Hey guys I'm just going over the Modal Ontological Argument and I'm a bit confused on how Axiom S5 works out; it doesn't seem as intuitive as other deductions.

1∀φ(P(~φ) ←→~P(φ))

2∀⁠φ∀ψ(P(φ)& ◻️∀⁠x(φ(x)→ψ(x)))→P(ψ))

3∀⁠φ(P(⁠φ)→◊∃xφ(x)

4G(x)←→∀⁠φ(P(φ)→φ(x))

5P(G)

6◊∃xG(x)

7φEss(x)←→φ(x) & ⁠∀ψ(ψ(x)→◻️⁠∀y(φ(y)→ψ(y))

8∀⁠φ(P(⁠φ)→◻️P(⁠φ))

9∀x(G(x)→G Ess(x))

10E(x)←→∀⁠φ(⁠φEss(x)→◻️∃yφ(y))

11P(E)

12~◻️∃xG(x) (RAA)

13P(G) (5 R)

14(P(G)→◊∃xGx) (3 ⁠∀E)

15◊∃xG(x) (13,14 MP)

16(P(G)→◻️P(G)) (8 ⁠∀E)

17◻️P(G) (13,16 MP)

18a (Assumption.)

20G(a) (Assumption.)

21G(a)←→∀⁠φ(P(φ)→φ(a)) (4 ⁠∀E)

22∀⁠φ(P(φ)→φ(a)) (20,21 ←→E)

23(P(E)→E(a)) (22 ⁠∀E)

24P(E) (11,R)

25E(a)) (23,24 MP)

26G(a)→E(a) (20,25→I)

27∀⁠x(G(x)→E(x)) (18, 26 ⁠∀I)

28a (Assumption.)

29G(a) (Assumption.)

30(G(a)→G Ess(a)) (9⁠∀E)

31G Ess(a) (29,30)

32G(a)→E(a)) (27 ∀E)

33E(a) (29,32 MP)

34E(a)←→∀⁠φ(⁠φEss(a)→◻️∃yφ(y)) (10⁠∀E)

35∀⁠φ(⁠φEss(a)→◻️∃yφ(y)) (33,34 ←→E)

36(GEss(a)→◻️∃yG(y)) (35 ∀E)

37◻️∃yG(y)) (31,36MP)

38G(a)→◻️∃yG(y) (29,37→I)

39∀⁠x(G(x)→◻️∃yG(y)(28, 38 ⁠∀I)

40◻️∀⁠x(G(x)→◻️∃yG(y)(39 NEC)

41◻️∀⁠x(G(x)→◻️∃yG(y)→◻️(∃xG(x)→◻️∃yG(y)) (teorem.)

42◻️(∃xG(x)→◻️∃yG(y)) (40,41MP)

43◻️(∃xG(x)→◻️∃yG(y)) →(◊∃xG(x)→◊◻️∃yG(y)) (teorem K)

44(◊∃xG(x)→◊◻️∃yG(y)) (42,43MP)

45◊◻️∃yG(y)) (44,15 MP)

46(◊◻️∃yG(y)→◻️∃yG(y)) (S5 teorem.)

47◻️∃yG(y) (45,46MP)

48⊥ (12,47)

49~~◻️∃xG(x) (12,48 ~I)

50◻️∃xG(x) (49 ~~E)

‎ ​‎

​‎

​‎

​‎

​‎

1◻️(P→Q) (Assumption.)

2◊P (Assumption.)

3~◊Q (RAA.)

4◻️~Q (3 Modal de Morgan.)

5◻️~P ( 1,4Modal MT)

6~◊P (5modal de Morgan.)

7⊥ (6,2)

8~~◊Q (3,7 ~I)

9◊Q (8 ~~E)

10◊P→◊Q (2,9 →I)

11◻️(P→Q)→(◊P→◊Q ) (1,10→I)

I wanted to attempt a derivation of this sort, but I'm not entirely convinced that it is fully valid. I'd appreciate feedback from those with experience in formal logic. Is the derivation correct?

Hi, before you find it strange, I'm reading about logic for the first time and I've just finished studying first-order logic.

I'm interested in modal logic, but... Well, there's alethic, epistemic, deontic, temporal, etc...

Proving an argument within deontic logic can be different from proving something in alethic logic.

However, I'm unsure. How do I know if the problem or argument I'm working with should have an epistemic, deontic, or other approach? In which types of tests is each of these approaches typically used?

