Why can't leprechauns be an abstract object? It may not be a mathematical object, but why not an abstract one? Being a set would be a definition of a mathematical object perhaps, but not all abstract objects are subsumed in that. Economics for example is an abstract object. creativity is an abstract object, etc. How does his definition differentiate between any of these? — schopenhauer1

They can be, I didn't mean to exclude this possibility; I was only explaining Frege's position on this. But yes, we could allow that fictional entities are real objects of some sort and that the number 3 would not be the same thing as a Leprechaun, because a number is a set, and a leprechaun is a fictional object but genuinely existing object (whatever fictional might mean). It would be an abstract object of some sort and not a physical object.

This would mean that names for fictional characters have a genuine referent. This is definitely a view that is held by many philosophers. You may be interested in the SEP article on fictional entities.

What is the object referencing? Presumably reference is a "real" thing, but how does he explain this without being self-referential? If he says it is somewhere in the world, then where is this "three"? But if he says it is in the realm of the imagination, then he once again has no way of differentiating it from the leprechaun. — schopenhauer1

An abstract object. In fact, for Frege it references as extension (of a second-level concept). To be a bit more modern: it would pick out a particular set. Leprechauns are not sets. Sets have properties leprauchauns don't have (and vice versa).

Frege's way of differentiating abstract objects is via definition. For him, a definition must settle all mixed identity claims. So, the definition of the number three would settle whether or not:

leprechaun = 3

is true or false (and it would come out false).

Let's be a bit more modern again, and not use Frege's explicit definition but a semi-analagous one:

$3 :=_{Df} \{\varnothing, \{\varnothing\}, \{\varnothing, \{\varnothing\}\}\}$

It follows from this, for example, that $\varnothing\in 3$ or that the empty set is an element (or part of) the number 3. But this is not true for leprechauns, i.e., empty sets aren't parts of leprechauns. So they can't be the same thing.

Hey Wayfarer, I do not have the time for a considered reply to this, but I can reply to the issue about imaginary or fictional objects, so I will do that first and will get back to this.

I'm not sure. Kornelius what would be the difference between numbers and leprechauns in Frege's conception of objects? I realize that question is funny as I write it :) — schopenhauer1

This is hitting on a point that is actually somewhat contentious and deeply philosophically interesting.

For Frege, the term "leprechaun" is an empty name (or, rather, an empty noun). It does not refer to an object.

The term "three", on the other hand, refers to an object.

But here is the issue. If I say, for example:

(1) "Sherlock Homes is a great detective"

This sentence does have a meaning. But, for Frege (as for many), the name "Sherlock Holmes" is an empty name. But, at the same time, "Pegasus" is also an empty name, and while the sentence:

(2) "Pegasus is a great detective"

Clearly has a different meaning than "Sherlock Holmes is a great detective", the referential semantic components of the two sentences (1) and (2) are the same, given that the names in each do not denote an object, and the rest of the sentence is identical! I say they have different meanings because many would believe (1) to be true, but (2) false, and it wouldn't be because they are mistaken about Sherlock Holmes or Pegasus (or what it means to be a detective).

So how do we make sense of this?

Frege has a differentiated notion of semantic content. That is, terms, sentences, etc., are have both a reference (as part of their semantic content) and a sense (as part of their semantic content). So while the names "Sherlock Holmes" and "Pegasus" don't differ with respect to their reference (they both do not refer to an object), they do differ with respect to their sense.

Now there is a lot of literature on Frege's views on this, and whether he had successfully clarified his notion of sense to properly account for empty names.

But, in short, that is the response. "Leprechaun" behaves semantically differently (so to speak) than number terms since the latter have reference and the former does not.

This is not the only possible view, however. While Frege would not have agreed, many philosophers have since argued for the existence of fictional characters, and thus for the existence of objects that would be the referents of the term "leprechaun". Mostly because it solves a lot of the technical philosophy of language/logic issues (but at the expense of not being consistent with our common sense views about these terms).

Hope this helped!

Recommended reading:

The Logic Book (good for practice exercises)

A Mathematical Introduction to Logic

Set Theory, Logic and their Limitations

Getting Clear on how to Formulate the Law of Identity, and why we need to be logically precise

In logic, it is important to note that arbitrary reference is equivalent to referring to all objects in a domain of discourse.

This is why it is crucial to be precise when employing the indefinite article. When one uses an indefinite article like "a", one must precisely articular whether one is using one of two quantificational statements: $\exists$ (for some, there exists a, etc.) or $\forall$ (for all, every, etc.)

The reason this is important is that some may want to argue that the law of identity does not employ the unversal quantification. If I say "a thing is identical to itself", do I mean to say:

1) $(\exists x) (x=x)$

OR

2) $(\forall x)(x=x)$

MU argues that the identity principle is just the statement (1), and he further states that someone who says it is (2) is confused:

Do you not recognize the difference between "a thing", and "all things"? — Metaphysician Undercover

It is very important to note that everyone agrees that (2) is the formulation for the law of identity. It is easy to show why. So here we go:

Proposition: (1) is not logically valid, where (1) refers to the proposition $(\exists x) (x=x)$

PROOF: By definition, a proposition is logically valid if and only if for any model $\mathcal{M}$, (1) is satisfied (is made true) in that model, i.e., $\vDash_{M} (\exists x) (x=x)$.

Consider a model $\mathcal{M}$ with domain $\varnothing$. Since the domain is empty, we have that $\vDash_{M} \neg(\exists x) (x=x)$ from which it follows that $\nvDash_{M} (\exists x) (x=x)$. Thus, (1) is not valid in $\mathcal{M}$, and thus not a logical truth. $\square$

INFORMAL ARGUMENT: Since the proof may use some technical devices we may not be familiar with, here is the essential idea. In order to show that (1) is not a logical truth, we just have to show that we can imagine a possible situation in which (1) is false. Indeed, in a universe where no objects exist, (1) is false. Therefore, (1) cannot be a logical truth.

What is interesting to notice as well, is that (2) does not imply (1). If this were the case, then any model in which (2) is true, (1) is also true. However, in the model I just showed you (see proof above) (2) is true (see one of my previous posts in this discussion if you want the technical proof) but (1) is false. Therefore, (2) does not imply (1).

