Tough modal logic formalization

Source.

Gamgee wrote:

In the Ontological Argument, Anselm refutes Guanilo’s Perfect Island criticism by stating that God has Aseity. Anselm does not, as far as i know, make any attempt to prove that God has Aseity, or even if Aseity is logically possible.

I argue that it is not, for the following reason:

1) We accept that God does not exist in the physical world.

2) Therefore, there must exist some realm outside of normal time and space (realm x, for conveniency)

3) God exists in realm x

4) Therefore, God requires realm x to exist so that he can exist in it.

5) Without realm x, God would not exist

6) Therefore, God is contingent upon realm x for his existence. Aseity is false.

Any obvious holes in my logic?

It’s not that bad but it’s badly structured. I’ll help you:

Definitions

D:x ≡ things

D:y ≡ things

Ex ≡ x exists

Fxy ≡ x exists in y

Cxy ≡ x in contingent upon y.

a ≡ God

b ≡ non-physical world

c ≡ physical world

Desired conclusion: God is contingent upon the non-physical world.

Desired route: Something to do with worlds.

Version one

1a. Ea

God exists. (Premise)

2a. ¬Fac

God does not exist in the physical world. (Premise)

3a. (∀x)(Ex→(Fxc∨Fxb))

For all x, if x exists, then x exists in the physical world or x exists in the non-physical world. (Premise)

4a. ⊢ Fab (1, 2, 3)

Therefore, God exists in the non-physical world.

5a. (∀x)(∀y)((Ex→Fxy)∧¬Ey)→¬Ex

For all x, if x exists in y and y does not exist, then x does not exist.

6a. ((∀x)(∀y)((Ex→Fxy)∧¬Ey)→¬Ex)→Cxy

If, for all x, for all y, if, if x exists, then x exists in y and y does not exist, then x does not exist, then x is contingent upon y.

7a. ⊢ Cab (5, 6)

Therefore, God is contingent upon the non-physical world.

But 5 is false. It says that if x exist, then x does not exist. I got stuck there. Trying to figure out how to formulate it in some other way to avoid this.

Version two

1b. Ea→Fab

If God exists, then God exists in the non-physical world. (premise)

2b. (∀x)(∀y)((Ex→Fxy)∧¬Ey)→¬Ex

For all x, for all y, if, if x exists, then x exists in y, and y does not exist, then x does not exist. (premise)

3b. ((∀x)(∀y)((Ex→Fxy)∧¬Ey)→¬Ex)→Cxy

If for all x, for all y, if, if x exists, then x exists in y, and y does not exist, then x does not exist, then x is contingent upon y. (premise)

And here I got stuck. I couldn’t find a way to get to Cxy without assuming ¬Eb.

Version three

1c. ¬Eb→¬Ea

If the non-physical world does not exist, then God does not exist. (Premise)

2c. (∀x)(∀y)(¬Ey→¬Ex)→Cxy

If for all x, for all y, if y does not exist, then x does not exist, then x in contingent upon y. (Premise)

3c. ⊢ Cab (1, 2)

Therefore, God is contingent upon the non-physical world.

This works. Premise two is analytic. Premise one is sometimes true per definition.

Comments

This argument was remarkably hard to formalize for me.

JTB+ and the first person perspective

Emil Kirkegaard

I have already expanded a bit on certain problematic aspects of the JTB+ theory. In this essay I will expand on a certain methodological feature: If you’re “looking” from the 1. person perspective, then there is little or no difference between justified belief and knowledge.

Just to recap, JTB+ is a theory of what knowledge is. JTB+ claims that knowledge consists of at least four jointly necessary and sufficient conditions:

JTB+. S knows that p iff
1. S believes that p.
2. S is justified in believing that p.
3. p is true.
4. [It is not a Gettier case]

I have assumed that JTB+ is true, that is, that JTB+ adequately captures what knowledge is; is a proper account of knowledge.

I have also assumed that there is no difference in the level of justification between justified belief and knowledge, that is, the same level or strength of justification is needed to be justified in a belief and to know a belief if the other conditions also apply (i.e. the believed proposition is true and it is not a Gettier case.).

Let’s look at the JTB+ set of conditions from then 1. person perspective. This is done by adding ‘S believes that…’ to the 4 conditions above:

JTB+ 1. person. S believes that S knows that p iff:
1a. S believes that S believes that p.
2a. S believes that S is justified in believing that p.
3a. S believes p is true.
4a. S believes that [It is not a Gettier case]

Most philosophers agree that one has some sort of special ability to (perhaps infallible) tell what oneself believes. And so if S believes that S believes that p, then it (reasonably; arguably) follows that S believes that p. In other words: (1a) implies (1).

Interestingly, (3a) implies (2a) given that one is epistemically rational. By epistemic rationality I mean that one only believes something if one is epistemically justified in doing so. This is contrasted with another form of rationality called pragmatic rationality which, roughly, goes like this: One believes that p if believing p is good for one. With ‘good for’ I mean something like: makes one happy. When I write ‘justified’ I mean ‘epistemically justified’.

Also it is assumed that one is not irrational, that is, believing things without any justification at all.

Remember that believing in p is logically equivalent to believing that p is true. This is what it means to believe in p.