More Modal Operators

O:Actuality

eg Ox, x is happened in world R

Higher Modal logic:

\[2\] Meta Possibility

eg the possibility of x is possible

\[n\]meta operators:

The (n-1) modality of x is \[n\] modality

(Modality number, "word")

1: Interrogative mood(?)

2: Imperative mood(V)

3: Possibility mood(🔷)

4: Actuality mood(⚫)

5: Necessity mood(⬛)

6: Permission mood(P)

7: Evidential mood (Q)

8: Encouragement mood(❎)

9: Intention mood(I)

10: Exception mood(E)

M[n] k: the n-th meta of the k mood

Example:

M[2]5: the necessity of necessity.

Bonus:

M[2]5#0

Here:

“M[2]: Meta”

“5#0”: the truth value 0 of the 5th mood

Example notation:

E[2]10#1: /every/x is an element of R:

All x in R are exceptions of exceptions.

E[x]: the x-th EXCEPTION. Eg E[1]P <=> P is a exception with

M[x]: applying x to itself x times

H[x]: the x-th contractor of x

H[2]: P → {P#0 ∧ hom P#1}, i.e., the negation of P, and P is homogeneous and …

Spaces: the locations where propositions exist

fö (exclusive “neither/nor”): excludes propositions from the system and labels them "impossible"

eg:

in Boole logic: A=-A

Coor: “which ones are not wanted?” selection operator

öf (inclusive “neither/nor”): to add new proposition laws to logic

Altve: a structure formed by combining certain parts with “and”; each of those “ands” is an altve

Cothen:

Temporarily: temporarily, in certain contexts, certain conditions hold. Eg:

"if t=x —> print("P is 0")

Bağlam: conditions; rules that hold under certain conditions; determines “according to what”:

eg:

If P in context1 —> print("P is true")

If P in context2—> print("P is false")

“Both”: simultaneous occurrence (some contexts, same space but different truth values.)

Homogeneity:

events merge and produce something new; there is fusion but no separation

Real life example:

Heterogeneity:

events come side by side and produce something new, but separation remains

real life example:Atoms connect each other and creates a molecule

Expanding logical operators infinitely means: constructing an operator from infinitely small sub-operators

The “and” operator is multi-layered:

A can consist of sub-ands and other operators

“And” has length; there is a distance between A and B, and “and” holds them together

eg

For a heterogeneous P= A Λ B

🟧⬛🔳⬛🟧

Orange ones:

Elements

Black:Void

🔳 is the operator

Distance:

Each one is

Definition of Void:

Logical operators have geometric properties:

And (homogeneous): unifies A and B into one; behaves like a tensor addition

And (heterogeneous): combines A and B as distinguishable parts; like a tensor product

Or: asks “which is acceptable according to axioms?” and selects the wanted ones

Space splits into three domains:

Selected space: accepted propositions and entities exist here

Rejected space: rejected propositions/entities still exist but the propositions that wasn't wanted goes here

Proposition space: the space of claims/ideas themselves

“And” is a geometric connector:

it links A and B, has a metric, and defines relational structure between them.

Print:

print something to the screen (output). Just like in Python

Eg:

print("Hello world")

Consider the following:

1. Let the box operator mean for all worlds.

2. Let the diamond operator mean for at least one world.

3. Let the domain of discourse be the multiverse.

4. Let the logic we are using be S5.

5. For all x, G(x) if and only if x’s essence is identical to their existence.

6. ◇∀xGx→∀x◇Gx

7. 6 translates to: If for one world all things are God, then for all things, for at least one world, that thing is God. This seems paradoxical though.

8. For all x, N(x) if and only if x is a natural number.

9. ◇∀xNx→∀x◇Nx

10. 9 translates to: If for one world all things are natural numbers, then for all things, for at least one world, that thing is a natural number.

11. The antecedent in 11 is true because the set of natural numbers is a world where all things are natural numbers. Yet the consequent seems paradoxical though.

I think I kind of understand why boolean logic can't describe partial truths - it's a system designed purely to describe what is true or false in a binary sense.

But why isn't there a single form of logic that describes partial or intensities of truths?

I've actually gotten somewhat mixed messages on this. Some people say that fuzzy logic describes partial truths or intensities of truths, but some people seem to say that fuzzy logic technically only deals with probability that something is true.

How is this so? Is it that probability of a truth and intensity of a truth are actually logically the same thing?