The reason (2) does not imply (1) is because (1) implies the existence of at least one objects. (2) does not imply the existence of an object. Here are some logical equivalences that might help you see this:

$\begin{align*}& (\exists x)(x=x) \Leftrightarrow \neg(\forall x)\neg(x=x)\\& (\forall x)(x=x) \Leftrightarrow \neg(\exists x)\neg(x=x)\end{align*}$

The Law of identity is held as a law that is logically true. Indeed, (2) is logically true, i.e., it is true in all models.

The proper way to state the law of identity is:

All objects are self-identical (whether many, one or no objects exist)

Or, alternatively, we can say:

There exists no object that is not identical to itself (whether many, one or no objects exist).

Both of these statements are logically equivalent and, most importantly, do not imply that an object exists.

Right! Hence my remark in the other thread that numbers (etc) are constitutive of thought. — Wayfarer

I think that in effect describing 'concepts' as ‘objects’ is a reification. — Wayfarer

Typically, we say that rules or, rather, norms are constitutive. If we think that mathematics is logic (logicist position), then we still have the problem of explaining logical objects.

Logic can be constitutive of thought while logical objects exist. That's no problem at all (rather, it presents no more issues that what realists about abstract objects face in any case).

These judgements are likewise constitutive of reason and rational inference, and they are being made whenever we assert or describe or argue anything whatever. They are the 'fabric of reason', so to speak. (For further elaboration on Freger's view of the 'laws of thought' in particular, see Frege on knowing the Third Realm, Tyler Burge.) — Wayfarer

Careful quoting Frege's view on this. While Frege likely held the constitutive thesis, he definitely was a realist about mathematical objects :) He would be a great example of holding both that logic's normativity is constitutive of thought, as well as the view that mathematical (logical) objects exist (and not in a metaphorical sense). Also, thanks for the reference. I am familiar with Burge's paper :)

I think that in effect describing 'concepts' as ‘objects’ is a reification — Wayfarer

I never did that, and the realist position does not endorse this. In fact, Frege wrote a whole paper on distinguishing concepts from objects.

Numbers, for Frege, are not concepts, however. They are objects. If you think that numbers are concepts, then you need to give an account of mathematics in which number terms occurs as predicate terms in the logical language, or as quantificational statements (whether they be first-order or second-order concepts).



The point is that realists think this is not possible, and so numbers have to be full-fledged objects.

What I mean to say here is that the position does not use the term 'object' in a loose, metaphorical sense, or in a sense that "reifies concepts'. It doesn't treat numbers as concepts. We know this because concepts have formal analogues in a precise logical language. Indeed, Frege's insistence that concepts and objects are not the same is reflected in the very syntax of first-order logic (and Frege's logic has a complete first-order fragment).

I accept the usage of the term ‘object’ as a linguistic convention, but I think this usage leads to a basic misunderstanding of the nature of what is being discussed. And the reason for that, is that modern thinking is overwhelmingly oriented towards the 'domain of objects' - the domain presumed fundamental and exclusively real by natural science . — Wayfarer

So I think your position is this: we make syntactic distinctions in our formal languages differentiating concepts (predicate expressions) from objects (terms). But the syntactic distinctions do not map onto any metaphysical distinctions between concepts and objects at all.

Ok, but the issue I see is this: empirical objects occur as terms in our formal languages. They are undoubtedly objects. Properties occur as predicate expressions in our formal languages, and properties are undoubtedly concepts (or the referents of concepts, though I take it you mean to use properties and concepts somewhat interchangeable. In any case that doesn't affect the discussion). But then why should terms that refer to abstract objects be taken to be "reification" of what are in fact concepts? Why not take it as evidence that we may have been doing the reverse, i.e., referring to abstract objects as mere concepts, when in fact they were not?

There's literally no conceptual space for it in modern naturalism, as what is real is regarded as existent, 'out there somewhere', as the saying has it (see the remark on 'animal extroversion' in the quotation below.) — Wayfarer

Correct: metaphysical naturalism is incompatible with mathematical/logical realism (in the ontological sense). Still, mathematics remains the most significant challenge to naturalism and one which the naturalists have yet to solve.

So that is the drift. It is not exactly what I set out to say when I sat down to write, but I hope it conveys something of what I'm getting at. — Wayfarer

It did, and I hope I have a better sense of your view so that my objections to it may seem more convincing! (or can be more easily dismantled :P)

That being said: I think we should dig out Frege's paper on Concepts and Objects.

The law of identity states that a thing is the same as itself. — Metaphysician Undercover

All things are identical to themselves. Which is exactly the formulation I discussed and exactly the principle that implies nothing with respect to the number of existant objects.

That is our point of disagreement. My claim is that the law of identity is not a law of logic, it's a metaphysical assumption. You think it's a law of logic. Because of this disagreement, I do no think we will ever find an expression of the law of identity which we both agree with. — Metaphysician Undercover

The law of identity is a law of logic. What was stated: an object is identical to itself, IS a law of logic. Period. This is why I think you may not be expressing what you really want to express clearly. I think you may have a different identity claim in mind that may, in fact, be a metaphysical claim. The one you expressed, however, is not.

My question to you is how do you proceed from the proposition "each thing...", to your formulation "for all x...."? Notice that the former refers to particular, individual things, and the latter refers to a group of things. — Metaphysician Undercover

They are equivalent. "For all x" and "each x" is logically equivalent. "Each" is a universal quantifier expression. I could easily have said that $(\forall x)(x=x)$ says: each x is self-identical OR
for each x, x is identical to x.

You must apply inductive logic to "each thing is identical to itself, to derive "all things are identical to themselves". — Metaphysician Undercover

Not at all. You are misconstruing the statement "each thing" for "each thing I observe now". That is NOT the law of identity. That is, "each thing observed up to this point has been identical to itself" is NOT at all what "each object is identical to itself" states.

For this reason, no induction is required AT ALL to get the law of identity. As stated, the law of identity is an axiom of logic. It is a logical principle through and through because, as I showed, it is true in every model.

That being said, if you want to discuss the empirical claim regarding the objects we've observed, then by all means. However, we do not come to the conclusion that an object is identical to itself via observation anyway.

Otherwise, I am not sure what you are getting at. I think, perhaps, you are trying to say that claims about persistent identity are metaphysical. Here I am in complete agreement with you. Identity over time is metaphysics.

