That is too strong. — hypericin

Yeah, agreed.

So can we always seperate out the affective and cognitive aspects of a belief? Is there a method, rule or algorithm that does this for us? I'm thinking not.

Checking if I've understood... are you suggesting that belief is a propositional attitude, and that we also have affective attitudes, and that these are unrelated?

You and he are talking about completely different aspects of emotion and reason than I am. — T Clark

Yep. That'll be 'cause we're on the topic. And read more than just the abstract.

There's no conceptual work to do here?

But what of the issues raised in and ?

And in the cited articles? Or the SEP articles on belief and emotion?

If Bob has an irrational fear of snakes, does he believe snakes are dangerous? — frank

If his fear is irrational - he refuses to touch a Child's Python, perhaps - despite knowing that he will not be hurt - then isn't he is afraid, but does not believe the snake to be dangerous?

Thanks for this reply.

You ask if all beliefs contain doubt. The obvious counterexample is the hinge beliefs @Sam26 has urged on us, and with which I mostly agree. Do we say that, because these are undoubted, they are not genuine beliefs? Or do we separate the cognitive view that such hinges are indubitable from the connotative view that nevertheless, I might be wrong...

So if someone does not doubt that 2+2 is 4, do we discount this as a belief becasue it is indubitable?

How do we represent that Balthasar agrees the glass skywalk is safe but refuses to walk on it? If we say that since he holds that "the skywalk is safe", that he believes the skywalk is safe? Or do we say instead that because he will not act on what he holds to be true, that he doesn't really believe?

Charu's decision, let us supose, is against the odds - a bookmaker would say the lover will stray again. But Charu wants the bookmaker to be wrong, and so apparently acts irrational. Except that there is no possibility of the bookmaker being wrong if Charu does not trust her lover. Given her desire to stay with her lover, the decision to trust is rational.

David's belief is not to be subjected to doubt. What are we to say here - again, that it's not a proper belief becasue it is indubitable? Or is it, as is so often supposed, the very epitome of belief precisely becasue it is undoubted despite the evidence?

Plenty of material here, plenty to consider.

I recall Steven Pinker stating that we justify beliefs using reason, but we form them based on our affective relationships with the world. — Tom Storm

Yeah, pretty much. The belief is prior to the argument. But are we amenable to rational persuasion with regard to our beliefs? And to what extent? Should a mental state that is not amenable to persuasion based on evidence or justification properly be called a belief? That's the direction this discussion might go.

This is a fact rather than an idea. — I like sushi

Everything in our minds is a blending of cognitive and non-cognitive states. — T Clark

As says, how do we know? Let's aim not to make pronouncements but to map out the territory - what part of belief is cognitive, what is connotative, and how do they relate?

...we consider Reason (including logic, cognition and representation) as inseparable aspects of the same phenomenon, and not as separable states — Joshs

The obverse and reverse sides of a coin are inseparable, but that does not prevent us considering them separately as required. we might map how they relate and how they differ. We take the blanket statements and map out the where or how.

"Being referred to by a name is not part of the logical property structure -- it belongs to the semantic interpretation." — J

More accurately, being a name is not part of the syntax, but part of the semantics, of the model being used. The model may be formal, or it may be part of a natural language. Modelling is a part of logic.

I'll have to take it for granted that there are no other successful interpretations. — Ludwig V

A good point. If there were alternate explanation, what might they look like?

There's a generalisation of possible world semantics that turns out to be predicate calculus. That is, what can be expressed in modal logic can be expressed just in predicate calculus, after Montague and van Benthem. Strictly, modal logic is not an extension of predicate logic, but a fragment of it.

So if there are other ways of formalising modal logic, if they are as strong as possible world semantics, they will still be a subset of predicate calculus.

Roughly, any other alternative interpretation would be equivalent to possible world semantics.

...an apparent asymmetry between referring to something and being referred to by something. — Ludwig V

Yep. There's a difference between using a name to refer and inaugurating a name. Two different speech acts. Referring to something is a different sort of activity to making it that some word is used to refer.

The inauguration of a name involves a status function, one way or another. The word comes to "count as" the thing.

So now the question whether the concept of a rigid designator is part of the formal system and must be assessed in that context, or part of natural language and assessed in that context. — Ludwig V

My inclination is to think about the interpretation as something we do with the syntax, as an activity. If that's on track, then we use a name as a rigid designator. The name is a string of letters until used as a rigid designator. The string of letters is a part of the syntax of the system, the use as a rigid designator, to name something, is a part of the semantics of the system.