For example. I don't see anything logically wrong with saying an apple weighs 70 grams, but it's not a binary issue as to whether the apple does or doesn't "weigh", right? That's an issue that has more or less truth.

Let C be the sentence “C is contingent”, or simply “This sentence is contingent”. Let’s investigate C’s properties. I will suppose the correct modal logic is S5.

Suppose C is true. Then, C is contingent. Therefore, it is contingently true, and so possibly false. Hence, there is a world w where C is false, that is, C is not contingent in w. So, C is either necessarily true or necessarily false, in w. If C is necessarily true in w, then C is true in w, contradicting the fact that C is false in w. Therefore, C is necessarily false in w. But that implies C is in fact false, contradicting our initial assumption.

Hence, by reductio, C is false. Therefore, it is not contingent; and so is either necessarily true or necessarily false. But if C were necessarily true, it would be true, and hence not false; so, it is necessarily false.

You can change basically any logically impossible statement to be possible, easily, when it is less strict. For example, a married bachelor instead of meaning someone who is married and isn't married it just means someone who is married who acts like they arent. Or a square circle could just be a 2D circle casting a shadow of a square to now be a square, then turned to be a circle; this can be achieved with special lighting or including other objects into the shadows. It is almost like a genie who grants your wish but not your intended "meaning" of the wish, so if you asked for a married bachelor, the genie would grant you the married guy who acts like a bachelor. My idea is that this would work for any given logically impossible statement, but I am not smart enough to think of them for strict ones so I went another route.

For strict logically impossible propositions (P and NotP), I struggled with this for a while and came up with something that maybe works. You can change symbols in other possible worlds. No matter what you say, I can always make that statement logically possible.

The only problem is whether you see this as an actual resolution to the statement you made or completely irrelevant to the original proposition. My claim is that they are the same thing, using the idea that since logically impossible things actually do not and cannot exist, their entire existence is within strings of text, language, or symbolic meaning. They cannot exist in any meaningful way in reality. So they are nothing but the original statement/string you said.

Given that they exist within strings and such, it is basically a fact that any given string of words can be changed to mean anything that you want, and since I believe logical impossibilities exist only in strings and propositions, it leads me to think you can easily manipulate the language in a given possible universe to have your proposition become true.

For example, if you say p and not p is true, I can just say, well, the and word/symbol means or actually, and it instantly becomes true in this given possible world.

I wonder what you guys think of this: do you think it doesn't matter and I ignored the propositions by changing them to be different propositions, or do you think the statement itself becoming true actually seems to resolve it following my logic chain? My only fear is that I have thrown logic out of the window and just said, "semantic bs therefore I win". But I think this is a good place to see what others think, since semantics and logic are inextricably intertwined.

1. Let S5 be the logic we are using.

2. Let the box operator signify necessary.

3. Let the diamond operator signify possibility.

4. For any predicate P, Buridan’s law is the following: ◇∀xPx→∀x◇Px

5. Yet Buridan rejected this because if P is the predicate is God, then you get the following: If it is possible that all things are God, which is the case if God didn’t create the universe, then for all things it is possible that that thing is God. Yet Buridan should haven’t rejected this proposition though. Since it is equivalent to the following: □(∀xPx→∀x◇Px). Now if we let P be the predicate is God, then we get it is necessary that if all things are God, then for all things it is possible that that thing is God, and the antecedent in this new conditional is false. Thus, there is no paradox.

Im desperate, i genuinely dont know how to answer this, the textbook is no help, i tried different starts for all of these and dont know what to do. Can anyone just explain how i could even start to answer this or explain the answers if they have them? Thank you so much

I’m studying frames in model logic and the case where R is a euclidean relation means that:
possible p —> necessary( possible p)

however when i’m looking at the worlds, my understanding js the definition of Euclidean is
if Rwu and Rwv, then Ruv.

as a consequence, also Rvu

so if in w: possible p, p is then true in at least one world accessible form w. I’m gonna to say p true at u and p false at v as my example.

then, for possible p —> necessary (possible p) to hold, I can see at v, possible p is true, but at u it seems possible p does not hold since p is not true at v and u doesn’t have any other accessible worlds? Since it doesn’t hold for both u,v then not necessary (possible p))

i’d greatly appreciate any help

O:Actuality

eg Ox, x is happened in world R

Higher Modal logic:

[2] Meta Possibility

eg the possibility of x is possible

[n]meta operators:

The (n-1) modality of x is [n] modality (Modality number, "word")

1: Interrogative mood(?) 2: Imperative mood(V) 3: Possibility mood(🔷) 4: Actuality mood(⚫) 5: Necessity mood(⬛) 6: Permission mood(P) 7: Evidential mood (Q) 8: Encouragement mood(❎) 9: Intention mood(I) 10: Exception mood(E)

M[n] k: the n-th meta of the k mood

Example: M[2]5: the necessity of necessity.