They believe that the Neural Activity is sufficient for us to move around in the world without bumping into things. This is insane denial of the obvious purpose for Visual Consciousness. The Conscious Visual experience is the thing that allows us to move around in the world. Neural Activity is not enough. We would be blind without the Conscious Visual experience. The Conscious Visual experience contains vast amounts of information about the external world all packed up into a single thing. — SteveKlinko

If I did not have the Conscious Visual experience I would not be able to pick up my coffee mug, or at least it would be much more difficult with just Neural Activity. — SteveKlinko

I guess I pressed for more explanation on these claims. I am not sure that they simply don't amount to the mere assertion that there is a difference between a conscious being, and one with "mere neural activity".

So: why would it be more difficult for an unconscious being (neural facts being equal otherwise) to pick up a cup? The response system you suggest is due to consciousness is actually due to our neural, optic, etc., system. We could get the same response output, without the subjective "inner movie" so to speak.

I want to be clear: I take the conscious experience at face-value and I think an explanation is needed. I certainly disagree with more radical naturalists who explain it away as an "illusion". That being said, the conscious experience might just be complex information processing (owed to complex neural systems/activity).

You can’t ask why the law of identity holds, or why elementary arithmetic proofs are valid. They are the basis on which judgements of validity are made. — Wayfarer

This is the view that logic is constitutively normative for thought. That is, the norms themselves make thinking possible, just as the rules of a game are constitutive for that game. They don't "regulate" how the game is played, they are part of what it is for the game to be the game that it is. Without the normative force of the logical laws, thinking is not possible. From the SEP:

Other philosophers have taken the normativity of logic to kick in at an even more fundamental level. According to them, the normative force of logic does not merely constrain reasoning, it applies to all thinking. The thesis deserves our attention both because of its historical interest—it has been attributed in various ways to Kant, Frege and Carnap[6]—and because of its connections to contemporary views in epistemology and the philosophy of mind (see Cherniak 1986: §2.5; Goldman 1986: Ch. 13; Milne 2009; as well as the references below).

To get a better handle on the thesis in question, let us agree to understand “thought” broadly as conceptual activity.[7] Judging, believing, inferring, for example, are all instances of thinking in this sense. It may seem puzzling at first how logic is to get a normative grip on thinking: Why merely by engaging in conceptual activity should one automatically be answerable to the strictures of logic?[8] After all, at least on the picture of thought we are currently considering, any disconnected stream-of-consciousness of imaginings qualifies as thinking. One answer is that logic is thought to put forth norms that are constitutive for thinking. That is, in order for a mental episode to count as an episode of thinking at all, it must, in a sense to be made precise, be “assessable in light of the laws of logic” (MacFarlane 2002: 37). Underlying this thesis is a distinction between two types of rules or norms: constitutive ones and regulative ones.

The distinction between regulative and constitutive norms is Kantian at root (KRV A179/B222). Here, however, I refer primarily to a related distinction due to John Searle. According to Searle, regulative norms “regulate antecedently or independently existing forms of behavior”, such as rules of etiquette or traffic laws. Constitutive norms, by contrast

"create or define new forms of behavior. The rules of football or chess, for example, do not merely regulate playing football or chess but as it were they create the very possibility of playing such games". (Searle 1969: 33–34; see also Searle 2010: 97)

From a formal point of view, it's a logical equivalence between orthogonality of two vectors and an equation between real numbers (vectors' lengths).

Should an equivalence of this kind be surprising? (or improbable)? The answer is NO: in fact, whatever angle you take between two vectors, there will be an induced equation between vectors' lengths, and vice-versa. — Mephist

Hi Mephist,

I am very curious about this. Doesn't the vector's length follow from the Pythagorean theorem? If so, then it seems it isn't a simple logical equivalence, since the equation between real numbers (a vector's length) depends on the application of the theorem. Or is there an independent way of measuring a vector's length?


As to the OP: I sincerely believe that the importance of mathematical theorems really is a function of the aesthetic tastes of mathematicians which is partly based on certain objective factors: like whether the theorem unifies seemingly disparate areas of mathematics, whether the theorem has far-reaching implications within mathematics, etc. , as well as subjective factors: like how surprising it is, how difficult the theorem may have been to prove, how necessary it felt to the mathematician (i.e. the extent to which the mathematician feels like they discovered something), etc.

And I'm saying that this usage amounts to a dead metaphor, that it's a consequence of the absorption of an empiricist or naturalistic point of view which then has un-acknowledged semantic and even metaphysical consequences. — Wayfarer

Why empiricist or naturalist? They would at least want to deny that there are abstract objects. One route to do this is to say that mathematical language that uses referential terms is a mere convenience or misleading, that the genuine semantics for such a language doesn't involve reference to numbers at all. They would, of course, have to construct such a language, or argue that terms can refer without taking seriously the objects to which they are referring.

I tend to see the view that objects are the reference of terms as a straightforward Platonist one (at least when it comes to abstract objects and properties/universals).

This is the problem then. That is not the law of identity. The law of identity does not allow that there is more than one X. When you say "for all X...", you have already allowed the possibility of more than one X, thus breaking the law. — Metaphysician Undercover

?

I am not sure where you are getting this and why you think it is true. Could you clarify? In no suitable formulation of the law of identity would it be valid only in a model with exactly one and only one object. How would you even formulate this? I take it something like this:

$(\exists x) (x=x) \wedge \neg(\exists y)\neg(x=y)$

But this is no law of logic, and certainly not a law of identity. It is fairly simple to provide a model for which the statement is false. Therefore, it is not a law of logic. Logical laws are true in every model, not just some models.

What is not clear is how you get from the law of identity, as commonly stated, to your formulation of it. And I'm sorry to be the one to inform you of this, but your example fails because it utilizes a formulation of the law of identity which is already itself in violation of the conventional law of identity. — Metaphysician Undercover

I am employing the very familiar and standard notion from logic.

In logic, the law of identity states that each thing is identical with itself. It is the first of the three laws of thought, along with the law of noncontradiction, and the law of excluded middle. However, no system of logic is built on just these laws, and none of these laws provide inference rules, such as modus ponens or DeMorgan's Laws.

In its formal representation, the law of identity is written "a = a" or "For all x: x = x", where a or x refer to a term rather than a proposition, and thus the law of identity is not used in propositional logic. It is that which is expressed by the equals sign "=", the notion of identity or equality. It can also be written less formally as A is A. One statement of such a principle is "Rose is a rose is a rose is a rose."

