I never find arguments to the effect of "axiom X is inescapable and even denying it affirms it" compelling. Most of the time such arguments just assume the axiom in the metalanguage and use that assumption to claim the axiom will appear in any language whatsoever, even though it only appears in the corresponding object language because it's being assumed in the first place... — MindForged
That logic could improve so much with the advent of Classical Logic via Frege, and improve over the prior Aristotelian Logic, motivates me to try not to assume that whatever logic is dominant at present is infallible or some such. — MindForged
No, a proposition is just an object. An object doesn't assign properties to itself, an object is just something with properties.
— MindForged
When you say "The dog is black" you assign the property of blackness to the dog.
Well, you have just said that an object (Goldbach Conjecture) both has the property of being true and doesn't have the property of being true. Again, you have violated the identity of an object.
The sentence "The dog is black" is about the situation of a dog having the property of blackness. Its referent is not just the dog, and not just blackness, but the whole situation.
"A referent (/ˈrɛfərənt/) is a person or thing to which a name – a linguistic expression or other symbol – refers. For example, in the sentence 'Mary saw me', the referent of the word 'Mary' is the particular person called Mary who is being spoken of, while the referent of the word 'me' is the person uttering the sentence."
Your initial objection here was the claim the Liars lack a referent in reality. The Liar sentences have a referent (themselves) and that's just the way it is.
— MindForged
The Liar sentence "This sentence is false" says that the sentence is both false and not false, so its referent is a situation where the sentence is both false and not false. But such a situation doesn't exist, because the Liar sentence is just false (like any contradiction). So the Liar sentence has no referent.
So at the beginning of the experiment the two photons are not identical because they have at least one different property - position in space: one is above the beam splitter, the other is below. — litewave
Also, whether each photon at the end of the experiment is the same photon as it was at the beginning of the experiment is a question of the preservation of identity through time. Identity doesn't have to be preserved in time; an object can be annihilated, or merged with another object, or separated from another object at some point in time. But at each point in time an object is identical to itself and different from other objects. — litewave
This is also connected with a second point. What exactly is meant when
we say that we deny a tautology (or a logical law, or a logical necessity)? In
denying that an axiom of classical logic is valid in general, don’t we have to
accept that this ‘axiom’ is false in at least one interpretation of an alternative
system in which the same formula may be expressed? Consider, for instance,
intuitionistic logic. In denying the validity of some instances of the law of
excluded middle, it is not the case that intuitionists accept its negation in its
place. However, they do accept that the law may be false sometimes (mostly
when we deal with infinite collections). For another example, consider some
paraconsistent logics, like those in da Costa’s Cn hierarchy. In denying the
Explosion Law, it is certainly not accepted as an axiom (or as logically valid
in the system) the negation of the Explosion Law, but some of its instances
must be false in some valuations. So, the argument could go, in denying the
universal validity of the reflexive law of identity in non-reflexive logics, are we
not committed to accepting that it may be false sometimes?
As we have said in the previous section, in non-reflexive logics we do not
accept the negation of the reflexive law of identity. Also, we don’t have to accept
that it must fail in at least some interpretations. Rather, we adopt its restriction
in the form of its inapplicability. Here, ‘inapplicability’ is couched in terms of
identity not making sense, not being a formula, for some kinds of terms. — The Received View on Quantum Non-Individuality: Formal and Metaphysical Analysis
That's a contradiction, not an identity violation. Without equality in the language identity isn't present within the language. — MindForged
No it isn't that's not what a referent is. I hate to quote Wiki of all places but it states it plainly:
"A referent (/ˈrɛfərənt/) is a person or thing to which a name – a linguistic expression or other symbol – refers. For example, in the sentence 'Mary saw me', the referent of the word 'Mary' is the particular person called Mary who is being spoken of, while the referent of the word 'me' is the person uttering the sentence." — MindForged
The referent of "The dog is black" is the dog in question, not "the whole situation". — MindForged
The Liar sentence does not say it is both true and false. The Liar claims, of itself, that it is false. That the Liar is also true is entailed by it being false. — MindForged
So any time that a photon interacts with something (say, the beam splitter or the detector), we could say that it is annihilated and created anew. But sometimes we want to consider the identity to have persisted (as with a photon in the double-slit experiment or things at a macroscopic level, such as humans). Even in the HOM experiment, there is some sort of continuity in that we started with two photons and ended up with two photons. But the history in terms of individual photon identity seems not to physically exist. — Andrew M
In other words, Identity (arguably) not applying to a certain class of objects is not the same as saying "An object has a property and does not have that property". — MindForged
So there is a language in which an object can have and not have the same property?
The Wiki quote just says that the referent of the word "Mary" is Mary and the referent of the word "me" is me, with which I agree. But the whole sentence "Mary saw me" is a linguistic expression too, and its referent (meaning) is the situation that Mary saw me.
