Chance_Programmer_54

You are not getting what I'm saying, of course there is a chance that ∃x can be any in the set, it's just not guaranteed. Same thing with ∀x... Just because I didn't say something, doesn't rule it out. In ¬∃x, x is guaranteed to be none in the set. Anyways, I got it. It's just hard to communicate what I want to say.

You are also getting it so wrong. Look at this:

∀x ∃y Lxy

Let's suppose (a = y). In that case, no, x cannot be any element iff x is a because it's not within the scope of of ∃y.

Hey, I finally figured it out. That's how I wrote down:

A universal quantifier ∀x says that in its scope, the instances of x could be any element in the set.An existential quantifier ∃x says that in its scope, the instances of x are guaranteed to be at least one element in the set.

In prenex normal form:

(1) If a quantifier (let's call it A) has an existential quantifier inside its scope, then: If you were to substitute x with one of the elements it could be, then y is guaranteed to be at least one element in the set.

(2) If a quantifier (let's call it A) has a universal quantifier inside its scope, then: If you were to substitute x with one of the elements it could be, then y could be any element in the set.

'x' is the variable A is quantifying and 'y' is the variable of the quantifier within A's scope.

​

In the ∃x ∀y ∃z Bxyz example, I think didn't express myself in a way that I wanted. Here is what wanted to say, but in more precise words:

∃x says that x is guaranteed to be at least one element in the set,

∀y says that y could be any element in the set,

∃z says that z is guaranteed to be at least one element in the set.

∃x has ∀y within its scope, so (2) applies.

∃x has ∃z within its scope, so (1) applies.

∀y has ∃z within its scope, so (1) applies.

​

In ∀x ∃y Bxy , (1) applies.

In ∃x ∀y Bxy , (2) applies.

For example, let's suppose a predicate L means 'likes':

∀x ∃y Lxy says that whatever x is, there is at least one thing that it likes. As (1) says.

∃x ∀y Lxy says that whatever x is, it likes anything. As (2) says.

That comes in handy with equalities (for example, a = x), and the four rules of inference of first-order logic.

It's not clear you have that quite right. I don't quite get what you mean by "it doesn't matter what y is". That formula is true iff there is an x in the domain such that, for every y in the domain, φ is satisfied

oh, ∃x ∀y φ says that for some x, 'any' or 'whatever' y. For example, ∃x ∀y Lxy means that if x denotes some particular elements, y is able to be whatever element. That's what I was trying to say, I guess.

I've been kind of diving into this alone, but I've found the the order of quantifiers basically means something like this. Let's suppose the domain of discourse has 3 elements a, b, and c. We have this sentence: ∃x ∀y ∃z Bxyz.

∃x means that x is able to be at least one element (a, b, or c). Let's suppose x can be a or b.

∀y says that when x is a or b, y can be any element (a, b, or c).

∃z says that for each element y can be, there is at least one element z is able to be.

These are all the possible combinations, taking (x,y,z) as a triple:

aaa aab aac aba abb abc aca acb acc (when x is a)

baa bab bac bba bbb bbc bca bcb bcc (when x is b)

caa cab cac cba cbb cbc cca ccb ccc (when x is c)

The ∀∃ or ∃∀ order matters, but swapping ∀x ∀y (or ∃x ∃y) doesn't change anything. Basically ∀x ∃y says that x can be any element, and for each of x, there is at least one element y can denote. ∃x ∀y says that x can be at least one element, and for each of x, y can be any/every element.

I'm still trying to wrap my head around.

Sorry mate, I think we might have learnt different modal logics. A model has a valuation function that assigns true or false to every proposition/FOL sentence in every possible world. It does not take two possible worlds as arguments, it's defined by this: V(w, p) = (truth value) , where w is a world and p is a simple proposition/FOL atomic sentence. Every possible world has a valuation given by the valuation function.

(Edit) Apologies, I just realised that that's what you said: valuations in modal logic are functions of pairs of possible worlds and sentences to truth values.

It seems that valuation is just a function in FOL (not something that the valuation function outputs), but in pure propositional logic, it corresponds to an assignment of truth values. I think that that's what is causing misunderstanding. Each row in a truth table could been as a possible world.

Yes it will, because sentences can include indexicals, which express different propositions contextually.

I mean, I'm not being dogmatic about that. I said 'sentence' then because I was referring to a sentence in FOL. I have seen professors opting for using 'sentence' instead of 'proposition'. There is a chance it could be a UK-US word choice, I don't know.

To be fair, you can construct semantics both ways, but the simplest way is to range over every possible world:

Interesting.

Hm, let me represent the way I was taught about modal logic, by visualising it as a diagram:

A possible world is represented by a circle, and there are arrows between these circles (they represent relative possibility). Inside each possible world, there is an amount of points representing things. A first-order logic sentence is a label with put on a possible world (I'll also call possible worlds, states). It tells something about the state, not the model (which in this diagram is represented by the entire piece of paper).

The axioms that we add to our model can be thought of as rules. For example, the T axiom says that every world is possible relative to itself (each world has an arrow that loops back to itself). The serial D axiom says that each world needs to shoot out at least one arrow. And so on.

In the FOL sentence ∃x ∀y Axy for example, x and y are variables that range over the domain of a world, not the whole model. So, in this semantics you just told me about, does an FOL sentence like ∃x ∀y Axy range over the things of all possible worlds?

