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.

Dialogue – Kennethamy and Emil

Kennethamy in response to something about certainty:

I did not say there was such a thing as objective certainty. I said objective certainty was what Descartes was aiming at, not subjective or psychological certainty. He did not care about that. People feel certain about all sorts of things, about which they later turn out to be wrong. And people feel certain about contrary things. Subjective certainty is of no epistemological interest. Descartes presented as his prime example of objective certainty, “I exist”. So, if you are going to deny there is such a thing as objective certainty, you have to deny you are objectively certain that you (yourself) exist. That is, that it would be possible for you to be mistaken about whether you exist. Do you think it would be possible for you to believe that you exist, and still not exist? For that is what it would be for you to be mistaken that you exist.

None of your pronouncements about certainty being a useful fiction really matter. You may think what you like. But you still have Descartes argument to wrestle with, and simply saying that objective certainty is a useful fiction, or the truth with a capital T is a fiction, will really not cut it. It is the argument that is the thing, and as Socrates said, “we must follow the argument wherever she leads us”. How do you handle Descartes’s argument that it is impossible to be mistaken about whether one exists, for in order to be mistaken, one must exist? Have you a reply?

Emil in response to the above:

Not quite sure that subjective certainty is of no epistemic interest, but otherwise I agree.

Kennethamy in response to the above:

Yes. I have been told over a trillion times not to exaggerate.

Emil in response to the above:

Hahahahaha. Priceless!

Source.

Beliefs as secondary truth bearers in a pluralistic proposition theory

It is common to speak of true beliefs. As an example think of the JTB analyses of knowledge. JTB, that is, justified true belief. One could see “true belief” as a shorthand for “a belief in a true proposition”. This seems to be the case. It is common to call the theory for the JTB analysis of knowledge, but when writing down the three necessary and sufficient conditions, one does not write “has a true belief” but “p is true”.

But perhaps it is a good idea to allow for some or all beliefs to be true/false while still maintaining that it is propositions that are the primary truth bearers. A reason not to think so is again parsimony similar to the case of allowed sentences to be true too. Suppose that it is a good idea anyway.

What are the truth-conditions for beliefs?

First we may note that there seems to be no problem with ambiguity as there is with sentences as truth bearers. Perhaps there are ambiguous beliefs. We will suppose that there are none. We may, then, introduce these simple truth-conditions for beliefs:

A belief is true iff the proposition believed in is true.

A belief is false iff the proposition believed in is false.

Sentences as secondary truth bearers in a pluralistic proposition theory #2

I have had some additional thoughts about this after discussing it with fast here.

First fast asks:

“You said, “a sentence is true [if and only if] it expresses exactly one proposition and that proposition is true. I don’t understand the reasoning behind the “exactly one” condition as you have worded it. An implication of what you said is that a sentence that expresses more than one proposition (hence, not exactly one proposition) is not true because you said, “if and ONLY if”, but I don’t see why you would think that.

[...]

Is it because if one of the propositions is false, then the sentence is both true and false and that’s a contradiction?”

I did reply to that in the thread but I think it deserves a longer reply.

First, yes, it is to avoid conflicts with bivalence about sentences, that is, for all sentences, a sentence is either true or false but not both. But then I realized that maybe one could drop bivalence about sentences but not drop it about propositions. Supposing that one drops bivalence about sentences, then one can adopt much broader truth-conditions of sentences:

A sentence is true iff it expresses a true proposition.

A sentence is false iff it expresses a false proposition.

However it is also possible to accept broader truth-conditions even keeping bivalence about sentences. One could just specify that all the propositions expressed by a sentence has to have the particular truth value. It doesn’t matter if it is one or more:

A sentence is true iff it expresses only true propositions.

A sentence is false iff it expresses only false propositions.

A justification principle about logical implication

[Update 11/22/09]

I note that Ben actually talked about this principle in a post on his blog, “if it’s reasonable to believe a bunch of premises, it’s also reasonable to (on the basis of the logical connection) believe the conclusions that can be validly inferred from those premises”,

[/update]

I have recently been discussing Gettier’s famous counter-examples to the JTB theory of knowledge. In his original paper Gettier argued that there are some cases where all the necessary and sufficient conditions of knowledge according to JTB theory are met, but the person in question fails to know. In the thread user ACB asked that:

If (1) the man who will get the job is Jones, and
(2) Jones has ten coins in his pocket,
then
(3) the man who will get the job has ten coins in his pocket.