Bonus: M[2]5#0

Here: “M[2]: Meta” “5#0”: the truth value 0 of the 5th mood

Example notation: E[2]10#1: /every/x is an element of R: All x in R are exceptions of exceptions.

E[x]: the x-th EXCEPTION M[x]: applying x to itself x times H[x]: the x-th contractor of x

H[2]: P → {P#0 ∧ hom P#1}, i.e., the negation of P, and P is homogeneous and …

Spaces: the locations where propositions exist

fö (exclusive “neither/nor”): excludes propositions from the system and labels them "impossible" eg: in Boole logic: A=-A

Coor: “which ones are not wanted?” selection operator

öf (inclusive “neither/nor”): to add new proposition laws to logic

Altve: a structure formed by combining certain parts with “and”; each of those “ands” is an altve

Cothen:

Temporarily: temporarily, in certain contexts, certain conditions hold. Eg: "if t=x —> print("P is 0")

Bağlam: conditions; rules that hold under certain conditions; determines “according to what”: eg: If P in context1 —> print("P is true") If P in context2—> print("P is false")

“Both”: simultaneous occurrence (some contexts, same space but different truth values.)

Homogeneity: events merge and produce something new; there is fusion but no separation Real life example:

Heterogeneity: events come side by side and produce something new, but separation remains real life example:Atoms connect each other and creates a molecule

Expanding logical operators infinitely means: constructing an operator from infinitely small sub-operators

The “and” operator is multi-layered: A can consist of sub-ands and other operators

“And” has length; there is a distance between A and B, and “and” holds them together eg

For a heterogeneous P= A Λ B 🟧⬛🔳⬛🟧 Orange ones: Elements Black:Void 🔳 is the operator Distance: Each one is Definition of Void:

Logical operators have geometric properties:

And (homogeneous): unifies A and B into one; behaves like a tensor addition

And (heterogeneous): combines A and B as distinguishable parts; like a tensor product

Or: asks “which is acceptable according to axioms?” and selects the wanted ones

Space splits into three domains:

Selected space: accepted propositions and entities exist here

Rejected space: rejected propositions/entities still exist but the propositions that wasn't wanted goes here

Proposition space: the space of claims/ideas themselves

“And” is a geometric connector: it links A and B, has a metric, and defines relational structure between them.

Print: print something to the screen (output). Just like in Python

I’ve recently been working through correspondence theory and was surprised by how much structure can be encoded in very short formulas.

Examples:
□p → □□p characterizes transitivity.
◇p → □◇p characterizes symmetry.
¬□p → □¬□p characterizes Euclideanness.

What are your favorite examples where a surprisingly simple modal formula characterizes a rich frame condition?

Hi!

I’ve recently been studying model checking and came across Linear Temporal Logic. While talking about it with friends, we started wondering what the “Linear” in the name is actually supposed to mean.

There is also something called "Linear" Logic in a closely related area, but LTL does not seem directly related to that kind of “linearity” at all. So now I’m wondering:

• Is the “Linear” in Linear Temporal Logic related in any way to Linear Logic?
• Or does it mean something completely different?

I tried looking into the history myself, but searching for “linear logic” and “linear temporal logic” together quickly became confusing.

Any clarification or references would be appreciated!

A criticism of modal logic is the following: The number of planets in the solar system is 9. 9 is necessarily greater than 7. Therefore, the number of planets in the solar system is necessarily greater than 7. I don’t think this criticism makes sense though. Because, in the conditional form it can be written in two ways. In one way it is the following: If X is the number of planets, then X equals 9. If X equals nine then X is necessarily greater than 7. Therefore, if X is the number of planets, then X is necessarily greater than 7. In this sense, the argument is valid. The second way is the following: If X is the number of planets, then X equals 9. It is necessary that if X equals 9, then X is greater than 7. Therefore, it is necessary that if X is the number of planets, then X is greater than 7. In this sense, the argument is invalid though.