I somewhat regret suggesting a third level, since the gap between a formal modal and a natural language is no where near at the level of the gap between a syntax and a semantics.

We don't always find it easy to have a clear definition yes. — Jack2848

That's not the argument. The argument is that there is no "clear definition".

With the exception of merely stipulated technical terms, the way we use words precedes any definition. The word is used before it is defined. It follows that definitions are post hoc, with all the issues that involves.

Certainly this is true of "know", with it's etymology going back at least to the PIE root *gno- "to know."

Justified true belief was never going to be the whole of how we use "knowledge".

And even worse for knowledge as true information.

Excellent essay.

The usual suspects are here, bending over backwards to pretend that it doesn't apply to them. What a sad lot they are... the self-made man has a fool for his creator.

Well, you are not the first to present this sort of account, a scientists setting to rights the poor benighted philosophers.

It's easy to provide an answer when you haven't understood the issue.

Hang around for a bit, see if you notice anything odd or problematic in what you've decided. Then we might have an interesting chat.

Ok. Since you have it all worked out, I'll leave you to it.

, , , ...so one might find oneself at pains to demonstrate something that is not at issue.

The first argument:

I don't think dualism is an assumption. It's a description of a state of affairs. — karl stone

A description. So you are saying that it's an empirical observation? What is it we are observing here...

I understand what it means to observe the sky, or the horizon, or the sound of the sea. I'm not sure of what it could mean to observe internal and external worlds. I see the sky with my eyes, hear the sea with my ears - what sense do I use to observe my own mind? And who is it that is doing the observing of that mind, if not my mind...?

That all seems very odd. A long stretch.

The second argument:

Senses that are evolved to enable us to survive; and thus, demonstrably accurate to external reality. — karl stone

I'm not at all keen on Donald Hoffman, a chap with whom you have some points in common, but one point he makes is that there need be no relation between what the evolved mind presents to us and what is "out there". Indeed, his conclusion is quite the opposite. Evolution selects not for veridical perceptions, but for fitness-enhancing ones — and these two are not only distinct but often incompatible. We cannot assume that perceptual accuracy correlates with survival success.

When one decides on one's enemy - subjectivism, perhaps, whatever that is - one tends to see them everywhere. One might find oneself criticising an argument that hasn't been presented.

Isn't it rather that in order to make an observation at all, you become an observer seperate from what you are observing?

banno If I may chime in with a related question. I seem to be missing an understanding about what a property is. I can see that whatever name that object (i.e. the eiffel tower) has is distinct from any of the properties that it has qua physical object. But if "a" refers to a, then it seems to me to follow that a is referred to by "a". So if the former is a property of "a" then the latter must be a property of a? You and J seem to agree that "a" does have a property of referring to a, but that although a is referred to by "a", that does not constitute a property. I don't see why not. I do see that one could not use it to identity the reference of "a", because it would be circular, but that's a different matter, isn't it? — Ludwig V

So what is a property? Fair question. Logicians use "property" in a slightly different way to Aristotle.

What is "'a' refers to a" about? At a naive level we might suppose that since the sentence predicates "'a' refers to..." to a, then a is an individual with the property of being referred to as 'a'. It's not much of a property. Aristotle and others added various ad hoc ideas here in an attempt to make it work. Formal logic does something quite different.

A property in extensional formal logic is just a set of things. So the property of being red is just the grouping of individuals that are red, the property of being in Australia is just the things that are in Australia, and that's all. The property of being red = the set of all red things. This captures only which objects fall under a predicate, not why or how.

The predicate “is referred to by ‘a’” defines a singleton set: {a}, so in this stripped-down view, it could be called a property — because a is in the set. However, singleton sets carry no information or weight. It's trivial, it's not discoverable by empirical means, it's neither intrinsic or necessary that "a" refer to a.

Do we count it as a property, then?

The point made earlier, in amongst a bunch of other stuff, is that in predicate calculus, properties are thought of as represented by the letters f,g,h... and so on. So f(a) says that a has the property f. Such systems are given an interpretation by assigning individuals to the individual variables.

We have the list of predicate variables, "f","g","h"; and the list of individual variables, "a","b","c"...

We assign a property, say f, to some individual, say a, by writing f(a).