In logical discourse, violations of the law of identity result in the informal logical fallacy known as equivocation.[1] That is to say, we cannot use the same term in the same discourse while having it signify different senses or meanings and introducing ambiguity into the discourse – even though the different meanings are conventionally prescribed to that term. The law of identity also allows for substitution, and is a tautology.

-Wiki

If your position requires you to reject a basic principle of formal logic, I would reconsider it carefully, especially since you admitted that you are not familiar with logic. We have an obligation not to be confident in our pronouncements if we are not entirely sure about all that goes into asserting them. Moreover, we should be open to reconsidering our position. So far in our discussion, everything is pointing to the conclusion that you (and not I) are confused about the law of identity. The formulation I have given for the law of identity is not mine, it is the one learned by everyone in the first course on logic in philosophy or mathematics.

I believe I have provided relatively clear explanations for what are elementary concepts in logic (i.e. the law of identity). If anything is unclear, please let me know.

Consciousness is definitely helpful for survival purposes, though, especially when you get to organisms like us, who are relatively complex and who aren't adapted to easily survive to reproduction age without a lot of assistance and without the benefits of being able to learn things (such as things in our environment that are dangerous). — Terrapin Station

This is an extremely interesting claim, and if we could make it precise, it would be very helpful in the debate on Consciousness. I am not well versed in these issues in philosophy of mind and cognitive science generally, but it seems to be a contentious issue whether or not consciousness would be something on which natural selection could operate.

It seems to me that I could picture the entire history of human/ape evolution, without the corresponding emergence of consciousness. Why would consciousness be of assistance to our survival? What type of actions and or responses would a conscious being be able to perform that an unconscious being would not be able to (or would not be able to with the same success)?

This is a genuine question. I have no idea at all.

IF there is a necessary being. The whole point is that IF there is a necessary being then the attributes of that being must also be necessary. — Janus

This pretty much sums up the appropriate response to the OP as well. The entire claim is a conditional claim of this sort, which is why it is not controversial.

If God exists, then he necessarily exists. If Jesus is God, then Jesus is necessarily God. If Jesus is not God, then Jesus is necessarily not God.

That's it really.

The definition of 'object' is: — Wayfarer

That is not a suitable definition for 'object' in philosophy at all. An object is whatever can be the semantic reference of a term. This includes abstract objects like sets, for example. Indeed, anything that could occur flanked to an identity sign in mathematics is an object, since only an object can be assigned to a term that flanks an identity sign.

I'm saying that logical and arithmetical truths are not reliant on objective validation, that they're true a priori - something which still has a connection to the thread, even if tenuous! — Wayfarer

But this is my point: you are misusing "objective validation". Arithmetical propositions are necessarily true and deductively certain. They are objectively validated since they are true in every possible world. That is, one cannot reject the proposition and still claim to be thinking rationally.

Or we could put it this way: mathematical propositions are deductively proven, therefore objectively validated.

He's just saying that if you use the variable to refer to something, then that thing exists as something, whether it's just an idea or description or whatever it is. — Terrapin Station

But the variable need not refer in certain models, since certain models may be empty. But it is true nonetheless since it is a quantified statement. Consider my previous post. But to be clear, a variable does not refer to a particular object. It is open for a semantic assignment (i.e. open to be assigned an object, though not in quantified statements, since the variable does not occur free). It has been a while for me, but I think all this is correct...

Now as it concerns "referring to ideas, descriptions, etc.", I am not too sure I follow and I am not sure how this would impact the discussion and the law of identity (which applies to objects, not descriptions, etc.).

I'm not familiar with your use of symbols, but there is an object assumed, or else there is nothing identified. The object need not be a physical object, are you familiar with mathematical objects? If your statement identifies a mathematical object, then this is an ontological statement, it gives reality to that mathematical object, as an identified object. Perhaps your symbol is the object itself, I don't know what your symbol symbolizes. And a model with no objects makes no sense to me, because the model is itself an object. — Metaphysician Undercover

Hi MU,

I apologize: I should not have assumed you were familiar with this; that is completely on me. I am employing standard first-order logic notation. The statement $(\forall x)(x=x)$ says "for all x, x is identical to x."

Yes, I do know about abstract objects but, no, the statement would be true in a model where there are no objects at all, whether abstract, physical, etc.

The symbol is also not an object.

You cannot claim that a specified object is identical to itself, and also say that there is no such object, without launching yourself into nonsense. — Metaphysician Undercover

This is incorrect. I will show this in my reply to this:

You can say that, but your claim is wrong. Try to demonstrate it, why don't you? Show me a model with no objects which validates the law of identity. — Metaphysician Undercover

This is easy to show, and it is something that would be taught in an introductory course in formal logic in every philosophy and mathematics department. I will try my best to elucidate the concepts as best I can since you mentioned that you are not familiar with first-order logic and model theory. I strongly recommend studying these topics; it is an absolute must for philosophy!

Let $\mathcal{L}$ be the standard first-order language in which $(\forall x)(x=x)$ is expressed.

(*This is just to say that we are talking about a sentence in first-order predicate logic, using the usual syntax of a first-order language, and only allowing quantification over objects (and not over predicates). None of this is actually important)

Let $\mathcal{M}$ be a structure whose domain is $\varnothing$ as well as the usual interpretation for the symbols in $\mathcal{L}$. There is no need to specify the interpretations for our purposes.

(*A structure in logic is a set equipped with functions that assign a semantic value (or interpretation) to the non-logical symbols in the language. So, for example, the symbol "=" in our language will get assigned the usual interpretation (equality), etc. This is the only assignment that is relevant here in any case, since the symbol $\forall$ is a logical constant, so does not get re-interpreted)

Now, a sentence $P$ is valid in $\mathcal{M}$ if (and only if) the structure/model $\mathcal{M}$ entails $P$. We write: $\vDash_{\mathcal{M}} P$. This would mean that $P$ is (logically) valid in $\mathcal{M}$.

Indeed, if for any model $\mathcal{M}^*, \vDash_{M^*} P$, then $P$ is a LOGICAL TRUTH. This is just to say that it is true in every model.

Now it follows, vacuously, that $\vDash_{M} (\forall x)(x=x)$ since there are no objects in $\varnothing$. If you do not see that it is vacuously satisfied, consider this:

$(\forall x)(x=x)$ is logically equivalent to $\neg (\exists x)\neg(x=x)$. That is, it is logically equivalent to the proposition that: it is not the case that there exists an object that is not identical to itself. It is obvious, then, that $\vDash_{\mathcal{M}} \neg (\exists x)\neg(x=x)$. Thus, $\vDash_{M} (\forall x)(x=x)$.

