of false things. However that term “infallible knowledge” (and it’s under-term “infallible foreknowledge”) is subject to some interpretation. Is it best understood as:

A. If something is known, then it is necessarily true.

Or?:

B. Necessarily, if something is known, then it is true.

Or equivalently, in terms of “cannot” instead of “necessarily”:

A. If something is known, then it cannot be false.

B. It cannot be false that, if something is known, then it is true.1

I contend that the second interpretation, (B), is the best. However suppose that one accepts the first, (A).

The assumption of the existence of a foreknower

Now let’s assume that there is someone that knows everything (which is the case), the knower. He posses infallible knowledge á la (A). Now we can work out the implications.

The foreknower exists and knows everything (that is the case):

1. There exists at least one person and that for all propositions, that a proposition is the case logically implies that that person knows that proposition.

(∃x)(∀P)(P⇒Kx(P))

Whatever is known is necessarily the case (A):

2. For all propositions and for all persons, that a person knows a proposition logically implies that that proposition is necessarily true.

(∀P)(∀x)(Kx(P)⇒□P)

Thus, every proposition that is the case is necessarily the case:

Thus, 3. For all propositions, that a proposition is the case logically implies that it is necessarily the case.

⊢ (∀P)(P⇒□P) [from 1, 2, HS]

Thus, everything that is logically possible is the case:

Thus, 4. For all propositions, that a proposition is logically possible logically implies that it is the case.

⊢ (∀P)(◊P⇒P) [from 3, others]2

Thus, everything that is logically possible is necessarily the case:

Thus, 5. For all propositions, that a proposition is logically possible logically implies that it is necessarily the case.

⊢ (∀P)(◊P⇒□P) [from 3, 4, HS]

This is called modal collapse. The acceptance of that all possibilities are necessarily the case.

Thus, the modal fallacy is no longer a fallacy:

Thus, 6. For all propositions (P) and for all propositions (Q), that a proposition (P) is the case, and that that proposition (P) logically implies a proposition (Q), logically implies that that proposition (Q) is necessarily the case.

⊢ (∀P)(∀Q)(P∧(P⇒Q)⇒□Q) [from 3]3

And so we can validly infer from a proposition being the case and that that proposition logically implies some other proposition to that that other proposition is necessarily the case.

Notes

1Or “cannot be not-true” to avoid relying on monoalethism (and the principle of bivalence) which means that truth bearers only have a single truth value.

2This follows like this: I. □P⇔¬◊¬P (definition of ◊). II. Thus, P⇒¬◊¬P. [I, 3, Equi., HS] Thus, ◊¬P⇒¬P. [II, CS, DN] Thus, III. ◊P⇒P. [II, Substitution of ¬P for P]

3This follow like this: P∧(P⇒Q)⇒Q is just MP, and from 3 it follows that any proposition that is the case is necessarily the case.

Language, the modal fallacy and the symbolic representation of a conditional


“[W]hat follows from a true premiss must be true” (The Problems of Philosophy, p. 60, link)

Wrote Russell as an example of a principle of logic that is more self-evident than the inductive principle. If we were to formalize this we would perhaps write it like this:

E1. □[([∀P][Q∧Q⇒P])→P]1

Or perhaps just just in propositional logic:

E2. □(P→Q) where “P” means P is true and P implies Q.

As the reader of my blog should know by now, the modal fallacy consists of trusting language and placing the modal operator of necessity in the consequence instead of before the conditional:

E3. P→□Q

However we could also put the modal operator somewhere else in our formalization:

E5. P□→Q

Operators solely ‘work on’ whatever is to the right of them.2 Thus the modal operator in (E5) works on the material conditional and not the proposition to the left of it. Similarly in (E2) the modal operator works on the parentheses-set which is treated as a single entity.

(E5) is closer to normal english (and danish) than (E2) which we can express in normal english like this:

E4. Necessarily, if P follows from Q and Q is true, then P is true.

Consider the sentence:

E5. If P follows from Q and Q is true, then P must be true.

(E5) is a reformulation of (E2). (E5) is worded like it would be by a normal english speaking person. In (E5) it may seem as though the modal operator is intended to work on the consequent. Indeed some people think this and commit the modal fallacy.

However the modal operator may also be thought of as working on the second part of the “if, then” clause, that is, the “then” part. The only problem with this interpretation that I know of, is that it makes the operator work on something that is to the left of it instead of to the right of it: Because “then” is to the left of “must” in (E5).

Notes

1Ignoring the complexities of bivalance.

2Monadic operators. Not dyadic operators like implication, consistency etc.

An appeal to skepticism about reasoning

Unaided and aided reasoning

Humans reason about many things. Some things are more complex than other things. The more complex a thing is, the more probably it is that one will reason wrongly about it. For simple things the probability of unaided reasoning reasoning wrongly is not high. For complex things the probability of unaided reasoning making a fallacy is high.(Making a fallacy is used interchangeably with reasoning wrongly.) By unaided reasoning I mean informal, non-explicit, intuitive like reasoning. The more aided a reasoning is, the more tools are applied to help it. Such tools are e.g. clarity, explicitness and logic.