We assign an interpretation to this syntax by assigning an individual to each of the individual variables, a to "a", b to "b", and so on.

So, assigning a property to an individual happens in a different part of the logic to assigning a name to an individual.

For these two reasons, having a name is not usually considered as having the property of having that name. Being referred to by a name is not part of the logical property structure — it belongs to the semantic interpretation.

So I still want a way to characterize the difference between saying "The Eiffel Tower is tall" and "That object [pointing] is called 'the Eiffel Tower'". Yes, the first is a property and the second is not, but where do these statements fall on the syntactic/semantic spectrum? — J

Seems fair to say that in natural languages the distinction between semantic and syntactic is fluid, far more so than in a formal language.

As I understand it, Kaplan's approach is to explain the character and the content of an indexical separately. The character is a function that in a given context yields the object being pointed to. So the character of "I" yields me; the character of "that" yields what is being pointed to, and so on for different indexicals. The content is the individual involved.

So “that object [pointing]” has an indexical with character λc. demonstratum(c), and content (in c) = Eiffel Tower. There's plenty more formalism that can be dropped in here, but the idea is basically that each different indexical has a character that returns an individual as the content. "I" returns the speaker, "that" returns the thing being pointed to, "you" returns the person being addressed, and so on,

The content of "that" is a rigid designator, and so in predicate and modal logic can be an individual variable.

This is all semantics. It's about the things, not about the strings.

So "That is tall", indicating the Eiffel tower, and "The Eiffel tower is tall", are about the same thing, referring to it using a rigid designator.

But "That object [pointing] is called 'the Eiffel Tower'" has an indexical that returns a rigid designation that is then equated to a name -- hence 'the Eiffel Tower' is in quotes. Being in quotes indicates that it is part of the metalanguage, that it's about the interpretation of the language and not a sentence int he language.

Is there something here that this misses?

Oh, boy. there's a lot in that post.

Relevant logic uses a variation of the accessibility relation from possible world semantics to model relevance relations between worlds. Together with Kripke's strict conditional this is used to group conditionals together in a way that seeks to overcome the so-called paradoxes of implication.

I'd have to do a bit more work before claiming to see how it works in the examples you give. My thought at the time I made the suggestion was that perhaps the relevance relation might be used to clarify the difference between, say, "heat is molecular motion" and "temperature is molecular motion"; that temperature might be relevant to molecular motion in a way that heat is not. it'd be something like that from a world in which we have temperature, we might only be able to access worlds in which we have molecular motion, but from a world in which we have heat, it might be that we can access a world without molecular motion. It might follow that temperature, and not heat, relates necessarily to molecular motion. I do not have sufficient grasp of the machinery of relevance logic to follow through on this.

What this shows is that we've been ignoring a bit of the theory of possible worlds that is becoming increasingly relevant here. We've been saying that ☐A is understood as "A is true in every possible world". This needs some qualification. These statements are indexed to possible worlds and to the accessibility relation. So a more accurate account is that ☐A in world one is understood as "A is true in every world that is accessible from world one".

We get away with the cut down version by assuming that we are working with the syntax of S5, in which every world is accessible from every other world.

And there's another qualification that is needed, as to the difference change in the use of "contingent" in Kripke's semantics. Previously, "contingent" had a sense of "dependent", so something was contingent if it's being true was dependent on some other fact. That is not so much the case in possible world semantics. A statement will be necessarily true in w₁, as explained, if it is true in every world accessible form w₁. A statement will be possible in w₁ if there is at least one world, accessible from w₁, in which the statement is true. It follows that if a statement is necessary then it is possible. A statement will be impossible in w₁ if there is no world accessible from w₁ in which it is true.

And a statement will be conditional in w₁ if there is at least one accessible world in which it is true, and at least one accessible world were it is false.

The dependence on "dependence" drops out, along with a whole lot of metaphysical baggage and a few proofs that god exists. That's part of the reason there is some resistance against this logic from those of a naive theological bent.

So to the metre rule, and area that is fraught with misunderstanding. Consider this, from your Malcolm quote: "Certainly there is no requirement to hold that if the sentence is not a contingent statement then it must be a necessary statement."

This might be so if we think of contingency as dependence. The length S is dependent on the length of the rod, in the base example.

But in a possible world semantics, if a statement is true, and not contingent, it follows immediately that it is necessarily true. Malcolm, from Kripke's perspective, is mistaken: there is a requirement to hold that if the sentence is not a contingent statement then it must be a necessary statement.


