This will not involve many science facts as the discussion is wholly philosophical in nature. This is an epistemological, not scientific essay, it just happens to use some facts of science.

I want to show the value of thinking of things as inconsistent sets of propositions (or whatever truth carrier you like, but I like propositions) or at least implausible sets of propositions (when at least one inference is inductive (or non-deductive, if you like that term better)).

Consider this set of propositions:

1. Newton’s physics is correct.
2. Things are in a such and such way at time t1.
3. That Newton’s physics is correct and things are in a such and such way at time t1, implies that things will happen in such and such way at time t2.
4. A study found that such and such did not happen at time t2.
5. The study is correct.
6. That the study is correct implies that such and such did not happen at time t2.

This set is plainly inconsistent; it cannot be true. At least one proposition in this set is false. Suppose we are around the time when Einstein introduced his relativity theories. At that time physicists had pretty good reason to believe (1) (among others: good explanatory power, lots of empiric confirmation), and I’m sure some people called physicists who did not stop believing in Newton’s physics even when some studies found results that are contrary to the predictions of Newton’s physics given some antecedent state of affairs dogmatic. I’m fairly sure such a claim of dogmatism is often thrown around in similar cases.

My point is that it is unwise to claim someone is being dogmatic quickly. For there are many things other than some widely accepted theory that could be wrong (in this case (1)). We could be wrong about the antecedent states of affairs (in this case (2)), or wrong about what the theory predicts (in this case (3)) given that state of affairs; perhaps the scientist that make the prediction from the theory made a calculation error. Something similar applies the the study that ‘challenges’ the accepted theory (in this case (5)). So there are many things that could be wrong without the accepted theory being false. It is wise to consider that before calling people that are being epistemically conservative for dogmatic.

The method of putting the relevant propositions in an inconsistent set forces us to be made aware of some perhaps not normally discussed propositions without which the set would be consistent (or not-implausible). Usually in a moderately complex case such as the one with Newton’s physics, a set of propositions that form as inconsistent set (or implausible) will contain 5-10 propositions. In more complex cases, the sets can be much longer (such as very complex cases involving the impossibility of an infinite past which involves temporal and modal logic). In general, the more propositions we can find that together forms an inconsistent set (or implausible), the better overview, and the easier to is to make a justified decision about which proposition(s) to stop believing in in the case that one actually believes all of them. If we are to avoid inconsistent beliefs (=inconsistent objects of beliefs), then we should think of the many potentially epistemically justified ways there are to deal with a such inconsistent set.

In the above case, rejecting (4) would probably not be a wise decision, neither would it be to reject (6). If there is only one study and it is not exceptionally well done, then rejecting (5) is probably not a bad decision to begin with. If more studies (by competent scientists) confirm the first study, then sooner or later we should begin wondering if not our beliefs would have better coherence were we to reject the theory (1). But before we do that we should consider other alternatives such as (2) and (3). It would not be good if we rejected some theory and later found it that we had no grounds to do that because we were wrong about what the antecedent state of affair was (2).

This way of solving problems (which usually involve an inconsistency of we add together the relevant propositions to a set), is applicable to every topic that I have thought of. It is especially useful to very complex situations where it is hard to get an overview and it seems hard to settle on a specific solution (that is, hard to find out which proposition is the epistemically most justified to deny).

Using the formalization system I wrote of earlier, let’s take a look at this famous question.

First we should note that this is a yes/no question which is different from the questions that I have earlier formalized. The earlier questions sought to identity a certain individual, but yes/no questions do not. Instead they ask whether something is the case or not. So this time I cannot use the (x=?) question phrase from earlier, since there is no individual to identify similarly to the earlier cases.

One idea is to simply add a question mark at the end. Like this:

F1. (∃x)(∃y)(Wxy∧Bxy∧Cxy∧x=a)?

Wxy ≡ x’s wife is identical with y

Bxy ≡ x beats y

Cxy ≡ x used to beat y¹

a ≡ you

But this fails to capture that it is not all of the things that are being asked whether they are the case or not. It is only the Bxy part that is being asked. The rest is stipulated as true. We can change the formalization to capture this, like this:

F1*. (∃x)(∃y)(Wxy∧Cxy∧x=a)∧(Bxy)?

