:wink:

This thread is for a read through of two SEP articles on possibility and actuality. — frank

I admire your confidence in being willing to tackle 60 intricate SEP pages about a generally controversial and deeply complex topic.

I know. After saying that it's probably just going to be me commenting on the SEP. :grimace:

I admire your confidence in being willing to tackle 60 intricate SEP pages about a generally controversial and deeply complex topic. — RussellA

Hey, how badly can we mangle it?

Hey, how badly can we mangle it? — frank

An actual possibility, but hopefully not.

The introduction conceptually orients; "possible worlds" means something like - that that is opposed to the "actual world" such as a historical counterfactual, or perhaps, an agent acting differently than she or he actually did.

The first section begins with a discussion of logic. The author presents the term "extension." This is not meant in the Cartesian sense as the length or width of an object; rather, extension in this context is what is being referred to, or denoted, by a "term." The extension of a "sentence" is its truth value; that is, presumably, whether the sentence is true or false.

1.1

Next we get a discussion of "substitutivity principles." I do not quite understand what is meant by "extensional logic" even though a definition is proffered. That being said, extensional logics seem to be characterized by being subject to substitutivity principles.

As I understand it, a substitutivity principle means the following: if two sentences are co-extensional, that is, they refer to the same truth values, that is to say they are logically equivalent sentences, then the addition of the same logical operators to both sentences will affect the truth value of both sentences in the same way, so that they retain the same extensionality and therefore each sentence can be substituted one for the other without changing the truth value.

The main point here, unless I am misreading, appears to be that modal logic (logic that uses the necessarily and possibly operators) is intensional, not extensional. Or in other words, logically equivalent modal sentences may not retain the same truth values if they are both modulated by the same operation.

Can I sugest

Boxes and diamonds: An open introduction to Modal Logic
as a companion to this thread.

HTML at https://bd.openlogicproject.org/bd-screen.pdf

Here's the definition of extensionality, from p.205, Appendix A:

Definition A.1 (Extensionality). If A and B are sets, then A=B iff every element of A is also an element of B , and vice versa.

It's the rule that these are all, extensionally, the very same: {a, a, b }= {a, b }= {b, a}.

And if the item a is also called "Fred", the they are extensionally the same as {Fred, b}.

The main point here, unless I am misreading, appears to be that modal logic (logic that uses the necessarily and possibly operators) is intensional, not extensional. — NotAristotle

Better, that it was thought to be intensional, until Kripke. Read on.

The introduction conceptually orients; "possible worlds" means something like - that that is opposed to the "actual world" such as a historical counterfactual, or perhaps, an agent acting differently than she or he actually did. — NotAristotle

Thanks for starting us out. I think you're right that we could think of possible worlds as opposed to actual, but in this context, we're following Leibniz, who allowed that the actual world is a possible world. What we would say is that the states of affairs associated with the actual world obtain. So they happened. This impacts the way we verify statements. We would say statements associated with the actual world will be verified by looking around us, so to speak. The statements of a possible world that does not obtain might have to be verified using logical or metaphysical possibility. For instance, in the possible world where Nixon lost the election, he wouldn't be living in the White House. You can imagine the various aspects of verifying that statement: "After Nixon lost, he wasn't living in the White House."

So we've brought up this terminology:

1. Statements
2. States of affairs
3. Verification
4. Obtain vs non-obtaining
5. Propositions

I'll note that I have an affinity for thinking in terms of propositions, which I think of referring to the elements of a community's bank of common ideas. I don't worry a lot about the ontological status of it, I don't know how it works. I just know it's part of how I think about ideas that count as abstract objects. They don't belong to me. They belong to the community. I can be wrong about them, and so on.

If we need to go back and explore any of the above terms, we can do that. Keep in mind that each one is a long rabbit hole, so we may start other threads if it becomes too boggy.

Thoughts?

I’ve always had a hard time understanding the value of the possible worlds way of thinking about things. I read the first section of the SEP article and a little bit of the second section.