In short, the universe of discourse in the structure $\mathcal{M}$ we just considered is empty, i.e., there exist no objects. And this structure satisfies the law of identity.

Indeed, the law of identity is true in EVERY model $\mathcal{M}^*$ with any domain of objects (empty or not).

I understand that this might seem overly technical if you haven't been exposed to logic, but I hope I made it as accessible as I could in such a post. Please let me know if there is any step that isn't clear!

Cheers.

No, I mean Aristotle's law of identity, which is an ontological principle. It states that a thing is the same as itself. It is ontological because it assumes the existence of the thing. Without the existence of the thing the principle makes no sense. So if any logicians make use of this principle, they are making use of an ontological principle. — Metaphysician Undercover

But we know now, because of mathematical advances in logic, that this principle does not assume the existence of anything. The statement $(\forall x) (x=x)$ is made true by any model that assumes no objects: it would be vacuously satisfied, and therefore true.

This makes sense, in any case, since it is a logically true proposition, i.e., it is true in every model, including all models in which no objects exist.

It is simply incorrect to say that the statement that every object is identical with itself implies (or presumes) that an object exists. It does not.

It may be the case that logicians make use of the principle, but to classify the principle itself, we need to see what validates it, and that is an ontological assumption about the existence of a thing. — Metaphysician Undercover

I am sorry to be blunt, but this is simply incorrect. As I said: every model validates it, no matter whether no objects, some objects or infinitely many (countable or uncountable) objects exist.

This way of thinking about necessary being really has little to do with the Scholastic or Spinozistic conceptions of necessary being. For one thing, for Spinoza, a necessary being must be infinite, because it must be independent of all contingent being. This means that it can be limited by nothing and nothing is "outside" it. Everything finite must ultimately be dependent upon it for its existence. It also follows form this logical that there cannot be more than one necessary being.

The idea that a necessary being is a being which must exist in all worlds is really not the same. It is rather the opposite from the Scholastic perspective; a necessary being is a being which all worlds must exist — Janus

My apologies if this other conception of necessity was being deployed. I know very little of it. I was using the contemporary idea of necessity (as it is understood in current metaphysics and logic).

Was this a historical discussion?

I think the law of identity is itself a metaphysical claim. So it's not a matter of me importing metaphysical claims into the law of identity, it already is a metaphysical claim. — Metaphysician Undercover

The law of identity is a law of logic; it is not an ontological principle. Perhaps you mean Leibniz's law of indiscernibles?

There is a notable and important difference between them.

The law of identity

$(\forall x)(x=x)$

The identity of indiscernibles:

$(\forall x)(\forall y)[(\forall F)(Fx\leftrightarrow Fy)\rightarrow x=y]$

The law of identity is most certainly a principle of logic, not of metaphysics.

The issue I see with calling these objective truth is, I am sure this is true to you, and I am sure you think this is true in general, but what if I don't know what these symbols mean? — leo

This has no bearing on whether or not a proposition is objective or whether or not it is true (or false). A proposition can be objective even if a person fails to understand it.

What if these arrows, chevrons and parentheses do not evoke anything in me beyond shapes drawn on a screen? Then these statements wouldn't be true to me, they would be drawings, and while I could say it is true to me that I see these drawings, I couldn't say these drawings refer to some independent truth. — leo

It doesn't matter whether you recognize the truth of said propositions. Again, a proposition can be true (or false) regardless of whether any particular person recognizes its truth.

and I too could create my own system in which I assign truth to such or such proposition, but that doesn't mean that the truth of these propositions would extend beyond the system they were formulated in. — leo

The point of the proposition I highlighted is that it is true on any possible truth-value assignments for its subterms $P$ or $Q$.

Perhaps you meant you could devise a logical system in which the logical constants behave differently, i.e., you could define different truth-functions. But no matter how you define them, there would be a corresponding truth-function (or complex of truth-functions) in the propositional logic that would behave exactly the same way.

Because the way I see it, such a system was created out of perceptions and thoughts, and it doesn't apply to people who have perceptions/thoughts incompatible with it — leo

Again, people may simply be mistaken. This happens all the time.

It seems inevitable to me that truth is personal, that we can't find a truth that applies to everyone — leo

What might be inevitable is that not all people will agree, but it doesn't follow from this that there aren't objective truths. Again, something can be objectively true despite the fact that many people disagree about the truth of the proposition in question.

What if some great catastrophe occurred in Africa very recently and I am not yet aware of it and it turns out all lions are dead? — leo

Then the proposition would, in fact, be false (but still objective). It would be objectively false. It would not matter what you knew or did not know at the time of asserting the given propositions.

Propositions are made true or false by facts/states of affairs, which are independent of the epistemological state of any person asserting a proposition.

Or what if I consider that it is meaningless to talk about what goes on in a place "at this moment" if I am not in that place? — leo

This would be a mistake. I can most definitely construct grammatically well-formed and truth-evaluable sentences about places I am not currently in.

People could very well disagree with that proposition in a reasonable way according to them — leo

The mistake you are making is in supposing that what is reasonable is internal to an individual. It is not. Norms of logic are constitutive of rationality and, thus, what is reasonable. One cannot deny these norms and claim to be thinking.

Consider a number - say 7. In what sense is that 'an object'? 'Well, there it is', you might say, pointing to it - but what you're pointing at is a symbol. Furthermore that symbol could be encoded in any number of media, written in a variety of scripts, - 'seven', VII, 00000111, and so on. But the referent, what the symbol '7' signifies, is always the same. And that's what I'm saying is not 'an object'; it's more like a constituent, than an object, of thought. — Wayfarer

Thanks Wayfarer for your response!

Indeed, I would say that the symbol "7" refers to the object 7. I do not use the term "object" metaphorically at all. The semantics of arithmetical language certainly attributes to semantic value (an object) to these terms. How else would they figure into identity claims?