The question mark is now understood as a predicate that works on whatever is before/to the left of it. (In parentheses for clarity.) Not to the right like with the other monadic predicates and propositional connectors (¬, ◊, □, etc.). In this case the question mark only functions on (Bxy) and not the rest of the formula.

Translated into LE:

There exists an x and there exists an y such that x’s wife is y and x used to beat y and x is identical with you and is it the case that x beats y?

Answering yes/no questions

When answered in the positive, the answered version simply removes the question mark. Producing:

F2. (∃x)(∃y)(Wxy∧Cxy∧x=a)∧(Bxy)

Answering in the negative removes the question mark and adds a negation sign to the part of the formula that the question predicate is working on. Producing this:

F3. (∃x)(∃y)(Wxy∧Cxy∧x=a)∧¬(Bxy)

If the produced formula is true, then the question has been answered correctly. However since this question is loaded. Both of the produced formulas are false, that is, it is both false that:

F2. (∃x)(∃y)(Wxy∧Cxy∧x=a)∧(Bxy)

There exists an x and there exists an y such that x’s wife is y and x used to beat y and x is identical with you and x beats y.

and that:

F3. (∃x)(∃y)(Wxy∧Cxy∧x=a)∧¬(Bxy)

There exists an x and there exists an y such that x’s wife is y and x used to beat y and x is identical with you and it is not the case that x beats y.

Since they both imply the falsehood:

(∃x)(∃y)(Wxy∧Cxy∧x=a)

There exists an x and there exists an y such that x’s wife is y and x used to beat y and x is identical with you.

Interpretation

I earlier wrote of the logical interpretation of subjects.¹ There I suggested, following Russell, that the subject of a descriptive, active, meaningful (DAM) sentence should be interpreted as an existential quantifier (∃x) but I now believe that that this seems to depend on who made the utterance and in which situation. Suppose for instance that a positive atheist² makes this utterance:

E1. God is omnipotent.

Do we really want to interpret this as:

E1′. (∃x)(Gx∧Ox)³

If we did, then the atheist would have contradicted himself since from (E1′) the existence of God follows. From this I conclude that this interpretation is implausible.

The utterer and the situation

One idea is to let (E1) represent a conditional when uttered by an atheist:

E1”. (∀x)(Gx→Ox)

Such a conditional is consistent with an atheistic position; It is not possible to deduce that God exists from (E1”). How should we think of sentences that are like (E1)? Should they always be interpreted as existential claims, should they always be interpreted as conditional claims or should the interpretation depend on the utterer and the circumstances in which it was uttered? The first option has already been dealt with and found implausible. Let’s consider the second option.

Consider this everyday sentence:

E2. The door is open.

If I said this to my roommate while we were both out in the garden, I think that he would think that I was silly or talking about some door far away. He would never interpret this sentence as a conditional which in that case is true. Is it true not because there is a door and it is open but it is true because there is no door at all in the garden. I imagine that it is like this in many other everyday situations. Suppose that is true, that is, everyday sentences like (E2) are most often best interpret as existential claims. We may allow that sentences involving non-everyday terms like “God” are often best interpret as conditionals.

These considerations indicate that the same sentence form Subject – sentence verb – subject predicate may yield different logical forms depending on which words are used. So, there is a disconnection between language form and logic form. This is undesirable.

Return to the first example. Suppose that a theist said the same sentence. Should it be interpreted as an existential claim or a conditional? I suppose that it is best to interpret it as an existential claim. But for the theist it would not make much of a difference since he also believes that God exists, and from that God exists, and the conditional, it follows that God is omnipotent.⁴

But even an atheist’s utterance of (E1) may best be interpreted as an existential claim. Suppose that the current american president is a closet atheist, that he is making a public speech and that the public believes that he is a theist. In that case it would be best for the public to interpret his words as an existential claim and not a conditional.

Notes

Abstract

I invent and explore a terminology about degrees of sentences, I explore how to negate sentences and sentence parts in english, I distinguish between verbs that can be used in sup-sentence parts and verbs that cannot, I discuss some problems with the verb “ought”, and lastly I explore the relationship between this terminology and predicate logic.