The referent of the word "Dog" is the dog, the referent of the word "black" is black (color), and the referent of the sentence "The dog is black" is the situation that the dog is black.
That's why I said in an earlier post that the Liar sentence says "implicitely" that it is true. But that doesn't matter. The meaning of the sentence is that it is both false and true, and that's what matters. That's why it is a contradiction.
But without identity there is not really an object. I don't know what it would mean that an object has no identity or what it would mean that the law of excluded middle does not hold. Paraconsistent logic and intuitionistic logic seem to characterize imperfect knowledge rather than objects in reality.
"Black" is not a referent though, it's a predicate, a proeprty an object may have or lack. In your sentence, the "dog" is the only referent in the sentence. — MindForged
You're confusing a predicate with a referent, and you're mistaking a state of affairs (or "a situation") with a referent. — MindForged
"false" is not a referent — MindForged
What do you think self-reference even means? That a situation refers to itself? — MindForged
So you say, and yet the entire point of this view (non-reflexive logics and the referenced view in QM) is that it might be the case that you can have an object without an identity. That just makes your post question begging against an opposing view. I've told you what it could mean: That identity only holds for some objects and not others, which is sketched out via a restriction in the logic as to what identity applies to. If you're looking for an in-depth semantics as to how this can work, well, I already reference the papers. Here, I'll even link them:
Classical Logic or Non-Reflexive Logic? A case of Semantic Underdetermination
The Received View on Quantum Non-Individuality: Formal and Metaphysical Analysis — MindForged
Properties are objects too - they are something that is identical to itself and different from other objects. I see no reason why words could not refer to properties.
Situations are objects too - they are identical to themselves and different from other objects. Situations are referents of sentences.
"False" is a property, so it can be a referent.
The sentence "This sentence is false" refers to a purported situation that includes the sentence, so in this sense it is a self-referential sentence. The part "This sentence" refers to the sentence itself.
In "My dog is black", you are attributing the property of "blackness" to the referent "dog". — MindForged
How are situations objects??? A situation (state of affairs) picks out how things are, it is not itself an object. — MindForged
And in so doing I am also referring to the property of blackness. The property of blackness is the referent (meaning) of the word "black".
An object is anything that has an identity. In other words, it is something (as opposed to nothing). In my view "objects" without an identity are nothing (so not really objects).
No, you are making a *reference* to the property of blackness, the *referent* of the sentence is the dog in question. — MindForged
Contradictions have properties (they are necessarily false, for instance), and therefore (on your view) they have an identity. So on your view there must be a contradictory situation, and hence, a contradictory object. — MindForged
When I am making a reference to the property of blackness then the referent is the property of blackness, no?
A contradiction has an identity as a sentence (a string words) but it does not have a referent. A contradiction refers to a contradictory situation but there is no such situation, so a contradiction has no referent.
It's not a situation, you can have a contradiction that makes no reference to a state of affairs (contradictions in math, for instance). — MindForged
Propositions aren't strings of words, and yet propositions can be contradictory. — MindForged
In theory - and also in reality if the theory is correct - there are two photons throughout the experiment, not one photon. They are numerically different, so there must be a property that ensures that they are two photons and not one. I would say that this differentiating property is the position of each photon in an abstract structure of the theory, because it is the abstract structure of the theory (including the definition of energy of a photon as a product of Planck constant and frequency) that differentiates the situation into two photons. — litewave
Example?
My understanding is that propositions are meanings of strings of words (if the string of words has a subject-predicate structure). In other words, propositions are referents of strings of words, or situations to which the strings of words refer. A contradictory string of words refers to a contradictory proposition/situation but such a proposition/situation would be an object without identity, which would be an absurdity, and therefore such an object doesn't exist and a contradictory string of words has no referent (meaning). We can talk about contradictory propositions or situations but ontologically they are nothing.
Propositions, we shall say, are the sharable objects of the attitudes and the primary bearers of truth and falsity. This stipulation rules out certain candidates for propositions, including thought- and utterance-tokens, which presumably are not sharable, and concrete events or facts, which presumably cannot be false.
[...]
Clearly, not all propositions can be possible states of affairs, because there are propositions that are not possibly true, whereas possible states of affairs must obtain in at least some possible world. We might wish to extend the notion of a state of affairs to include impossible ones.
The problem is that quantum mechanics would seem to rule this out. The reason is that if there were a differentiating property such as position while in the beam splitter then, per figure 1, quantum states 2 and 3 would be physically distinct states and therefore would not destructively interfere (cancel out). But, as experiments show, they do. — Andrew M
Disjunction introduction is a valid rule of inference and it is not a valid rule of inference. That's a contradiction yet clearly it's not making reference to a state of affairs. — MindForged
This is especially the case if you think logic is fundamental to reality, because then the validity of the inference rules varies in different states of affairs (or at least in different possible worlds), which seems to prevent logic from being fundamental. — MindForged
A proposition is not a situation. The SEP summarizes this well:
Propositions, we shall say, are the sharable objects of the attitudes and the primary bearers of truth and falsity. This stipulation rules out certain candidates for propositions, including thought- and utterance-tokens, which presumably are not sharable, and concrete events or facts, which presumably cannot be false.