If they are not objects, we must explain arithmetical propositions in some other way. For example, $2+2=4$ is true, but what makes it true? What are the referents of each expression? The standard replay is that the semantic value of "+", "=" are relations/functions, and "2", "4" refer to objects. However, we could give a different semantics: one which takes more seriously the adjectival use of number expressions. Consider the sentence: "There are four chairs in the room" v.s. "The number of chairs in the room is four". In the first, "four" is an adjective, and maybe we treat "four" as a second-level concept (this has been tried, but the view has problems). The second sentence is a straight-forward objective use, so the term "four" is an object.

If we want to say that our number talk, whether in everyday language or in arithmetical language, is misleading, and that the true nature of our numerical expressions is that they refer to second level concepts, or perhaps to constituents of thought as you suggested, then we need to realize how difficult such views actually are to sustain.

Take your position: a numerical expression actually refers to a constituent of thought. This is problematic, since it cannot mean the constituent of my thought or your thought. This would violate the universality of mathematical propositions. Then you might mean it is the constituent of some Thought, with a capital T, but what would that mean exactly? I am not sure this view doesn't also have pretty thought metaphysical and epistemological assumptions we're being asked to swallow.

As regards objectivity - I'm inclined to say that arithmetical proofs, and so on, are also likewise 'objectively true' only by way of metaphor. The point about an arithmetical proof is that it is logically compelling - again, the means by which we determine its veracity are purely internal to the nature of thought, they're not 'objective' in the strict sense of 'pertaining to an object or collection of objects'. In fact we often appeal to mathematics to determine what is objectively true; there's a sense in which mathematical reasoning is "prior" to empirical validation, in that the mathematics provide a reference to determine what is objectively happening. — Wayfarer

Even if we determine the veracity of logical and mathematical propositions by reason alone, this doesn't make these propositions any less objectively true. I think you are trying to redefine objectivity to suit a particular position, i.e., it must "pertain to objects" and you are refusing to admit that mathematical propositions involve objects. Even if they don't involve objects, there is no way mathematical propositions are not objective. They are paramount objectively true propositions.

Here is a view, let me know what you think. Objective propositions are propositions that are truth-evaluable. They do not have to refer to objects at all. This is has to be a mistake. If I say "the use of the copula in English is to ...." or "the word "English" starts with the letter "E"", then I I am using objective propositions. They can be determinately true or determinately false, but they do not seem to "involve objects" in the way that you are describing. The same would be true of many propositions that do not involve objects.

That's the thing, is it true that "The sky is orange" is false? What if I'm watching a sunset and I see the sky orange? What if someone perceives differently and see the sky orange when others see it blue? What if someone doesn't perceive a sky (in which case the sentence wouldn't be truth-evaluable for that person)? How could we say that "The sky is orange" is false for everyone? How could we find anything that is necessarily false (or true) for everyone?

My point is we can't find anything that is true for everyone. And that even the sentence "we can't find anything that is true for everyone" wouldn't be true for everyone. And so on in an infinite regress — leo

Hi Leo,

Thanks for the reply, but I have to admit that as it stands this does not seem to be a defensible position. I used that sentence as an example for what is fairly obvious. We could make our sentences as precise as we need to in order to avoid any issues with reasonable disagreement. There will always be disagreement, but it would likely be unreasonable.

I can enumerate for you an infinite number of objectively true propositions. I will start with one:

$(P \wedge (P\rightarrow Q)) \rightarrow Q$

This is objectively true. It is objective because I can easily create a truth-table to show its relevant truth-conditions for any truth-value assignments for $P$ and $Q$. It is true because this sentence is true on any possible truth-value assignments. That is, it is a logical truth.

Now let $M_1, M_2, M_3,\dots$ be an enumeration of infinitely many propositions with different contents. It is easy to see that:

$((P \wedge (P\rightarrow Q)) \rightarrow Q) \vee M_1$

Is a logical truth and, so, both objective and true. Further,

$(((P \wedge (P\rightarrow Q)) \rightarrow Q) \vee M_1)\vee M_2$

is also objective and true for the same reasons. By $\vee$ introduction, it is easy to see that I can generate infinitely many of these. In fact, I didn't even a list of infinitely many proposition to do that. I could have started with a logical truth, and simply generated infinitely many propositions by applying $\vee$ introduction infinitely many times with the proposition $P$.

I can equally generate many empirical sentences that are both objective and true. For example, "More than two lions exist in Africa at this moment", where we can qualify "this moment" with the precise time of my writing. There are so many propositions of this sort.

Radical skepticism is fine, but I am not bothered if this is the only way you find to deny that propositions can be objective and true.

I also think that radical skepticism tends to work only with conditions we need not accept (on what constitutes knowledge), but I am out of time so I will leave this point to another reply.

I have to admit I was not able to read through all the posts carefully here, but I believe there is something we all seem to be conflating: truth and objectivity.

Truth and objectivity are distinct properties. For example, a proposition can be both objective and true, but it can also be both objective and false.

Sentences express propositions. Truth is a property of sentences. There are various theories about truth, but a popular one is correspondance. A sentence is true in so far as the proposition corresponds with what is the case. For example:

$\lt\lt\mathrm{Snow\ is\ white}\gt\gt is\ true \Leftrightarrow Snow\ is\ white$

This says that the sentence "snow is white" is true if and only if snow is white.Objectivity, on the other hand, says only that a sentence has a determinate truth-value. This is independent of whether it is true or false and also independent of whether or not we know it is true or false.

The sentence "The sky is orange" is objective. It is objective because it is truth-evaluable. It happens to be false, but so be it.

The concept of objective truth seems incoherent to me. If we say objective truth is something everyone agrees on, it seems that there is nothing everyone agrees on, and not everyone agrees that "there is nothing everyone agrees on", and so on and so forth. — leo

I believe the explanation I gave above settles this issue. Objectivity and truth are two different things. Moreover, a sentence can be objective if it is truth-evaluable and even if no one agrees on whether it is true or false.

A sentence is not objective if it is not truth-evaluable, i.e., it does not have determinate truth-conditions (or is not true or false). An easy example of a sentence that is not objective is the following: "I don't like art". This is not a truth-evaluable sentence, in so far as it is merely expressing personal taste. "Kornelius doesn't like art" is, however, an objective sentence. Notice that this is the case whether or not I in fact like art.

A more contentious example would be ethical claims. Consider a rather morbid example:

"Raping and cannibalizing a person is morally impermissible".

I am an objectivist here. I think this sentence is true or false (indeed I think it is true).

To say that such a statement is not objective is to say that it isn't truth-evaluable: that it is making no claim about the world. This seems extremely hard to swallow.