Negating sentences in english (PDF, 15 pages)

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.

Formatting requires a PDF file.

modal-logic-formalization

This is a translation of my earlier article on the subject. Link.

The modal fallacy

By Emil Kirkegaard, Deleet.dk

This fallacy is rather common among persons who are not well versed in logic (especially modal logic). Consider these two not logically identical sentences:

I) If there exists at least one subject S that knows which outcome U situation F will have at time t₁, then outcome U will necessarily happen at time t₁.

II) Necessarily, if there exists at least one subject S that knows which outcome U situation F will have at time t₁, then outcome U will happen at time t₁.

An unlucky property of the natural language is that it does not distinguish between these two sentences and that one normally almost always uses (I) if one is not aware of the difference. This problem is actual in Danish and English and maybe other languages as well. What happens is that ‘necessary(-ily)’ gets misplaced. It gets placed in the consequence of an implication but in reality it speaks (or should speak) of the entire implication. Logically the sentences can be formalized like this:

I’) P→ □Q

II’) □(P→ Q)

Now the difference should be clear and it should also be clear that (I) is false and that (2) does not support what one normally believes that it supports. Let us consider two arguments where the first is very common among young atheists who are not well versed in logic:

Argument a

If God knows which outcome situation F will have at time t₁, then the situation will necessarily have the outcome he knows it will have. If the situation necessarily will have the outcome, then all humans who are involved in situation F have no free will in F. Moreover, if God knows the outcome of all situations, then no man has a free will.

The argument is a little complicated, let us just look at the first part of it:

1. God knows which outcome situation F will have at time t₁. (premise or hypothesis)

2. If God knows which outcome situation F will have at time t₁, then the situation will necessarily have that outcome. (premise)

3. The situation will necessarily have that outcome. (1, 2)

4. If the situation will necessarily have that outcome, then no human in the situation F has a free will.

5. No human in situation F has a free will. (3, 4).

[snip a bit about the Danish language not having a future case]. Note that ‘necessarily’ will typically not be placed in the start of a sentence in natural language like one does in philosophy to reduce ambiguity (cf. (3)). The argument (a) can me formalized like this:

1′. P

2′. P→ □Q

3′. □Q

4′. □Q→S

5′. S

The argument is valid but the problem is that (2) is false. Defenders of the argument or similar will typically argue (2) by noting that if someone knows something, then it is necessarily true, because otherwise they would not know it. This is also false in this specific formulation. That whatever one believes is true is a necessary condition for knowledge (cf. JTB+) but from here it does not follow that it is necessarily true.

Remember that a necessary truth is true in all worlds and therefore it follows that if a person knows something then that something is true in all logically possible world. But this is false because there is a logically possible world where Earth is flat but still I know that Earth is round.

Recall sentence (I) and (II) from earlier. The analogue sentences for knowledge are these:

I”) If someone knows p, then p is necessarily true.

II”) Necessarily, if someone knows p. then p is true.

(I”) is false and (II”) is true. But for argument (a) to be sound, then it is required that (I”) is true. If we substitute (I”) with (II”), then the argument is no longer valid because □Q doesn’t follow from (1) and (2).

Let’s consider another argument.

Argument b

If my mother knows which education I will choose after high school, then I will necessarily choose that education. My mother knows which education I will choose after high school, therefore I will necessarily choose that education.

The we can again spot the problem where ‘necessarily’ is misplaced. It is true when it speaks of the entire implication but false then it only speaks of the consequence. If this argument was sound then it would prove that I cannot change my mind, but this is false.

References

An awesome source by professor Norman Swartz (Indiana University Ph.D., 1971 (History and Philosophy of Science)):

http://www.sfu.ca/philosophy/swartz/modal_fallacy.htm

Internet Encyclopedia of Philosophy about the modal fallacy in divine foreknowledge.

http://www.iep.utm.edu/f/foreknow.htm#section6

The Fallacy Files about modal fallacies in general. There is a subpage about the specific fallacy I have in mind.

http://www.fallacyfiles.org/modalfal.html

Stanford Encyclopedia of Philosophy about modal logic. They mention the fallacy just before the section on deontic logic.

http://plato.stanford.edu/entries/logic-modal/