But does it logically follow that if Smith is justified in believing (1) and (2), then he is justified in believing (3)? [followed by a proposed counter-example]

I and another person thought that it did follow. In other words we subscribed to the following principle about justification:

For all persons, for all propositions, P, and for all propositions, Q, that a person is epistemically justified in believing that P, and that P logically implies Q logically implies that that person is epistemically justified in believing that Q.
(∀x)(∀P)(∀Q)(Jx(P)∧P⇒Q))⇒Jx(Q)

The above case seems to me to be a true instantiation of the justification principle. ACB disagreed with the principle and proposed a counter-example with the alphabet which did not convince me. He then tried another counter-example that involved some mathematical propositions. That proposed counter-example did not convince me either, but it did make me think of an example that did convince me. Here’s my counter-example:

1. 1+1=2

2. 456·789=359784

Both of these propositions are true, they are even necessarily true. According to the definition of logical implication they imply each other (and themselves), since any necessarily true proposition imply any (other) necessarily true proposition.1

Now suppose that a child is learning elemental math. Say that she has not even learned multiplication yet, however she has learned that 1+1=2 is true and she knows this. That implies that she is epistemically justified in her belief that 1+1=2. But it clear to me that she is not epistemically justified in believing that 456·789=359784. This is a counter-example to the justification principle and the principle is therefore false.

It seems to me that one could perhaps save the justification principle with some relevance logic understanding of “logical implication”. However I shall not pursue that here.

Notes

1The definition of “logical implication” is: a proposition logically implies another proposition iff in all possible worlds where the first proposition is true, so is the second.

The sentence theory of truth bearers – the problem of ambiguity #2

Ben Burgis over at (Blog&~Blog) has commented on my essay about the monist sentence theory of truth bearers. I have some comments on his comments. Aha! Let the comment wars begin.

Ben makes three somewhat related points. I have comments only for the two first.

The first point

Here’s what he had to say:

(1) The indexical phrasing might make things a bit confusing in this specific case. On one level, it’s surely contingent that Ben Burgis exists, but one might argue that it’s logically impossible that any instance of “I exist” tokened by anyone could ever be false. What one thinks about what to ultimately make of this might depend on what one thinks about the widely alleged essentialness of indexical claims–if “I exist” really *means* Ben Burgis exists, that’s one thing, but given that I could forget that I’m Ben Burgis but still be quite sure that I exist, there are tricky issues at play here.

I certainly did not try to get into problems by using indexicals (such as pronouns). It seems that I can avoid this issue by simply choosing another example (more about this in the second comment) or avoiding indexicals at all. I suppose I could just change it to:

S. It is logically possible that Emil Kirkegaard exists and that Emil Kirkegaard does not exist.1

(Though as for the problems with being wrong about “I exist” (the proposition!), see this discussion over at Philosophyforum.com. There is something curious about the phrase “cannot be wrong” when applied to truth bearers. It is not clear how to properly understand it. I made two quick analyses of the concept in this essay.)

The second point

Ben’s second point:

(2) Another complicating factor about the example is that existence is being treated as a predicate, which seems to assume “noneism,” the view that there are objects that have some properties (like being referred to) but which don’t exist. Anyone who agrees with Quine’s claim in “On What There Is?” that the answer to the question of ontology (“what exists?”) is “everything” would, while agreeing that it’s possible for there to be no object that Ben-Burgisizes, strong object to ◊¬Ei.

I do not believe in “noneism” (never heard of it). I only write it like that because it is simpler and not confusing in most cases. Here are two other ways to formalize the same sentence (original (S)):

1. ◊(∃x)(Ux)∧◊¬(∃x)(Ux)

2. ◊[(∃x)(Ux)∧¬(∃x)(Ux)]

(Where “Ux” is some unique description of me. I will just translate it to “is Emil Kirkegaard”, alternatively it could be “fits the unique description of Emil Kirkegaard”.)

So, in predicate logic english-ish:

1*. It is logically possible that there exists at least one person such that that person is Emil Kirkegaard and it is logically possible that it is not the case that there exists at least one person such that that person is Emil Kirkegaard.

2*. It logically possible that (there exists at least one person such that that person is Emil Kirkegaard and that it is not the case that there exists at least one person such that that person is Emil Kirkegaard).

The sheer length of this is why I usually use ‘simplified predication’ when formalizing.