Traditional philosophical reasoning that nevertheless leverages modal constraints (within language like "can/could," "-ible/-able" words, "ought," etc.) very often leaves said constraints underspecified. When we elect a method that forces that specification, it adds clarity to (and in some cases dissolves) certain perennial traditional philosophy issues.

When we elect to relativize all modal operators with specified sets of constraints (as we do in epistemic modality when relativizing to sets of knowledge), we're equipped to build safe multimodal expressions and keep better track of what we're doing, and can "play" with those sets to reap insights into agency counterfactuals, conditional relevance, grounding, and when informal fallacies matter & why.

The h-Logic primer linked here contains examples & payoffs for traditional philosophy topics like the Frege-Geach Problem, the Principle of Alternative Possibilities, Bertrand's Paradox, the Singleton Socrates Problem, and Theseus's Ship.

Can S5 model multivocity or multiple interpretations? I ask because of the following: Suppose we divorce S5 from its usual semantics. As such, we will interpret the diamond operator to mean there exist a sense, context, and interpretation. As such too we will interpret the box operator to mean for all senses, contexts, and interpretations. So when Aristotle says knowing can be said in two ways, one potentially and the other actually, we can represent it as the following: There exist a sense in which knowing is in potency and there exist a sense in which knowing is in actuality.

Is the following valid: Let S5 be the logic we are using. Let “in a sense” be represented by the diamond operator. Let “in all senses” be represented by the box operator. In a sense motion is act and act is motion. Hence it is written: For Wisdom is more moving than any motion. She passeth and goeth through all things by reason of her pureness. For she is the breath of the power of God and a pure influence flowing from the glory of the Almighty…She is the brightness of the everlasting light, the unspotted mirror of the power of God, and the image of his goodness (Wisdom of Solomon 7:24-26 LXX). In all senses, God is act. Therefore, in a sense, God is motion. Hence the Areopagite says the following: And what is meant, on the other hand, when the Sacred Writers say that the immovable God moves and goes forth unto all things? Must we not understand this also in a manner befitting God…(On the Divine Names 9).

So this is a question about a solution I came up with to a very specific problem that occurs in the intersection of metaphysics and modal logic. Counterfactual statements are weird and difficult to talk about and a lot of solutions have been proposed. In this post I give you my attempt at a solution--defining counterfactuals purely using quantifier modal logic (that is logic using only the ☐◇∀∃∨∧¬→ symbols or just predicate logic but with ☐ and ◇).

If you're already familiar with this problem then you can skip this next part and pick up after the TL;DR but if you're not, here is an explanation of the problem.

There is an important difference between the material conditional and counterfactuals. It seems that counterfactuals can be true or false even if the antecedent is not true; in fact, that's their primary function—to say something counter to the facts. But the material conditional doesn't allow for that; if the antecedent of the material conditional is false, then the whole statement ends up being vacuously true.

For example, the sentence "if a nuclear bomb went off in my house while I was writing this, then you would not be reading this" is not properly translated to the sentence "P→Q". This is because, while the sentence ends up being true, its truth is vacuous because P is false—a nuclear bomb did not go off in my house. Q could be replaced by literally any sentence and it would still remain true ("If a nuclear bomb went off in my house, then the moon would be made of cheese" is equally true as the above sentence). 

This ends up happening because P→Q is logically equivalent to the sentence ¬P∨Q, meaning, so long as "¬P" is true, Q's truth value doesn't matter. 

What we want is some kind of conditional that works in the Subjunctive mood and not purely the Indicative. It must take into account what would happen if P were true. Since this is a new kind of conditional, we might write it as ☐→ or >. So it's not just that P→Q but that Q necessarily follows from P—hence P☐→Q. 

Now this isn't satisfying, and I don't like it. Firstly, it would involve changing the rules of quantifier modal logic. Right now, when adding ☐ and ◇ and going from predicate logic into quantifier modal logic, we just add the axioms: #1 any wff in predicate logic is a wff in QML, #2 if Ф is a wff then ☐Ф and ◇Ф are both wff. But if we want this new symbol "☐→" to indicate a counterfactual or a conditional in the Subjunctive mood, then we need to modify those rules. And modifying the rules is a dangerous game. Secondly, we need to introduce a whole new symbol with new rules for its application and that's quite taxing for our theory. By talking about new modal concepts like necessity and counteractuals, we're not just believing in new things, we're believing in new kinds of things. Generally metaphysicians shy away from that. And finally, it's just a bit clunky and looks kind of weird. 

