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The oft-given example to demonstrate the failure of substitutivity in an intensional context goes as follows:

(P1) Lois Lane believes Superman can fly

(P2) Superman is Clark Kent

(C) Lois Lane believes Clark Kent can fly

Another, more novel example, goes as follows:

(P1) Max knows that the dictator of Russia in 1950 is (was) the dictator of Russia in 1950

(P2) The dictator of Russia in 1950 is (was) Joseph Stalin

(C) Max knows that the dictator of Russia in 1950 is (was) Joseph Stalin

In both examples, it seems evident that the conclusion dosen't follow; that substitutivity fails. Yet there seems to be great discord amongst philosophers of language and linguists alike as to why substitutivity fails in instances akin those provided. Taking a cursory look at both syllogisms, it seems to me evident that the identity operator "is" is not adequately scoped. In order to demonstrate this, let us reformulate the premises in both arguments so as to clarify the use of the term "is".

Syllogism 1:

(P1) Lois Lane believes Superman can fly

(P2)* Superman is identical to Clark Kent

(C) Lois Lane believes Clark Kent can fly

Syllogism 2:

(P1) Max knows that the dictator of Russia in 1950 is (was) the dictator of Russia in 1950

(P2)* The dictator of Russia in 1950 is identical to Joseph Stalin

(C) Max knows that the dictator of Russia in 1950 is (was) Joseph Stalin

Given the above transformations, it becomes clear that the notion of identity used is central to whether or not the above syllogisms prove themselves to be valid. I hold that the reason the above syllogisms prove to be invalid is due to the confusion that stems from the second premise, namely, that the notion of identity understood encompasses extrinsic properties (properties that do not inhere in the subject) when in reality it only encompasses intrinsic properties (properties that inhere in the subject).

To elucidate, upon reading the identity operator as encompassing all intrinsic and extrinsic properties, the syllogisms prove to be valid. This can be demonstrated as follows:

Syllogism 1:

(P1) Lois Lane believes Superman can fly

(P2)** Superman has all the same intrinsic and extrinsic properties as Clark Kent

(P3) Superman has the extrinsic property of "being believed of as capable of flying by Lois Lane"

(P4) Clark Kent has the extrinsic property of "being believed as capable of flying by Lois Lane" [P2, P3]

(P5) If (x) has the extrinsic property of "being believed as capable of flying by Lois Lane", then Lois Lane believes (x) can fly

(C) Lois Lane believes Clark Kent can fly [P4,P5]

Syllogism 2:

(P1) Max knows that the dictator of Russia in 1950 is (was) the dictator of Russia in 1950

(P2)** The dictator of Russia in 1950 has all the same intrinsic and extrinsic properties as Joseph Stalin

(P3) The dictator of Russia in 1950 has the extrinsic property of "being known by Max as having the same identity as the dictator of Russia in 1950"

(P4) Joseph Stalin has the extrinsic property of "being known by Max as having the same identity as the dictator of Russia in 1950" [P2, P3]

(P5) If (x) has the extrinsic property of "being known by Max as having the same identity as the dictator of Russia in 1950", then Max knows (x) has the same identity as the dictator of Russia in 1950.

(C) Max knows that the dictator of Russia in 1950 is (was) Joseph Stalin [P4, P5]

Yet it is clear that we do not intend to express the syllogisms presented above, rather what is intended by the identity operator is that (a) and (b) have the same intrinsic properties, though not necessarily the same extrinsic properties. If I am correct in ascertaining that to be the case, then it seems that the invalidity of the argument stems from the tacit assumption that if (a) and (b) have the same intrinsic properties, then they have the same extrinsic properties; something which can, ironically, be demonstrated to false by the syllogisms presented above (as it seems obvious that each pair of notions in consideration have differing extrinsic properties).

So, considering the reasoning laid out above, is the failure of substitutivity in an intensional context simply due to a lack of clarity in terms of the identity operator?