Thus, denying objectivity is actually an extremely radical position. You would need to explain a whole lot of things. Why would such a sentence have no determinate truth-conditions, for example? It seems to express an obvious truth-evaluable thought.

If we go into more straightforward examples, i.e., "The sky is orange", a radical subjectivist would need to explain why this sentence is not truth-evaluable. It seems obviously truth-evaluable to me. It expresses a thought that is either true or false.

One obvious question is whether there are any actual objects to which this this applies. Take your example of the Euclidean triangle - it can be demonstrated by a physical drawing, which is an object, but the principle itself can't be said to be 'an object' in any sense but the metaphorical, can it? — Wayfarer

(Thanks for the reply!)

Of course it can. While it is still an open dispute in the philosophy of mathematics, ontological realists argue for the existence of abstract mathematical objects. This is not a metaphor at all. There are very good reasons to think that abstract objects do exist. Mathematical objects would exist necessarily.

Other logical principles and laws and 'arithmetical primitives' (foundational concepts in arithmetic which cannot be further defined) are likewise not objects in any sense other than the metaphorical. They can be applied to objects, insofar as the attributes of the objects in question can be made to conform to them, which is fundamental to modern scientific method. — Wayfarer

Treating mathematical objects as mere metaphor is not a very easy position to defend. One would have a difficult time reconstructing mathematics from this starting point. I am not saying things are obvious here at all, but you are too quick to settle on this position.

Moreover, mathematical objects and structures are typically not constructed/invented/discovered (I want this proposition to be philosophically neutral) for science. Quite the opposite is true: the application of mathematics is typically after the fact, and the vast abstract structures of mathematics are not necessary for science at all. Science requires a very small, strict subset of the mathematical structures and objects we already know about.

That's the sense in which an a priori truth is a necessary truth, is it not? And that also is assumed by modern scientific method, which seeks mathematical certainty in respect of those matters it investigates. — Wayfarer

I agree with the first sentence: if there are such things as a priori truths, then yes they would most certainly be necessary truths.

I disagree with the second sentence: science is successful precisely because it does not demand that scientific knowledge meet deductive standards (standards which are held in mathematics, philosophy/logic or the formal sciences more broadly). There is nothing deductively certain in the sciences. Scientific knowledge is falsifiable, and thus not certain. Only formalized theories that follow struct deductive reasoning like mathematics or logic can claim certainty.

This makes sense, as no scientific proposition is a logically necessary one. All scientific propositions, if true, are only contingently true. And even if contingently true, we cannot be certain of their truth. At best, we can only be highly confident that the proposition is true.

So the point of all the above is that 'necessity' in this sense, is a logical, not an empirical, matter. Bearing this in mind, caution is required when we talk of 'objects' and 'beings' in this context, as it is not altogether clear that what we are discussing is an objective matter. — Wayfarer

Of course necessity is a logical matter. But this does not preclude the possibility of necessarily existant objects and beings at all.

I also do not understand what you mean by "discussing an objective matter". Logic is entirely objective. In fact, logic and mathematics are not only objective, they are deductive. We can be certain of the truths of logic and mathematics in a way we cannot about empirical propositions.

There is nothing senseless about the proposition "God is a necessary being". It could very well be true, as it could very well be false (on the assumption that the word "God" has a sense, i.e. is meaningful -- but that is a separate matter).

In fact, I would say that I am fairly certain about the following proposition (on the commonly understood meaning of 'God'): "If God exists, then God necessarily exists". The proposition "God exists" is still not established, however.

Moreover, it seems to me that you want to have a very constrained interpretation of the word "objective". Objectivity has only to do with truth-values. That is, a proposition is objective if it is true or false, whether or now we know whether it is true or false. A proposition that isn't objective is one that does not have a determinate truth-value.

We should not take disagreement as a sign of subjectivity. We can disagree about objective propositions (we all do!). We should also not think that empirical propositions are the only set of objective propositions. This is not true at all. Not only are logical and mathematical propositions clearly objective, I would extend the category of objective propositions to any set of propositions where reasons can be brought to bear. I would certainly include ethics in this account. Aesthetics most likely (though I know nothing about aesthetics). Propositions that would be excluded, for example, would be propositions of personal taste, for example. "I like this movie", "this hamburger tastes awful, yuck!", etc. I think (hope) this all makes sense? Let me know what you think.

That being said, I think that most of the disagreement and issues encountered in this thread turns on a mistake about what we think is actually being said in the OP. What is being said is not controversial at all. It is acceptable to all theists, atheists and agnostics.

The idea is that if a necessary being is necessary, then necessity precedes being. Further, if absent a necessary being, then nothing can be, then there is necessarily nothing if not something; again, the priority of necessity. Perhaps it is not God - whatever that means - that is first cause, first mover. & etc. - that is primary, first, foundational, but necessity. But in that case just what exactly is necessity?

i suspect this is just a rabbit hole in language; still, though, it is necessary to deal with it.

As it sits, the idea of a necessary being is a kind of nonsense. Who will make sense of it? — tim wood

Hi Tim,

Thanks for the response. I am not quite sure I know exactly what you intend to mean by your first sentence. That is, I am not sure what exactly "necessity precedes being" means, and why this should follow from the mere fact of a being that necessarily exists.

It seems to me that should something exist necessarily, all we are saying is that it is logically inconceivable to imagine a possible world in which such a being does not exist. Maybe we can do better with less technical language: if a being exists necessarily, then what I am saying is that I simply cannot imagine a situation in which such a being cannot exist. If I think I am imagining such a situation, I must be mistaken, since such a situation is simply incoherent.

Mathematics is typically the best way to understand necessity. It is necessary that a Euclidean triangle have interior angles that sum to 180 degrees. That is, I cannot conceive of a triangle (in Euclidean space) with interior angles that sum up to more (or less) than 180 degrees. If I think I am doing so, then I am simply mistaken, or I am not imagining a triangle at all.

I am not convinced that there is anything problematic about speaking of necessity in this simple way, and extending our talk of necessity to the existence of objects. If I imagine a situation where one object has the property $P$, and some object that does not have this property, then there necessarily exist at least two distinct objects.

Now if I want to say that an objects exists necessarily, I am simply saying that this object exists in every possible situation, and that I cannot conceive of a situation in which such an object does not exist.