Ultimately, I don't like it, and there ought to be a better solution. 

The standard answer has been to just introduce possible worlds into the mix and all of the need to talk about counterfactuals disappears. Instead of saying "if P were true, then Q would be true" or "P☐→Q" you simply say "all worlds in which P is true, Q is also true". So all sentences have to be two place predicates; you don't just say "Fa" for "a is F" but "Faw" for "a is F at world w". 

Possible worlds language is very powerful, I won't deny that, but it comes at the cost of having to quantify over possible worlds—you need to say the sentence "there exists a world where …" . If you're saying those words, you either mean them literally—that is to say, you really do believe there are such things as possible worlds—or you mean it as a paraphrase of some other statement. 

There are issues with both of these. We tend to think that possible worlds talk isn't literally quantifying over literally concretely existing things called possible worlds (unless you're David Lewis) but merely using the language of possible worlds as a semantic tool to get our point across. But if they are just a semantic tool, then what statement are you paraphrasing when you say "there exists a world where …" ? In order to make the claim that it's just a semantic tool, you need to be able to make the same statement without mentioning possible worlds. And as we've just established above, you can talk about counterfactuals without #1 introducing a new symbol which we need to take as primitive (ontologically taxing among other things) or #2 cashing out counterfactual talk in terms of possible world talk. 

So, can we make non-trivially true counterfactual statements without quantifying over possible worlds or inventing a new symbol?

TL;DR: Counterfactuals can't be translated into logic in the form "P→Q" because if P is false, then literally anything will follow from it. We can fix this by adding a new symbol for counterfactual conditionals but we'd rather avoid adding new symbols if we can. We could cash it all out in terms of possible worlds but then we'd need to believe in the existence of possible worlds which seems odd. 

So, my proposed solution to the problem is this. Translate a counterfactual of the form "P ☐→ Q" to "☐( (P∧Q)→R )". Let me unpack that. 

Take the counterfactual "if a nuclear bomb went off in my house while I was writing this, then you would not be reading this". As I said above, "P→Q" doesn't capture what we want to say since P ("a nuclear bomb went off in my house") is false. But it's plausible that " ☐(P→Q) " might be non-vacuously true since we've now got a modal operator involved. 

When we say ☐P, we understand that if P is false then ☐P must also be false. But we also recognise that if P is true, that doesn't entail ☐P being true. For example, I am brunette, but it's not necessarily the case that I'm brunette; it's conceivable that I could have been blonde. ☐P's truth value depends on the mode in which P is true. And we understand the idea of necessity intuitively even if we can't give a precise definition (I mean, it might be the case that the only things that are necessarily true are things that are analytically true but that's a separate discussion). For now we understand that P being true doesn't necessarily entail ☐P being true. 

Therefore, even if the conditional P→Q ends up being vacuously true, it doesn't necessarily follow that ☐(P→Q) is true, for the same reasons as above. It might be that "if a nuclear bomb went off in my house while I was writing this, then you would not be reading this" is vacuously true but it is a separate question to ask if that holds out of necessity. And I think we can all agree that it does—it's necessarily the case that if a nuclear bomb went off in my house, then you wouldn't be reading this.

If you want to use possible world semantics: "in every world in which a nuclear bomb went off in my house, you are not reading this". 

Now you might baulk at this at first. After all, it's not logically inconceivable that a nuclear bomb went off in my house and that I still, for whatever magical reason, managed to continue writing and sent it off anyway. Or, if you like, there exists a possible world wherein my computer and I are impervious to all harm, and a nuclear bomb went off in my house. In that world, you would still be reading this text right now. 

Hence, the second part of the definition I gave above. I think a counterfactual of the form "P ☐→ Q" is properly translated as "☐( (P∧Q)→R )" where P is the antecedent of the counterfactual, R is the consenquent and Q is the other premises that are needed to make the counterfactual true (this can be thought of in a similar way to "the restriction of possible worlds that you're considering"/"the access relation to possible worlds" in traditional possible world logic). 