Clarity could be to make distinctions between two meanings of a word/phrase. It would also be to avoid using ambiguous words/phrases and stick to non-ambiguous ones (or relatively non-ambiguous ones). Clarity is also to make it clear when one is using two terms or phrases synonymously/interchangeably as I just did before, and also to note when one uses two words as antonyms of each other. Consider this example of that: If a man is fat, then he is stupid. Peter is smart. Therefore, Peter is not fat. Here it is plausibly interpreted that “smart” should be taken to mean “not-stupid”, i.e. as an antonym. If done, then the argument is valid.

Explicitness is to type down the argument. Much reasoning happens in the mind without writing it down at all. It is also explicitness to write down more premises of an argument, so called hidden premises. Most of the time when we present arguments, both verbally and in written text, we do not write down all the premises and we have to guess or interpreted what the hidden premises are. Consider this argument: Since Peter is smart, then he is not stupid. This is best interpreted as: If Peter is smart, then he is not stupid. Peter is smart. Thus, Peter is not stupid. Sometimes we accept an argument that is based on a premise (or premises) we would not accept if we had noticed it/them or had been more conscious of it/them.

For a reasoning to be aided of logic would be to translate the argument from normal language into a formal language, or rephrase the propositions such that they are closer to the standard translations of formal logic. Logical aid would also be to learn logic. The fact that one knows logic helps one reason even when formal logic is not used because it helps one being more aware of potential fallacies. Learning about the various fallacies would also be a form of logical aid.

A general skepticism about reasoning

It is perhaps clear from some of my remarks about that I think one ought to apply a general skepticism about reasoning. We can strengthen this skepticism by considering how many times we have reasoned wrongly in the past. I will bet that for any person it has happened an innumerable number of times. From this we may infer with good inductive certainty that we will probably reason wrongly in the future too. We may also observe that the fallacies in reasoning often happen with unaided reasoning. Indeed the more unaided it is, the more often it goes wrong (i.e. a fallacy is committed). This is hardly surprising.

I shall give an example of unaided reasoning going wrong. The modal fallacy strikes me as the best example of unaided reasoning going wrong. The modal fallacy very often happens when people try to reason about something complex, that is, the consistency of foreknowledge and free will. This issue involves both temporal concepts and alethic concepts. Explaining the fallacy would take us too far away from the subject of this essay and thus I will leave it unexplained, but see N. Swartz for an explanation.1

Some issues are so complex and strange that the probability of our unaided reasoning not making a fallacy is so little, that we ought not to bother trying with it at all. This is the case with much of mathematics, and most people accept this. Nearly no one tries to reason informally about differential or integral math, and there is a good reason not to. It would not work very well. It would be a waste of time. A similar case is complex issues about or involving infinities. Though here the situation is different. Some people do try to use unaided reasoning to think about it, and not surprising the results are not good. They should have used the special kind of mathematics that has been developed to deal with it. I hold that these mathematical cases serve as a good analogy for various philosophical problems. The problems are so complex that it would, and it is, a waste of time to try to reason about such issues without help, that is, unaided.

Determining how complex an issue is

Suppose now that you’ve accepted that it is a waste of time to attempt to reason unaided about some philosophical problems. How do we determine which issues are too complex to be handled by unaided reasoning? Suppose we use this essay as an example. We can determine what level of general skepticism we ought to have about it by considering two things a) how complex the issue is, and b) how unaided the reasoning in this essay is.

How might we go about answering the first question? One good idea is to look at the issue and try to identify modal terms in it. Modal terms are terms such as “ought”, “can”, “necessary” etc. There are a few of these in this article but only a few. The presence of a few of these gives a weak reason to be skeptical about the reasoning in the essay.

How about the second question? We ought to keep the things I mentioned before in mind. Is there any formal logic in the essay? No. So there is no help from this to be had. Does the author know logic and fallacies? Yes. This is some aid. Are the arguments presented in clear language? Yes, to a good degree. How complete are the arguments formulated? To a medium degree. So, we can conclude that the reasoning presented in the essay is somewhat aided.

Now we have to consider the answer to the first and the second question together. Is the issue complex enough and the reasoning unaided enough to warrant extreme skepticism? No. I submit that the issue is complex enough to warrant a medium level of skepticism but that the aids used reduce the warranted level of skepticism a bit. It seems unnecessary to require formal logic to be used to trust the reasoning in this essay.

Similar reasoning to the above can and should be applied to any form of argumentation.

Aiding our reasoning

Supposing that you have accepted my reasoning above. Suppose now that we encounter a really complex issue. We now see that our unaided reasoning about it would not be very useful because we ought to have a high level of skepticism about it. To avoid this skepticism, how might we aid our reasoning? Basically we can do the things I mentioned in the beginning: We might improve clarity by avoiding ambiguous words and phrases. We might improve explicitness by writing out the complete arguments. We might get help from logic by formalizing the arguments and checking that they are formally valid. (Or something equivalent in the case of inductive arguments.) We might learn of the various modal logics to help us think about issues involving modalities. This last one I think is very important, cf. the example of the modal fallacy.

Consider the issue of examining whether a being being omnipotent implies that that being is eternally omnipotent.2 This issue involves lots of modalities and different kinds too. There is both the temporal kinds and the alethic kinds. I submit that this issue is so complex, that our unaided reasoning about it is close to useless; If we ought to reason about it, then we ought to reason highly aided about it. I also think that that issue is too complex for us to properly reason about it without formal logic. I base this on observed discussions concerning this issue and similar issues. The discussions almost never get anywhere without the aid of formal logic or at the very least the knowledge of propositional, predicate and the various needed modal logics.