More ambiguity?

The sentence may have been more ambiguous than I originally thought. How about this interpretation?:2

3. 1. ◊(∃x)(Ux)∧¬(∃x)(Ux)

3*. It is logically possible that there exists at least one person such that that person is Emil Kirkegaard and it is not the case that there exists at least one person such that that person is Emil Kirkegaard.

That’s just applying the predicate “it is logically possible” to the first part and not the second.

Notes

1Philosophers have some weird history for using their own names in examples. I shall follow their example. Just for kicks.

2Is there any convention about what to do when both asking a question and mentioning things that require a colon (:)?

Sentences as secondary truth bearers in a pluralistic proposition theory

It seems to me that monist sentence theories are too implausible, but might it not nonetheless be the case that some sentences are true/false? In this essay I will discuss sentences as secondary truth bearers.

Pragmatic value

I can see that it has some pragmatic value to say that sentences are also sometimes true/false in addition to propositions. The pragmatic value is that it makes it easier to talk about certain things without having to use complex phrases like “the proposition expressed by (the sentence) is true (or false)”. Perhaps this is a good enough reason to posit that sentences also in some cases have the properties true/false.

An alternative solution is to invent some shorthands for talking about propositions expressed. See (N. Swartz, R. Bradley, 1979).

Parsimony

The problem I see with it is that of parsimony. “Entities must not be multiplied beyond necessity” (Wiki). Is that not exactly what we are doing? At least if properties are entities. I think they are since entity is the most inclusive set (similar to “thing”)1. But perhaps it is not as problematic to multiply properties as it is to multiply other kinds of entities in an explanation. I don’t know.

What are the conditions for a sentence being true/false?

This is how I see understand the position:

A sentence is true iff it expresses exactly one proposition and that proposition is true.
A sentence is false iff it expresses exactly one proposition and that proposition is false.

The phrase “ expresses exactly one proposition” seems to avoid the ambiguity problem that I wrote about earlier.

Notes

1Yes, I am aware of Russell’s paradox that may arise when defining sets like this. I’m working on a ‘solution’.

Truth bearers

The truth bearers are the kind of entities that have the property true. It is thought that it is the same kind of entities that have the property false too. They are sometimes referred to as the bearers of truth/falsity. I shall just refer to them as “truth bearers”.

Theories of truth bearers

There are multiple theories for what kind of entities truth bearers are. Some think it is sentences that are true/false but I think that there are too many problems with these theories. I shall call such theories for sentence theories of truth bearers.

Others, like me, think that it is propositions that are true/false, proposition theories of truth bearers.

Some presumably think something else, perhaps that it is beliefs that are truth bearers, belief theories of truth bearers.

Multiple truth bearers

It is often written “the bearers of truth/falsity”. Note the definite article “the”, it seems to imply (implicature (SEP), not implication) that there is only one kind of entities that are true/false. But could it not be that there were multiple kinds of entities that are true/false? I can see no good reason not to think this possible. The only objection that I can think of is parsimony/Occam’s Razor (Wiki); “entities must not be multiplied beyond necessity” and this presumably applies to properties too. One should not multiply properties if unnecessary. “Necessary for what?”, one might ask. “Necessary to explain truth and falsity”, I answer.

Proposed terminology

We may call theories that restrict the properties of truth/falsity to a single kind of entities for monist theories of truth bearers. Theories that allow for multiple kinds of entities may be called pluralistic theories of truth bearers.

If there entities that are always true/false, they may be called the primary truth bearers. Other entities that only in some cases bear truth/falsity may be called secondary truth bearers.

The sentence theory of truth bearers – the problem of ambiguity

I think there are numerous problems with the sentence theory of truth bearers. Here I will touch on one problem, that is, the problem of ambiguity. I start by assuming the sentence theory of truth bearers.

The problem

Consider the sentence:

S. It is logically possible that I exist and that I do not exist.

Is (S) true or false? I can’t tell because it is ambiguous. If you don’t see how it is ambiguous try deciding whether the predicate “It is logically possible” applies to only “I exist” or to both “I exist” and to “I do not exist”. Which is it? Logic helps us see the difference. We may formalize the two interpretations like this:

1. ◊Ei∧◊¬Ei
2. ◊(Ei∧¬Ei)

(Where “Ex” means x exists, “i” means I.)

We can translate these into english-ish:

1*. It is logically possible that I exist and it is logically possible that I do not exist.