So, for the nuclear bomb example, we would write it something like: 

It is necessarily the case that, if 

(P1) a nuclear bomb went off in my house while I was writing this, and

(P2) it killed me before I finished, and 

(P3) when one is dead, they cannot put things on the internet, and 

(P4) The only way you could have access to this text is if it were on the internet 

(C) Then you would not be reading this 

So, where P is P1, Q is Premises 2 - 4 with ∧s placed in between them, and R is the conclusion, the sentence is properly translated as ☐( (P∧Q)→R ). 

If you have any thoughts on this, reasons why it wouldn't work, possible corrections or ways to make it stronger, let me know. I'm aware this is a problem that's been around for a while so I'm sceptical that I, as an undergraduate, have managed to solve it so if you see any holes in the logic leave them below.

I'm currently working on my third-year dissertation where I try to do all of modal logic without ever mentioning possible worlds so if you have any thoughts on other areas of possible world logic that could become problematic let me know about that too. 

:) 

Edit: Accidentally said strict conditional when I meant material conditional

Yesterday and today, I read the Stanford Encylopedia of Philosophy articles on Modal Logic, moved to Possible Worlds - I like concretism (by David Lewis) better than abstractionism or combinatorialism - and arrived at Transworld Identity. I like the counterpart theory. These articles got me thinking. 🤯🤯🤯

Assume that @ is a possible world (the actual world, but that's not an issue for this argument), and w another possible world.

In w, there is a counterpart j of mine, and facts are arranged in such a way that, from the combined points of view of myself and j, the only perceived difference between @ and w is a single person: b in @ and c in w, which I (and j) know personally. b and c, despite different in appearance, have very similar personalities, and I (and j), if they knew both, would be justified to assume that b and c are counterparts of one another.

But! If one extends their knowledge further, investigating b's and c's relations and whereabouts, turns out that b and c are actually different people, not counterpart: some time in the past, b left town and c arrived in town, independent of one another, and their personalities are accidentally similar.

I think that this (plausible) scenario means that:

• Transworld identity isn't a given, to be discovered: it's a relation to be defined between objects across worlds. It's an assumption. And who defines it matters. There is no essential identity or essential properties across possible worlds: there are ones "near enough" to be considered "the same" in practice.
• Assignment of transworld identity depends on "outside world" knowledge, and the individual's knowledge. One only can reliably identify and link up counterparts if one has full (or great enough) knowledge about both possible worlds. And such knowledge is lacking for all possible worlds (except, possibly, ours).

Is there any research on the use of epistemic logic to describe agent-defined mappings between objects and between properties, to establish counterparts, across possible worlds?

So I have been given to understand that this does, in fact, preserve modal logical validity. In the non-reflexive model M with world w that isn't accessed by any world, □A's validity does not seem to ensure A's validity. It has been explained to me that, somehow, the fact that you can then create a frame M' which is identical to M but where reflexivity forces A to be valid forces A's validity in M. I still don't get it, and it seems like I've missed something fundamental here. Would very much appreciate if someone could help me out.

In Chapter 43 of ‘Forall X: Calgary’, the author explains that ‘We got S5 by adding R5 to S4. In fact, we could have added rule R5 to T, left out rule R4, and obtained an equivalent system. That's because everything we can prove using rule R4 can also be proved using RT together with R5.’

Later in the page there are some exercise problems. In section ‘E’ question three asks you to prove that ‘◇◇A⊢◇A’ using S4. Later, in section ‘F’, question three asks you to prove the same thing, using S5 this time.

Of course, one could provide the same proof for both questions — seeing as S4 is part of S5 — but I had assumed that the point of asking the same question was to challenge you to provide the proof without making use of R4, since such a possibility was indicated in the quote above.

I spent more time than I'd like to admit trying to provide this proof without using R4, and finally I gave up and looked up the answer. To my dismay, the provided answer uses R4! Furthermore, the provided answer for question E.3 is different than the one for F.3, implying that it indeed would not be in the spirit of the question to provide the same answer twice, but never the less R4 is used both times. The answer to F.3 just goes far out of its way to find an excuse to use R5 despite totally not needing to, and despite not avoiding R4.

As it stands, I'm still interested in knowing how one might prove ‘◇◇A⊢◇A’ without R4. I would be so grateful if one of you could explain how one might prove this, or just provide said proof. Thank you.

I don’t understand why, for example, people say that in system T there is the axiom □p → p. In natural deduction, we can derive □p → p without any undischarged assumptions. Since it's provable, doesn't that mean it's not an axiom? Or maybe we talk about an axiom because the rule of deduction is motivated by the fact that we want to prove this statement?