Then, assuming the person in question is epistemically rational, then it follows that (3a) implies (2a).
Also interestingly, (2a) implies (3a). This is because, at least generally speaking, if one believes that p and p implies q, then one believes that q. In other words: If one has justification to believe something, then one believes it. Beginning to believe that p is, at least generally speaking, an automatic process resulting from the belief that one is epistemically justified in believing p. There is, broadly speaking, no choice involved. This denies a view about belief formation that is called voluntarism.1 Quoting philosopher Theodore M. Drange:

Sobel assumes the voluntarist outlook that it makes sense to speak literally of “choosing to believe or not to believe.” I myself have strong misgivings about such an outlook. It does not seem to me that such an action is performed frequently or that it is performed by psychologically normal people. Rather, I find that normal people usually just believe automatically in accord with their assessment of the evidence available to them, and do not make choices to believe or not to believe.2

So, since (3a) implies (2a) and (2a) implies (3a), then they are logically equivalent. This implies that from the 1. person perspective truth and justification are conflated or equivocated given the premises and assumptions I have mentioned.

Does this establish the thesis I mentioned in the beginning? Almost. There is still a (perhaps set of) anti-Gettier case conditions to be met.

1However, this term is also used for other things.

2Taken from Theodore M. Drange, Review of Jordan Howard Sobel’s Logic and Theism, 2006; www.infidels.org/library/modern/theodore_drange/sobel.html

A quick argument for the principle of simplicity

Suppose there is a world where there are facts F1, F2, F3, … Fn that need to be explained. Suppose further that someone advances an infinite amount of theories that aims to explain the facts. Suppose even further that all the theories presented happen to explain the facts equally well.

The first theory implies the existence of one entity, E1. The second implies the existence of E1 and E2. The third implies the existence of E1, E2 and E3. … The N’th theory implies the existence of E1, E2, E3, … En entities. How should one choose which theory is more likely to be correct? The intuition I have is that the first is the most plausible and the one with the most implied entities (there is no one with the most though) is the most implausible. In other words: the more entities implied, the less probably the theory is. So, if we are to formulate this as a general reasoning principle we could do it like this:

Of two equivalent theories or explanations, all other things being equal, the simplest one is to be preferred.1

More reading:

en.wikipedia.org/wiki/Occam%27s_Razor

plato.stanford.edu/entries/simplicity/

www.iep.utm.edu/o/ockham.htm#H2

1Common phrasing of the principle. en.wikipedia.org/wiki/Occam%27s_Razor#Variations

Incomplete propositions, time parameters

Dialogue

Deleet:

Reality changes all the time such that the truth value of propositions change. For instance, the proposition “Deleet is alive” is true right now but it will become false when I die. Similarly, the proposition “John F. Kennedy is alive” is false because reality has changed and he died. When he was alive, the proposition was true. Before he was born the proposition was also false.

Ken:

The proposition that Deleet is alive in 2009 is true 1,000 years ago, and will be true 1,000 years hence. All you have to do is to insert the missing time parameter. One true, always true. Reality doesn’t change, what we say about it does.

Deleet:

It was my intention to leave the time parameter out. In this way the truth of the proposition changes when reality changes (correspondence theory of truth).

Reality; the objective world changes. I will not discuss that here.

Ken:

Yes. Your leaving it out is why you think that truth changes with time. You should put it back in so that you will not think that.

Deleet:

My view is that your conflating two propositions: “Subject S is alive” and “Subject S is alive at time t”. These are not identical; They are not the same proposition. They sometimes share truth values and they sometimes don’t.

Ken:

But the first is an incomplete proposition, which is why you think that truth is temporal.
It is like the difference between:

1. Dogs are mammals, and,
2. All dogs are mammals.

1 and 2 are not different propositions. 2 is just a complete proposition, and 1 is incomplete.
You would not say that Dogs are mammal is a vague fact because the sentence, “Dogs are mammals” is incomplete, would you?

Deleet:

1.
There is no such thing as an incomplete proposition. A proposition is “[w]hat is conveyed by a declarative sentence [...]“, source.

There are incomplete statements in the sense that the statements do not fully express the meaning they was intended to convey and so we fill in the rest of the meaning by interpretation. Your example with dogs is an example of us filling in: (1) can convey the meaning of “All dogs are mammals.” (2) or “the meaning of “Some dogs are mammals.”.

2.
Neither (1) or (2) are really propositions although in practice we treat declarative statements as propositions. They are not. The meaning they convey are. (1) does not convey any specific proposition because it can, depending on interpretation, have multiple meanings; (1) is ambiguous. (2) conveys a single proposition.

3.
I hold that my two statements (3) “Subject S is alive” and (4) “Subject S is alive at time t” are specific proposition conveying statements i.e. they convey precisely one proposition.

Ken:

Only if (3) is uttered at time t, when, I suppose, it will be construed as (4). I don’t know how you use, “statement”, “proposition”, or “sentence”. But you see what I mean. The time of the utterance allows us to understand what the complete statement is, just as the context will allow us to construe “Dogs are animals” as, “All dogs are animals”, but “Dogs are brown” as, “Some dogs are brown”, since we’ll assume that the speaker has the same knowledge as we do in the absence of any evidence to the contrary.

Deleet:

1.
The general assumption is to insert “at time t and t is now”. This still makes (3) and (4) different. In (4) t does not have to be now; It could be 1960.

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Source.

Propositions and time

A time parameter ≡ the part of a proposition-conveying sentence that conveys information about the time. E.g. “The Earth is round in 1960.“. The time parameter is the text marked with italic.

Ken seems to think that all truths are timeless; independent of time. I agree that some truths are timeless. All truths that have a set; non-relative time parameter (e.g. “in 1950″) are true no matter what the time is now. However, not all truths have a set time parameter. Normally when one makes an assertion, it is missing the time parameter. We interpreted it as having a present time parameter–”now”. E.g. when one says “The Earth is round”, we interpreted the proposition-conveying sentence to be ” The Earth is round now“.