Now it very well may be that no object satisfies this criterion. But I am not sure that there is anything problematic about it. Maybe you can clarify your position?

Also:

if absent a necessary being, then nothing can be

I must be misunderstanding this statement, because on the face of it, it seems obviously false to me. Surely, it is entirely possible that there exist only contingent objects, and that there is no object (persons/beings, etc. included) that exists necessarily. Thus, if there is no object that exists necessarily, it does not follow that nothing exists.

Your last remarks also give me the impression that we are both using the word 'necessity' in different ways, and that there is no dispute here. This may be why I am not understanding your position. Necessary existence has nothing to do at all with causation. An object can exist necessarily and be entirely causally innocuous. That is, the object need not enter into any causal relations at all.

Some argue that certain abstract objects exist necessarily. Mathematical structures/objects would be a prime example.

The law of identity should stated as: $\forall x (x=x)$. However, when you write:

because despite the fact that the thing is changing it still remains the thing that it is, i.e.the same as itself.

I believe you are importing metaphysical claims into the law of identity. The law itself is completely neutral with respect to whether or not an object is the same (or different) after undergoing certain change.

But you are right to point out that the OP also seems to add content to the law of identity. That is, it seems the OP wishes to say that the law of identity only applies to things that do not change. But this is also incorrect, as things can change and still remain the same object.

For example, let's consider the following description $D(x)$:

$x$ is a lawyer in a start-up firm in Brooklyn.

Now let's suppose that John ($j$) is a lawyer in a start-up firm in Brooklyn. It would follow that $D(j)$ is true for some time $[t,t+n]$ (i.e., from the moment that John started his job at the start-up firm, to the moment he either left that job to work elsewhere, or the moment the firm was established and no longer a start-up).

But consider that this was not always true about John, and that, indeed, there is an interval of time for which $\neg D(j)$ is true. Despite this, $j=j$, and there are certainly no doubts about this. In short, change can be captured by what set of sentences are true (and false) about a given object at different times.

We could take a radical metaphysical position and insist that objects can only be self-identical for any given time slice $t$. But here too, the law of identity would apply at any given time slice. The law is completely neutral here.

I think this is right, in any case. If someone has any references to philosophers who argue that the law of identity is not neutral with respect to these metaphysical questions, please send me the reference! Thanks!

Hey Tim,

It is absolutely true that there is the possibility for there to exist multiple, distinct necessary beings. However, neither claims (1) or (2) from the OP are affected by this. That is, there could exist more than one necessary being and both (1) and (2) can still be true.

As to the more recent question, i.e., on the status of 'necessity'. A necessary being is a being that exists in every possible world. That is, a world in which such a being did not exist is not logically possible, i.e., it would be inconceivable.

Given that this is the nature of necessity, the person who wishes to argue for the necessary existence of a being has a daunting task ahead of them.

I hope this helped!

You could say that if Jesus is God then Jesus is a necessary being. But that's not at all the identification claim you're asking about.

Interesting. As I read the OP, this was the only controversial claim I came across. Could you re-state what you think the intended claim is? You would be correct, I take it, that the original argument would not establish a deeper, intended proposition.

Given how uncontroversial the claims seem to be, I think you are right to say that the intended claims are different from those mentioned. I just am not sure what they would be.

Thanks!

Hey fresco, you are right to point out that @CurlyHairedCobbler presented an argument that seems to take the form:

${P\rightarrow Q, \neg Q}\vdash \neg P$

However, this is different from affirming the consequent, which has the form:

${P\rightarrow Q, Q} \vdash P$

This is a fallacy, since $P$ could be either true or false (given $Q$ is true). The first, however, is a valid inference. The contrapositive $\neg Q\rightarrow \neg P$ is true. In fact, the contrapositive is logically equivalent to the conditional $P\rightarrow Q$.

I hope this helps!



This is interesting. I think (1) and (2) are uncontroversial.

However, there seems to be an issue with the claim "Jesus is necessarily God". This says something different. Namely, the necessity operator, here, applied to an identity. When I say that God exists necessarily, what would follow is that if Jesus is God, then Jesus exists necessarily.

I think it is a further step in reasoning to speak of Jesus being necessarily identical to God. That is, if God is a necessary being, then God exists in every possible world. Should Jesus be God, then Jesus is identical with God in every possible world. So, yes, necessarily Jesus is God. Again, however, I am not sure that this is a controversial claim. It seems controversial, but claims of the sort would apply to any necessarily existing being.

But perhaps I missed something...

Great. Thank you!

I'm assuming that this is a trivial question, but nonetheless it just occured me and I can't think of a way to approach it. If my mathematical premises are wrong (which could very well be the case) please correct me.
Let's take two infinite sets of numbers:
1. ...1,2,3,4,5,6,7,8,9,10,11 .....
2. .... 2,4,6,8,10,12,14 ......
So both sets follow a very obvious pattern.
Question:
Does the first set constitute a bigger infinity than the second one, as, let's take the interval (1;4) for example, the first set includes four numbers (1,2,3,4,) of this interval whereas the second one only includes two (2,4)?
Furthermore, applying this reasoning to the whole sequence, can the equation: Infinity (set 1)= Infinity (set 2) x 2 be infered from that?

Hey Gilbert,

I think this question has already been answered quite well by fdrake and ssu. I thought I would offer a less technical answer (though hopefully not any less correct) that might add to what has already been discussed.

I always thought of the following situation as a helpful way to understand bijections. Imagine you are setting up a large banquet. Let's further imagine that each guest gets one fork and one napkin. If I am sure to set up every seat with a fork and a napkin (no more and no less for either), then I know that the number of forks and napkins in the banquet are the same. I know this even if I don't know exactly how many forks and napkins there are. I don't need to count them, I just need to know that they are paired.

So, to your original question, is there a way to pair the numbers in line 1 with the numbers in line 2 in a similar way, i.e., in a way that I know that they are matched up exactly without needing to count any of them and being sure I've left none of them out.

There sure is! And this was alluded to in ssu's post. Let x be any number from list (1). Pair x with 2x. In this way, each number in (1) will be paired with each number in (2) and it is easy to see that it will be paired with one and only one number in (2) and that none of the numbers from the list (2) will be left unmatched. Thus, I know that the number of numbers in (1) is the same as the number of numbers in (2), in the very same way that I know I have the same number of forks as napkins at the banquet.

Hope that added something.