I am a self-avowed pragmatist. Can somebody explain how I might use model logic to solve problems or clarify concepts.

Uses for modal logic

In philosophy
In engineering

I’ve always had a hard time understanding the value of the possible worlds way of thinking about things. I read the first section of the SEP article and a little bit of the second section.

I am a self-avowed pragmatist. Can somebody explain how I might use model logic to solve problems or clarify concepts. — T Clark

Extending back to Socrates, there's an aspect of philosophy that is essentially a reflection on how we humans think. The highlights are points where we feel like we may have pulled the veil back on the underpinnings, or clarified something. The reasons a person might be compelled to explore this sort of thing are probably going to be personal, but with a connection to cultural events.

If you don't have this sort of need to understand, it's probably going to seem pretty useless. Your question reminds of the lament of the founder of Manichaeism that he was trying to make a new religion, but the local authorities were only concerned with whether or not he was healing people of disease. Diverging priorities. What can we do?

Thanks

Thank you

As a self proclaimed naturalist and a zealot follower of Wittgenstein, if you interested in Kripkean modal semantics and how its rigidity distorts what science actually discovers and how language is actually used, I would most happy present my lengthy criticisms on this type of thinking and its application. But i think this thread wants to enlighten these views, not critique them.

1. Possible Worlds and Modal Logic

In this first paragraph, the author of this article, Christopher Menzel, lays out the problem that possible world semantics was supposed to address:

In addition to the usual sentence operators of classical logic such as ‘and’ (‘∧’), ‘or’ (‘∨’), ‘not’ (‘¬’), ‘if...then’ (‘→’), and, in the first-order case, the quantifiers ‘all’ (‘∀’) and ‘some’ (‘∃’), these languages contain operators intended to represent the modal adverbs ‘necessarily’ (‘□’) and ‘possibly’ (‘◇’). — Possible Worlds, SEP

So the point was to add modal logic to first order logic. The problem was that modal logic had never been rigorously developed in the way first order logic had been, plus there was skepticism about its content:

A concomitant philosophical consequence of this void in modal logic was a deep skepticism, voiced most prominently by Quine, toward any appeal to modal notions in metaphysics generally, notably, the notion of an essential property. (See Quine 1953 and 1956, and the appendix to Plantinga 1974.) — ibid

Thoughts?

You are saying that a proposition is a statement that we all agree on? I have heard the term proposition applied in a more neutral sense. "The cat is on the mat" might be a proposition. It could be true; it could be false; it is not necessarily something we agree on. I think that is what you mean by "statement" however.

Thoughts? — frank

Sounds right to me. To use the language of the article, I think "possible world semantics" is supposed to change "modal logic" from an "intensional" into an "extensional" language (EDIT: Or as I read further, to subject modal logic to an "extensional semantic theory"). Or, put differently, to change modal logic so that it is subject to "substitutivity principles."

The term "semantics" is a question mark for me here because semantics has to do with meaning, right? So how does meaning factor into a formal logical system?

Semantics in a logical system seems like a somewhat difficult prospect. Would be interested to hear your criticisms of Kripkean (possible world?) semantics after I have digested the article.

1.2

The problem we ran into with the extensionality of modal logic concerned the fact that modal logic appeared to not be subject to classical substitutivity principles. An ostensibly more accurate statement would be that classical substitutivity does not hold when first order logic Tarskian interpretation is attempted to be translated into modal logic's possible worlds interpretation.

A Tarskian interpretation appears to apply only to the actual world. Thus, the author says that the Tarskian interpretation: "fixes the domain of quantification and the extension of all predicates." Tarskian interpretation, with its own semantics, does not appear to allow for possibility. That is why a possible world semantics was proposed for modal logic.

To try to simplify some of the symbolism in the article, the "possibly" quantifier: quantifies over a set of statements that themselves refer to states of affairs that are true about at least one possible world (w). The "necessarily" quantifier: quantifies over a set of statements that refer to states of affairs that are true of the set of all possible worlds (u).

