Undergrad Predicate logic and modal logic

[Q-1]

I have a question about predicate logic and modal logic.

Namely, do any of them overlap with one another? To give an example, does existential quantification apply to counterfactual statements?

A counterfactual statement can be something like "A possible world where I won the lottery." I wouldn't be able to assign an existential quantifier in the same manner that I could to a statement in this world, because that statement simply does not obtain or isn't instantiated in this world contrary to a possible world where I did indeed win the lottery.

Thank you.

 

[andrewkirk]

I believe that modal logic can be encoded in predicate logic simply by adding to the alphabet of our language constants that refer to all the possible worlds, and adding an additional argument to every predicate, that denotes the world to which the predicate is being applied.

For instance isPres(n, 2018, France, Macron) could mean that Macron is the president of the country called France in the year called year 2018 in world n. If we were not doing modal logic, we wouldn't need the first argument, and would just write isPres(2018, France, Macron).

With this approach we can encode the 'possibly' quantifier $\diamond$ by writing $\diamond F(x,y,z)$ where $F$ is a ternary predicate, as $\exists w:\ G(w,x,y,z)$ where $G(w,x,y,z)$ means that $F(x,y,z)$ is true in world $w$.

Similarly, we can encode the 'necessarily' quantifier $\square$ by writing $\square F(x,y,z)$ as $\forall w:\ G(w,x,y,z)$

 

[Q-1]

I believe that modal logic can be encoded in predicate logic simply by adding to the alphabet of our language constants that refer to all the possible worlds, and adding an additional argument to every predicate, that denotes the world to which the predicate is being applied.

I see. But, the accessibility relation between possible worlds and assigning new existential quantifiers to them is wholly dependent on our world. Is that correct?

 

[Q-1]

This is relevant. Quine was opposed to quantification of modal relations such as "counterfactuals".

https://plato.stanford.edu/entries/logic-modal/#QuaModLog

 

[TeethWhitener]

I see. But, the accessibility relation between possible worlds and assigning new existential quantifiers to them is wholly dependent on our world. Is that correct?

The accessibility relation is defined over a set of worlds, regardless of whether that set contains our actual world or not.

The model theory of modal predicate logic (combining modal operators and existential quantifiers) generally uses Kripke semantics nowadays:
https://en.m.wikipedia.org/wiki/Kripke_semantics

Quine might not have loved it on metaphysical grounds, but mathematically it’s all well-defined

 

[Q-1]

We can only "measure" (quantify) counterfactuals by an accessibility relation to our own world (stipulation or instantiation). Therefore how can we assert something as necessarily true in all possible world's if quantification of modal relations (counterfactuals) is/are restricted to only our world?

I posted this question on Stack Exchange and got the following responses:

https://philosophy.stackexchange.co...accessibility-relations-satisfied/58954#58954

I'm also interested if frame conditions have anything to do with this?

 

[Q-1]

The above post was meant to be a separate thread; but, the moderators merged it into this one. I have no issue with that; but, confusion about what its doing here might arise.

That is all.

 

[fresh_42]

The above post was meant to be a separate thread; but, the moderators merged it into this one. I have no issue with that; but, confusion about what its doing here might arise.

That is all.

It is still about the comparison of the same logical systems. Furthermore there is a risk, especially in your last post, that this will enter philosophical realms, which we will not discuss here. To open another thread in order to avoid a possible closure of one of them due to philosophy is a nice trick, but nevertheless merely a measurement of risk reduction. Please stay within the boundaries of logic as a mathematical framework.

 

[Q-1]

It is still about the comparison of the same logical systems. Furthermore, there is a risk, especially in your last post, that this will enter philosophical realms, which we will not discuss here. To open another thread in order to avoid a possible closure of one of them due to philosophy is a nice trick, but nevertheless merely a measurement of risk reduction. Please stay within the boundaries of logic as a mathematical framework.

I shall try. Thank you.

 

[TeethWhitener]

Therefore how can we assert something as necessarily true in all possible world's if quantification of modal relations (counterfactuals) is/are restricted to only our world?

Again, mathematically, there’s nothing inherently special about our world. Quantification simply describes an accessibility relation between worlds, regardless of whether one of those worlds is the actual world.

Spoiler: Philosophical digression

Even philosophically, your statement about restricting quantification to our world is an epistemological statement, rather than a metaphysical one (IOW, restricting quantification as you suggested is a reflection of the scope of what we know, rather than relations which may or may not exist between sets of possible worlds which do not contain our own).

 

[Q-1]

Quantification simply describes an accessibility relation between worlds, regardless of whether one of those worlds is the actual world.

I believe you mean to say "stipulation" here and not quantification (I'm using Kripke's terminology if I'm correct here wrt. to stipulations). Sure I can stipulate a possible world where an event might have happened otherwise; but, the framing condition for doing so, to sound technical, will always be restricted to the world where the stipulation was made with respect to that event or state of affairs. A roundabout way of positing counterfactuals.

As for the philosophy. I'm going to avoid that as it's a tautology in regards to what you stated.

 

[TeethWhitener]

Sure I can stipulate a possible world where a counterfactual might have happened otherwise; but, the framing condition for doing so, to sound technical, will always be restricted to the world where the stipulation was made with respect to that event or state of affairs

You might be interested in interpretations of the Barcan formula: https://en.m.wikipedia.org/wiki/Barcan_formula

 

[Q-1]

You might be interested in interpretations of the Barcan formula: https://en.m.wikipedia.org/wiki/Barcan_formula

Yeah, so the next question is how can you state a necessary condition if you can't quantify beyond the actual world.

 

[TeethWhitener]

Yeah, so the next question is how can you state a necessary condition if you can't quantify beyond the actual world.

Yes, this is a fine philosophical question. But the only way it relates to the math is in choosing which set of axioms to allow (for instance, Barcan’s formula).

 

[Q-1]

Yes, this is a fine philosophical question. But the only way it relates to the math is in choosing which set of axioms to allow (for instance, Barcan’s formula).

Do you have any papers where I can explore this matter further?

Thanks,

 

[TeethWhitener]

Do you have any papers where I can explore this matter further?

Thanks,

For the math Hughes and Cresswell’s “Introduction to Modal Logic” is a fantastic resource.

Spoiler: Philosophical resources

For the philosophy, two classic resources are David Lewis’s work on modal realism (I suppose Lewis would be considered “anti-Barcan”) and Saul Kripke’s response in “Naming and Necessity.” Counterfactuals are a big topic.

 

[Q-1]

So, I've been doing some more reading, and I think the de re and di dicto distinction if worth mentioning here. In one instance we can talk or stipulate counterfactuals, but that's a de dicto. A de re cannot obtain when talking about counterfactuals. And, that's all there is to it I suppose.

Thoughts?

 

[TeethWhitener]

So this is pretty deep into the weeds here, and I think we’ve probably reached the end of the math and are sailing into pure philosophy of language. The moderators tend to shut down philosophy discussions here (but for some reason we seem to have thread after thread of experimentally indistinguishable quantum mechanics interpretations ). But these are good philosophical questions, and I don’t want to discourage you from asking them—but this forum isn’t really suited for these discussions. I would recommend a close study of Kripke’s “Naming and Necessity.” In the case above, I imagine Kripke would argue that the impossibility of de re counterfactuals only obtains for rigid designators. (Of course, one might reasonably speculate that de re statements are a fortiori rigid, but this is even deeper into the weeds.)