[...]
Clearly, not all propositions can be possible states of affairs, because there are propositions that are not possibly true, whereas possible states of affairs must obtain in at least some possible world. We might wish to extend the notion of a state of affairs to include impossible ones. — MindForged
Also, that you refer to a contradiction as an object (I agree it is) would seem to imply that it has properties (it does). But if it has properties, it cannot be "nothing ontologically". — MindForged
So it is the theory, the structure of its definitions and rules, that differentiates the stuff into two photons and thus gives each of them a separate identity. In this theory, in its abstract space or structure (and also in the corresponding abstract structure of reality), the two photons have a different position. But physicists cannot measure this position; it's not a position in physical space. — litewave
OK, so granting that there are two photons throughout the experiment, are you saying that the property distinguishing the photons would not be physically measurable but still be physically real (a hidden variable)? Or not physically real and just part of the abstract structure (instrumentalist)? Or something else? — Andrew M
This contradiction is referring to a purported situation where disjunction introduction is a valid rule of inference and not a valid rule of inference.
But if an axiom is valid (true) in one possible world and not valid (not true) in a different possible world then it is not a contradiction to say that the axiom is both valid and not valid. A contradiction arises when we affirm and deny something in the same sense, but here we are not doing it: were are saying that the axiom is valid in a possible world and is not valid in a different possible world. It would be a contradiction to say that an axiom is valid and not valid in the same possible world.
A proposition is not a situation. The SEP summarizes this well:
Propositions, we shall say, are the sharable objects of the attitudes and the primary bearers of truth and falsity. This stipulation rules out certain candidates for propositions, including thought- and utterance-tokens, which presumably are not sharable, and concrete events or facts, which presumably cannot be false.
[...]
Clearly, not all propositions can be possible states of affairs, because there are propositions that are not possibly true, whereas possible states of affairs must obtain in at least some possible world. We might wish to extend the notion of a state of affairs to include impossible ones.
— MindForged
I mean "situation" not as a fact but as an arrangement of an object and its property. Such an arrangement may hold in some possible world and thus be true in that world, and in another possible world it may not hold and thus not be true.
Contradiction in the sense of a string of words without a referent has its identity as a string of words. But contradiction in the sense of the purported referent itself - a contradictory situation/arrangement/proposition - does not have identity (does not have the properties it has) and therefore does not exist in my view; it's ontologically nothing even though we can talk about it as if it were something.
That doesn't make sense. If, as you have said, that logic is fundamental to reality there is no possible situation where disjunction introduction is invalid. — MindForged
Speaking of "situations"/states of affairs in this way is a mistake. They are not the same things as propositions. — MindForged
You already said that you believe logic is fundamental to reality in your first post, on the first page. So this precludes you from dealing with possible worlds where different inference rules hold. — MindForged
A proposition is distinct from a state of affairs, so you can have a proposition (which is an object) that is a contradiction, yet that doesn't entail there is some state of affairs (or a possible state of affairs) which corresponds to the contradictory proposition. — MindForged
A situation (or state of affairs) is some way the world is that makes a given proposition true. It is not the arrangement of an object because objects are part of a state of affairs. — MindForged
I've already dealt with this. Propositions aren't strings of words (that's a sentence) and yet they can have a referent in reality, and a truth value. — MindForged
Contradictory propositions (under most views) are precisly those propositions which cannot correspond to a possible state of affairs. They are not ontologically nothing, even on your view, because you said that have a property. If you say a contradictory proposition has no properties, that means they don't have the property of falsity. Which is just ridiculous because propositions are necessarily false, which is a property. — MindForged
Depends on what you mean by "physical". Is "imaginary momentum" that a particle has in quantum tunneling "physical"? It cannot be measured, even in principle, but physical theory implies it is there. — litewave
The issue I'm thinking about with the HOM experiment is this. Suppose we name the two photons that are measured at the end of the experiment P1 and P2. Can P1 be identified with the photon that was originally above (or, else, below) the beam splitter? — Andrew M
Well, since the two photons have no measurable differentiating property at the end of the experiment, not even a different position in physical space, we cannot find out which one was above the beam splitter at the beginning of the experiment and which one was below the beam splitter. — litewave
We might at least measure (if it is technologically feasible) whether their frequency didn't temporarily change during the experiment, to rule out that they temporarily merged into one photon (which would manifest as temporary doubling of frequency, since total energy should be conserved). If they merged into one photon and then separated again it seems that their identities were terminated at the merger and new photons came into existence at the subsequent separation. — litewave
If there was no merger into one photon then the identity of the particles was preserved but since they have no measurable differentiating property at the end of the experiment we can no longer say which one is which. — litewave
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