It seems to me that substitutivity principles can be maintained within a possible world semantics applied to modal logic, but that a Tarskian interpretation of first order logic cannot be reconciled with possible world semantics.

Sounds right to me. To use the language of the article, I think "possible world semantics" is supposed to change "modal logic" from an "intensional" into an "extensional" language (EDIT: Or as I read further, to subject modal logic to an "extensional semantic theory"). — NotAristotle

I think your EDIT is the proper interpretation. It makes modal logic the subject of an extensional logic. Here's a quote from the referenced supplement at the end of 1.2:

"As noted, possible world semantics does not make modal logic itself extensional; the substitutivity principles all remain invalid for modal languages under (basic) possible worlds semantics. Rather, it is the semantic theory itself — more exactly, the logic in which the theory is expressed — that is extensional."

You are saying that a proposition is a statement that we all agree on? I have heard the term proposition applied in a more neutral sense. "The cat is on the mat" might be a proposition. It could be true; it could be false; it is not necessarily something we agree on. I think that is what you mean by "statement" however. — NotAristotle

No, I don't think a proposition is something we necessarily agree on. It's a truthbearer. It's the content of a thought or belief, for instance if John believes that grass is green, then the content of his belief is that grass is green, and that's a proposition. The alternative to propositions would be Davidson's system, which uses sentences.

The term "semantics" is a question mark for me here because semantics has to do with meaning, right? So how does meaning factor into a formal logical system? — NotAristotle

I think the point was to be rigorous about the meaning of certain statements, and that was lacking for modal statements. So they wanted a logical system that would specify the meaning of "It is possible that grass is green" or "It is necessary that grass is green."

but that a Tarskian interpretation of first order logic cannot be reconciled with possible world semantics. — NotAristotle

Why is that?

Alright, then by statement do you mean a token of some proposition in some possible world?

Why is that? — frank

I was thinking it is because the Tarskian interpretation of the extension(s) referenced by a quantifier does not account for possibility.

That is, the sentence: "Necessarily, all John's pets are mammals" is not a sentence that can be parsed by a Tarskian interpretation that converts the first order logic to that sentence.

And that is because there is no way of making sense of "necessarily" under a Tarskian interpretation and without possible world semantics.

Sure thing, my critique would begin with natural kinds, and the “infamous” example “water is h2o”.

Alright, then by statement do you mean a token of some proposition in some possible world? — NotAristotle

"Statement" and "proposition" are often used interchangeably. You just have to determine what the author means.

The problem was that modal logic had never been rigorously developed in the way first order logic had been — frank

From the article:

And even though a variety of modal deductive systems had in fact been rigorously developed in the early 20th century, notably by Lewis and Langford (1932), there was for the languages of those systems nothing comparable to the elegant semantics that Tarski had provided for the languages of classical first-order logic. Consequently, there was no rigorous account of what it means for a sentence in those languages to be true and, hence, no account of the critical semantic notions of validity and logical consequence to underwrite the corresponding deductive notions of theoremhood and provability. A concomitant philosophical consequence of this void in modal logic was a deep skepticism, voiced most prominently by Quine, toward any appeal to modal notions in metaphysics generally, notably, the notion of an essential property. — SEP | Possible Worlds | 1.0

There is a notable presupposition occurring here. It is the presupposition that natural language yields to formal language in terms of semantic rigor. The author, Christopher Menzel, is saying that Tarski's "elegant semantics" provided "truth" with semantic rigor in first-order logic, and that there was a desire to mimic this move by providing an "elegant semantics" for natural-language words like "necessarily" and "possibly," thus furnishing them with the same sort of semantic rigor that Tarski achieved.