And to god. So for Malcolm, the concept of God, like seeing a material thing, is not inherently self-contradictory. While specific reasoning may be invalid, a general demonstration of non-contradiction is not possible. Both concepts are integral to human thought and life.

See my comments in for a bit more on Anselm. While we might not agree that the idea of a something a greater than which cannot be conceived is self-contradictory, it's not clear that it can be made coherent, either. There is the problem of how to deal with a necessary being without the consequence of modal collapse.

But that's enough for now. Thanks again for your posts.

He's trying to lay out the requirements for a consistent picture, not choosing among pictures — J

That's the idea.

It's not that we can't, or even shouldn't, choose amongst the metaphysical theories. It's that if some metaphysics is not in harmony with the best thinking on modal logic or logic more generally, we ought treat it with scepticism; it has issues to be addressed.

He did at one stage describe himself as an anti-realist, but that seemed to be a result of his toying with truth, a seperate issue to possible worlds.

The women in this party are so assertive now that we may need some special rules for men to get them pre-selected. — Alan Stockdale, to NSW Liberal’s Women’s Council

:roll:

,

Kripke: Identity and Necessity

Kripke asks if this lectern could have been made of ice. His answer is that it is entirely possible that the lectern before us is made of ice, but that if this were so it would be a different lectern. — Banno

I don't think I've bought into Cartesian assumptions — karl stone

I'll try to explain the assumption again. It's ubiquitous, and so can be difficult to see.

Descartes supposes that we have on the one hand, mind, and on the other, the stuff of the world; roughly the presumption is that of dualism, of a divide between what is subjective and what is objective. Immediately on making this supposition he, and we, are faced with the problem of how the stuff of the mind interacts with the stuff of the world. Descartes solution was god, your solution is observation.

But what if that supposition, that schism between thought and thing, were a mistake?

I'll leave that hanging. Thoughts?

This makes the issue much more precise, thanks. — J

Cheers.

The difference between syntax and semantics is very clear in formal logic. Less so in natural languages. The following is probably familiar.

Let's look briefly at propositional logic. It includes just letters, p,q, and so on, as many as you want, and a couple of symbols, usually v and ~. To this we add formation rules that tell us what we are allowed to write. First, we can write any letter by itself. So we can write "p", or we can write "q". Then, we are allowed to put a "~" in front of anything else we are allowed to write; so we can write "~p" and "~q" and so on. Then, for two things we can write, we can join them with a "v". So we can now write "~pvq". From this, we can set out a system that shows how some strings of letters are well-formed - they follow the formation rules - while others are not... if we follow those rules we can never write down "pvvq~", for example. (I've left out brackets just to keep things simple. Also, ^ and ⊃ can both be defined in terms of v and ~, so they are not needed here)

All we have here is a system of syntax. It is purely a set of rules for stringing letters together in a specific way. In particular, it tells us nothing of what "p" and "q" stand for, and so nothing of which of our strings of letters might be true or false.

We add an interpretation to this syntactic system by ascribing "T" or "F" to each of the letters, together with a rule for the truth functionality of "v" and "~". A string beginning with "~" will be T if and only if the stuff after the "~" is F, and a string joined by a "v" will be T if and only if the stuff on either side of the "v" is also T.

A useful way to understand this is that p,q, and so on denote either T or F. We've moved from syntax to semantics.

We can expand our syntactic system by allowing ourselves to write not just p's and q's, but also "f(a)" and 'g(b)" and so on, in the place of those p's and q's. We can add rules for using ∃(x), but still at the syntactic level - just setting out what is well-formed and what isn't. This gives us a bigger system.

And to that we add more interpretation, were the letters "a","b" and so on stand for a and b, respectively, and "f" stands for some group of such letters, perhaps "f" stands for {a,c,e} while "g" stands for {b,c} or whatever. We then get that f(a) is true - "a" is in the set {a,c,e}, while f(b) is false - "b" is not in the set {a,c,e}. Similar rules apply for interpreting the quantifiers. And this gives us predicate calculus.

We can then expand the system once more, adding the operator "☐" outside of all of the stuff in the syntax for predicate logic, together with a few rules for how we can write these. This gives us the systems S1 through S5. These are just ways of writing down strings of letters, with ever more complicated permutations.