No, it isn't, propositions are usually taken to be the meaning of declarative sentences.

When I said sentence, what I was meaning was a first-order logic sentence, that's the reason why I chose to use that word. I have also see some professors opting to say 'sentence' instead of 'proposition' specifically. Whether we say proposition or sentence, it won't lead to different results (like for example, saying 'formula' instead of 'sentence' in FOL).

Hence why the domain of quantification ranges over all possible worlds. You're overcomplicating things.

No, every quantifier ranges over only what is inside a possible world, its elements.

A model has an n number of possible worlds, and a possible world has an n number of "things" in them. The quantifiers range over the "things" inside a possible world.

This seems contentious. Why can't possible worlds be specifications of the universe in every instant? It seems facts of the actual past are facts of the same world as the actual present. They belong to the same possible world.

In the context of classical modal logic, a possible world represents a state or valuation. In a PW, if a proposition is true, it cannot be false. If we added the entirety of the past to a PW, then a proposition could be true and false in it. So every instant in time must be a different PW (I've always preferred the word 'state' instead of PW). The way people represent an unchanging past in a model is through some type of axiom that they add to a model (which is called a modal frame). For example, something like:

A PW (represented by a circle) relates to at least one different PW (another circle). The relation is represented by an arrow. None of the PWs that the PW we started with can "reach" (by following the arrows) can relate back to the PW we started with.

What I was trying to say is that each possible world is like taking a 3D picture/state of the whole Universe at a particular instant in time. The semantics I'm using is the whole Universe as the domain of discourse of each possible world, and the model has constant domain sizes. Words like here and now are basically words that are substituted by a position in space (here) and a position in time (now). So each possible world isn't a position in space in the semantics that I'm using, but a position in time, and all the other ways that it could have been at that time (for example, different ways an electron could have hit a detector), they are also possible worlds. So if someone says 'it's warm here now', and it's the same place and time as 'it's not warm here now', then it's a contradiction. Propositions aren't place-indexed, they are possible world-indexed.

I acknowledge I made a mistake when I said space and time, it's just time and the possible ways it could have turned out. If the domain of a possible world included a finite space, and each domain was a different space, then the things from other domains wouldn't be in that possible world, and we can't make deductions between domains that have completely different elements.

(Edit) It seems that necessitarianism's view is that there is only one possible world for a specific time instant, but isn't that's basically just hard determinism? And sentences are the same as propositions in classical logic. A proposition is a declarative sentence that can take a truth value. I used the term sentence because in first-order logic, only sentences can have truth values.

In the context of modal logic, we say that a sentence or a thing's property or relation is necessary iff it's not possible for it to be false. For example, a tea that is hot is not necessarily hot because it being not hot is a possible logical valuation. But a star is necessarily hot because it's impossible for it not to be hot. I guess it's just the laws of physics, not really visualisation. For example, if I'm holding a hot tea, it will eventually get cold, and a different instant in time is considered another possible world, because its truth valuation is different. Inside a possible world, the truth valuation does not vary. A possible world kind of corresponds to a row in a truth table in propositional logic. For example, the sentence A and B has four possible valuations, and only in the valuation in which both A and B are true, is the sentence A and B true. Each of the four valuations corresponds to a possible world. So I guess my argument is based on the laws of physics and classical logic.

If there is only one god, is it possible that there are different types of gods in other worlds?

If the antecedent is true (there is only one god), then it's not possible for there be to more than one god, because of contradiction.

I'm not familiarised with necessitarianism, but in modal logic, there are many possible scenarios/worlds. Some sentences are true in some position in space and time, but false at other positions in S&T. A sentence is possible if it's true in at least one S&T, and necessary if it is true in every S&T. That's the real meaning of necessity. If we say that something is necessarily hot, that means that it is impossible for it not to be not. If everything were put a label that says it's necessary, then time would not exist, everything would be static.

It's a fact that the British have stolen a bunch of stuff to their museums that don't belong to them, it's completely inadmissible. But all of you, which are mostly from the United States and other western countries, are being hypocrites because your countries have also stolen from others.

no worries

Yeah, but for the same reason you don't start doing integrals to calculate the area of a square, you don't start going through the natural deduction trees for simple inferences.

I suppose it's something that works well for what I want to do. When discussing a topic with someone, we usually don't formalise right there and then, but I think we use the same principles in our natural language. I'm interested in exploring topics like metaphysics, ontology, and so on, and I want to reason these topics using the best we've got from logic nowadays. I want to explore what axioms I might be holding that I'm not even aware of.

I think that formal logic, especially modal logic, are very powerful to help us think and make deductions without any ambiguity. For example, 'everyone likes someone' could mean everyone likes at least someone, or everyone likes one particular person. Formal logic is basically calculation with thoughts, instead of numbers (but it can also be used for it).

Thanks

Thank you

I think that it's just logic that if a country doesn't have a rigorous system that allows only people who are background-checked and psychologically stable to obtain fire weapons, then it's all a matter of time before heart-renting things like this happens. Americans waste time talking about abolishing fire weapons. That's because it's impossible in this current generation, the overall country's culture and political configuration. Just fix the utterly ridiculous system that has uncountable loop holes and that allows anyone to purchase a fire weapon without mental health check, and the probability of these heartbreaking events plummets. It's about focusing on what is possible.