The underlying question has to do with the relative value of natural and formal languages (see for example the edit to ). A second question has to do with the role of metaphysics in language and logic, and Quine's "skepticism" pertains to this second question. The formalization of the concept of "truth" vis-a-vis (formal) language was driven in large part by anti-metaphysicalists who wanted to reduce truth-questions to language-questions. Those same anti-metaphysicalists, such as Quine, were presumably less comfortable with modal terms given how much more difficult it is to maintain an anti-metaphysicalism while at the same time taking such terms seriously. Put more simply, modal language is too metaphysical for Quine's liking, and this in turn signifies that modal language will be even less amenable to formal semantics than truth language, namely because metaphysical inquiries and claims cannot be reduced to formalisms. These metaphysical "incursions" become more and more obvious as this tradition progresses, and especially so in your second SEP article. The rub lies in the fact that not everyone involved in these movements was an anti-metaphysicalist, at least to the same extent.

I think this is a good idea for a thread. The difficulty for my part lies in trying to understand how to interact with an encyclopedia entry.

The term "semantics" is a question mark for me here because semantics has to do with meaning, right? So how does meaning factor into a formal logical system? — NotAristotle

Formal logic clearly differentiates semantics and syntax. At the core it's the difference between strings of letters in an accepted order and what those strings of letters stand for.

What follows is greatly simplified. The full story can be found in any introduction to logic.

So we can set out the whole of the syntax of predicate logic - the p's and q's - in a few brief rules.
1. We are allowed to write any letter p, q, r...
2. We can put a ~ in front of any thing we can write. So we can write ~p, ~q, and so on.
3. We can put a ^ between any two things we write. So we can write p ^ q, and we can write ~p ^ r
....and so on. If we stick to these rules we can't write things such as "p~^~".

A few simple rules like this sets out the syntax for propositional logic - the strings of letters that are allowed. These are called the well-formed formulae (WFFs). But it says nothing at all about the semantics, what those letters stand for. Thats the place of semantics. The p's and q's are taken to stand for sentences or propositions, the "^" stands for "and", the "~" for "not" and so on.

In predicate logic, we adopt the whole syntax of propositional logic, and add that were we write the p's and q's, we might write f(a) and g(a, b) and so on. This again is syntax, telling us only what we are allowed to write down, how the symbols used can be ordered, and nothing about what they stand for. We also include x, y, and so on as variable for individuals.

In predicate logic we give an interpretation by assigning individuals to the a's and b's. So "a" might stand for "NotAristotle" and "b" for "Australia" and so on. The process of giving an interpretation is more or less the process of constructing a model for the language. It's what determines what is true and what is false. So if we add that "g(x,y) is interpreted as "x is in y", then g(a,b) is just "NotAristotle is in Australia", and we have that g(a,b) will be true if and only if "NotAristotle is in Australia".

Again, note that syntax tells us what we can write, but not what is true and what is false. To decide what is true and what is false, we need a semantics, an interpretation of the symbols.

So we have the syntax, which is bunch of symbols and the rules for writing them. Then we have an interpretation, the rules for what those symbols stand for, in a given domain, a set of all the individuals. And together these make a model, a system that tells us which strings of letters are true and which are false. The model gives the truth values of the sentences or propositions of the language, the semantics of the formal language.

I see. I think my questioning about semantics in relation to logic was in reference to propositional logic that deals with formal languages only; in that case, maybe you would see why I would wonder how semantics fits. But I understand how an interpretation can add a semantic component to predicate logic.

we need a semantics, an interpretation of the symbols. — Banno

By this I understand you to be saying that the symbols need to refer to something (or predicate something) in the world (or in a possible world if we are using possible world semantics).

Yeah I am still confused about why modal logic itself is not extensional, but possible world semantics is apparently extensional.

By this I understand you to be saying that the symbols need to refer to something (or predicate something) in the world (or in a possible world if we are using possible world semantics). — NotAristotle

Roughly, yes. But it's freer than that. It's fine in a formal system to say things like "a" stands for a, perhaps in explaining what the " does in separating mention form use. The symbols do not have to denote actual-world entities.

So they can, but need not.