In order to give a coherent interpretation to these systems, Kripke taught us to use possible world semantics. In a way all this amounts to is a process to group the predicates used previously. So we said earlier that "f" stands for {a,c,e}, and to this we now add that in different worlds, f can stand for different sets of individuals. So in w₀ "f" stands for {a,c,e}, while in w₁ f stands for {a,b}, and so on in whatever way we stipulate - w₀ being world zero, w₁ being world one, and so on. Now we have added a semantics to the syntax of S4 and S5.

So reference-fixing is giving an interpretation, yes? — J

Exactly.

We have two levels, if you like, for each system. At one level we just set out how the letters can be written out, what sequence is acceptable. That's the syntax. At the next level, we add an interpretation, what the letters stand for. That's the semantics. So for propositional logic, the letters stand for T or F, and for predicate logic, we add individuals a,b,c... and for modal logic we add worlds, w₁, w₂ and so on, in order to get out interpretation, our semantics.

And to this we might add a third level, where we seek to understand what we are doing in a natural language by applying these formal systems. So for propositional logic, we understand the p's and q's as standing for the sentences of our natural language, and T and F as True and False. For predicate logic, we understand a,b,c as standing for Fred Bloggs, the Eiffel tower and consumerism, or whatever. And in modal logic, we get Naming and Necessity, where we try to understand our talk of modal contexts in natural languages in terms of the formal system we have developed.


I left out brackets, truth tables, domains, and accessibility, amongst other things, and only scratched the surface of extensionality. But I hope I've made clear how clean the distinction is between syntax and semantics in formal systems.

If reality's nature is not such that eyes can give us valid information about it, then I would expect reality to have evolved some other system to do so. — Patterner

There's something deeply problematic about using evolution to explain away ontological problems. Evolution assumes a degree of realism in assuming that there is a deep past in which there are things that could evolve, so of course it is consistent with realism. But it would be a mistake to think that evolution demonstrates realism.

None of which is to say that evolution didn't occur. It's just not methodologically a good way to try to dispose of solipsism or idealism.

Now it seems to me that despite his protestations against Cartesianism, @karl stone is buying in to many of the assumptions that Descartes made. He wants to find firm foundations and build a system from those foundations, a very Cartesian method. Sure, instead of the cogito he wants to use perception as that foundation, but it isn't going all that well.

Of course I agree with @Sam26 that a response is found in a treatment of what it is to doubt, along the lines of Wittgenstein's discussion of hinge propositions, but unlike Sam I reject idealism, along with certain sorts of realism, as a false juxtaposition.

But the devil is in the detail, and the way forward is to keep struggling with the analysis.

I wonder how we make sense of such claims as "if I were you then ...." — Michael

@J gave a pretty clear account of this, don't you think? Together with @frank's account of how we identify an individual with their origin, which is the approach Kripke is arguing for in Naming and Necesity.

Seems to me you are correct that @karl stone hasn't succeeded in casting aside the sceptic.

I'm pleased to see so much analytic work going on. Working through the issues is the only way to work out how to fit all these pieces together - if that is possible.

This is core analytic philosophy - looking closely at how the terms involved are being used, comparing them with formal systems we know are consistent, seeing what works and what does not. Bread and butter stuff. It's hard conceptual work.

Good analytic arguments.

You'll get a pushback against "you know it is real because you can see it" from the idealists and solipsists, who will claim that it might be an hallucination or other phantasm.

My response is that I don't much know what it means to say that it's a real screen until you tell me what the alternative is - a cardboard screen or a result of the acid drops or whatever. But I'll point out that you and I can agree that it's a screen and get on with the conversation on that basis. That is, we can agree not to subject the screen to doubt. Hence the independent party bit.

The point that we find ourselves embedded in the world is a good one - even if you put it in terms of being an "evolved creature".


So, unfortunately, I'm not seeing a substantive point of disagreement between us. Except the foot pain... I can't feel the pain in your foot.

Cool.

f(a, (b,c)) is of course malformed, and even if we charitably allowed some sort of well formed interpretation - perhaps f(a,b,c) - it doesn't even address the issue of extensionality.

Ok.

I'm not overly thrilled with Descartes. Been a while, other ideas came along. So arguing against him is superfluous. Unnecessary.

Knowing that you have a pain in your foot is not exactly like knowing there is a screen in front of you becasue you can see it.

Seem to be a few things being treated as one here.