2*. It is logically possible that (I exist and that I do not exist).

The first is true since my existence, anyone’s existence is a contingent matter (except contradictory entities). The second is false since it is not logically possible that I both exist and not exist (at the same time). That’s a contradiction. The problem is with deciding whether or not (S) is true or not. It can mean either (1) or (2), but which? It seems that there is no way in principle to tell whether (S) is true or not.

Both true and false

Another idea is to accept that it means both and simply say that (S) is both true and false. That doesn’t strike me as a good solution. It is basically giving up classical logic and accepting dialetheism.1

Neither true or false

One more plausible solution is to say that (S) is neither true or false; adopting this principle: All ambiguous sentences are neither true or false. The problem with this is that lots of sentences that we normally use are ambiguous, but maybe not in the context that they are used in. This is the best solution that I know of to the problem of ambiguity. Though it runs into methodological problems. When is a sentence ambiguous and when is it not?

Notes

1Maybe Priest would be happy about this? Another true contradiction discovered!

Psychological and epistemic certainty

A rewrite of an earlier article “two kinds of certainty”.

-

A quick explanation of two types of certainty that people tend to confuse.

Psychological certainty

The first is the one we typically mean in normal language. It’s called psychological certainty. It’s a feeling of certainty; A confidence in something. This is the one we’re talking about when we say things like “Are you 100% sure?”. It is possible that someone is 100% psychologically certain that something is true and that the something is actually false. Psychological certainty comes in degrees. Good examples of psychological certainty and false beliefs are found in religious people and various sport fans.

Epistemic certainty

The second is epistemic certainty. This is the one that philosophers usually talk about. It’s the inability to be wrong type of certainty. If one is epistemically certain, then one cannot be wrong in some sense. This type of certainty is also called cartesian (after Descartes) certainty, infallible certainty and absolute certainty. This type of certainty does not come in degrees; Either one is epistemically certain or one is not. It is not entirely clear how to explicate this kind of certainty. Here are two proposals:

1. (∀x)(∀P)[Bx(P)∧□P⇒Cx(P)]
For all agents and for all propositions, (that an agent believes a proposition and that proposition is necessarily the case) logically implies that that agent is epistemically certain of that proposition.

2. (∀x)(∀P)[Bx(P)⇒P]
For all agents and for all propositions, that an agent believes a proposition logically implies that proposition.

Translation keys

Domains. x is agents. P is propositions.
Bx(P) means x believes that P.
Cx(P) means x is epistemically certain that P.
⇒ is logical implication.

For convenience, it smart to type p-certain and e-certain to distinguish between them.

References

philofreligion.homestead.com/files/CertaintyandIrrevisability.htm (About psychological and epistemic certainty.)

Psychological and epistemic certainty

A quick explanation of two types of certainty that people tend to confuse.

Psychological certainty

The first is the one we typically mean in normal language. It’s called psychological certainty. It’s a feeling of certainty; A confidence in something. This is the one we’re talking about when we say things like “Are you 100% sure?”. It is possible that someone is 100% psychologically certain that something is true and that the something is actually false. Psychological certainty comes in degrees. Good examples of psychological certainty and false beliefs are found in religious people and various sport fans.

Epistemic certainty

The second is epistemic certainty. This is the one that philosophers usually talk about. It’s the inability to be wrong type of certainty. If one is epistemically certain, then one cannot be wrong in some sense. This type of certainty is also called cartesian (after Descartes) certainty, infallible certainty and absolute certainty. This type of certainty does not come in degrees; Either one is epistemically certain or one is not. It is not entirely clear how to explicate this kind of certainty. Here are two proposals:

1. (∀x)(∀P)[Bx(P)∧□P⇒Cx(P)]
For all agents and for all propositions, (that an agent believes a proposition and that proposition is necessarily the case) logically implies that that agent is epistemically certain of that proposition.

2. (∀x)(∀P)[Bx(P)⇒P]
For all agents and for all propositions, that an agent believes a proposition logically implies that proposition.

Translation keys

Domains. x is agents. P is propositions.
Bx(P) means x believes that P.
Cx(P) means x is epistemically certain that P.
⇒ is logical implication.

For convenience, it smart to type p-certain and e-certain to distinguish between them.

References

philofreligion.homestead.com/files/CertaintyandIrrevisability.htm (About psychological and epistemic certainty.)