I don't see how the knowledge that Tully wrote X is something about the body. — frank

It's not not about the body either. Your body wrote the reply, making use of what you knew about Tully, in a way not that dissimilar to how you ride a bike, making use of what you know about peddles and wheels.

The classical approach is to divide "know how" from "know that", and treat of each with an utterly different account. I want to consider an alternative: that knowing involves doing, including doing speaking and thinking.

Sure. We agree on much of this.

Are you comfortable saying you know things that cannot be justified? Saying you know you are in pain looks to be unjustifiable, so is it the sort of thing we can use as the epitome of "know"? Or is it more a fringe use?

Pain, being prior to thought, cannot be doubted. — karl stone

Ok. Can it be known?

Have a look at Sam's answer. it's not just pain that cannot be doubted. Can you coherently doubt that you are reading this, and that it is a reply to your own post? Not if you are going to answer me.

You're not wrong, but there is more here...

If you like. "relationship" would not be my choice, for the reasons given.

"Object of thought" is loaded. "Content" might be preferred.

But also, what is a "thought"? A proposition in one's mind? Is it distinct from a feeling, or an intuition, or a belief?

These are the problems with the classical approach - might call it the cognitive theory of knowledge, that are addressed by treating knowledge as embodied, as an activity.

Trouble is, saying you know you have a pain is problematic. How do you justify such a claim? The only justification seem to be that you are in pain - the justification is the very thing being justified.

Thoughts? Are you happy to claim to know things that are unjustifiable?

What's wrong with saying knowledge is a relationship between a knower and a proposition? — frank

"relationship" has a particular baggage - f(a,b) is a relationship.

But the proposition isn't related to the knowledge, so much as part of it's content.

This can be set out again in terms of substitution. if it were a relation, then substitution should be allowed - if f(a,b) and c=b then f(a,c).

But if you know that Cicero wrote De Officiis, it does not follow that you know that Tully wrote De Officiis, despite Tull=Cicero.

So knowing is substitutionally opaque. Relations, not so much.

Not so sure about this. First of all, I don't take "If I were Barack Obama . . . " as a genuine reference to a possible world. For me, this is loose talk for "Barack Obama should have. . . " If we insist on pressing this hypothetical, we run up against Kripke: "You can't be Obama; he was born of different parents." And I think this is right. "If I were Obama . . . " etc. reads like a meaningful sentence but that's an illusion. — J

Yep.

Is it simply a stipulation? — Michael

Pretty much. I use that idea.

Others might picture a logical space in which all possible worlds are listed, and think instead of selecting those worlds that match some criteria from that set.

It amounts to much the same thing.

Importantly, logically possible worlds have no relation to the possible worlds of quantum mechanics. They are very different activities. Trying to join them will lead to confusion.

, , oh, and the continuation of my previous post is that fixing the name - with an original baptism or whatever - is a very different sort of thing to using the name.

Naming a thing and using a name are very different speech acts.

So we might say that using a name involves a rigid designation, while setting aside the way in which that rigid designation came about for seperate investigation. Th utility of possible world semantics does not depend on our having an accepted theory of how things are named.

Right, this is the same question I'm raising about whether something about reference needs to be included in a list of X's properties. — J

Good stuff. If I've understood, there is an answer to your puzzle.

In a formal system, there is a difference between the syntax - S4, S5 and so on - and the semantics, the interpretation given to the system. In the syntax the letters don't stand for anything, and the formulae are neither true nor false. To show completeness and coherence we need to give the system an interpretation, also called a model. For modal logic there is an interpretation that works, using possible world semantics.

Now the predicates "f","g", "h"... are understood as properties of "a","b","c"..., and so we can write f(a), g(a), g(b) int he usual fashion, but this is just stringing letters together until the "a", "b", "c" and so on are given an interpretation.

Giving an interpretation is assigning "a" to a, "b" to b, and so on, and also assigning f={a,b}, and so on. So we have the name and the predicate in the uninterpreted system, and the corresponding individual and property in the interpreted system.

Notice the difference between saying that a is f, f(a), which happens within the interpretation, and saying that "a" stand for a, which is giving (stipulating) the interpretation?

That's why it's not quite right to say that a has the property of "being a". Being a is a part of the interpretation, not of the system of properties.

It’s like saying: “Is the fact that we call the Eiffel Tower ‘La Tour Eiffel’ a property of the Eiffel Tower itself?” Obviously not — that’s a fact about us, our language, not about the tower as such.