ADDENDUM

Some of the comments below state that the identity operator is in fact clear and that its operation is to denote the terms (such as 'Superman' and 'Clark Kent') as coreferentiality, not to denote two notions as identical. I contend under such an interpretation that the syllogisms above are in fact valid and can only be construed as invalid if the following form is taken as what is meant (I will use only the first syllogism for the sake of brevity):

(P1) Lois Lane knows 'Superman' refers to (x)

(P2) Lois Lane belives that (x) flys

(P3) Lois Lane believes, when heard or read, the statement "[Superman] can fly" to be true

(P4) 'Superman' and 'Clark Kent' both refer to (x)

(C) Lois Lane believes, when heard or read, the statement "[Clark Kent] can fly" to be true

  • Note that the first two premises were added to disambiguate the context

Clearly, in this instance, the syllogism is invalid, though this is a deviation from the original syllogism, which, when construed to represent the identity operator as merely denoting coreferentiality (as was claimed), takes the following form:

(P1) Lois Lane believes Superman can fly

(P2) 'Superman' and 'Clark Kent' refer to the same referent

(C) Lois Lane believes Clark Kent can fly

Under the above interpretation of the identity operator, there is no controversy to be had, as the non-quotation mark name is meant to be a direct denotation of the referent object, meaning (P1) and (C) are identical disregarding the syntax.

Ostensibly, one of the drawbacks of holding such a position is that one cannot make sense of quantification in modal contexts such as the following:

(P1) 8 is necessarily greater than 7

(P2) The number of planets is 8

(C) The number of planets is necessarily greater than 7

Despite the claim, it seems to me that clearly one can. That is to say, the second premise, if understood as denoting the coreferentiality of the labels, then the modal property is established as applicable to the sentence when such a label is provided (as its referential value is identical to the initial label). To illustrate this once again, the interpretation of the modal operator in such a manner gives us the following:

(P1) 8 is necessarily greater than 7

(P2) 'The number of planets' and '8' refer to the same referent

(C) The number of planets is necessarily greater than 7

Hence the term [the number of planets] here is simply transformed into another label for the number 8, not a reference to the amount of planetary bodies. Of course, if it is not the intention of the formulator to use the identity operator in such a manner, the formulator can explicate their intentions by converting their syllogism into its explicit form (just as I have done for the last two syllogisms), clearly defining the identity operator.

Of course, the most common interpretation of the planet syllogism takes "the number of planets" in the second premise to not simply denote a mere label that takes on a rigid meaning, and hence is often taken to be invalid. Though if it is not seen as a matter of labeling, the understanding of the function of the identity operator in the context of the syllogism must be different than a mere denoting of labels to a specific referent, and hence I take it to be denoting identity in terms of properties or not denoting identity at all.

If that is the case, one could potentially construe it as a problem of scoping, where the phrase "the number of planets is 8" should be specified rather than left unrestricted. Hence the phrase would be "The number of planets in word (x) at time (t) is 8", resulting in the conclusion that follows being a necessary fact (namely that the number of planets in world [x] at time [t] is greater than 7; note that this would be the result of a tacit modal shift, rendering the second premise necessary). Yet even in this case the second premise would be false (though the conclusion still true) if the “is” operator is understood as an operator of identity. Rather the second premise (the number of planets in world (x) at time (t) is 8) must be understood as saying “there are 8 planets in world (x) at time (t)” (a far more natural reading of the premise that we shall refer to as the “natural sense”) if one wishes for the syllogism to be sound.

-Note that in the syllogism that interprets premise (2) in the natural sense requires an implicit premise that states “necessarily, if two sets of objects of a kind differ numerically, then the set with the larger cardinality is numerically more”. The last premise must also be understood in a natural sense, i.e., as stating “necessarily there are more than 7 planets in world (x) at time (t)”.

In its unrestricted form (which must be done if a shift in modality is to be avoided), with “is” functioning as an identity operator serving as a claim relating to the intrinsic properties of the number of planets, the syllogism would be not invalid, but rather unsound due to the "the number of planets" and the number "8" having different intrinsic properties (necessarily being greater than 7 being one of them). In the instance that the syllogism with its unrestricted scope is understood in its natural sense, the syllogism becomes invalid (there being 8 planets does not entail that there are necessarily 8 planets).

Regardless, in both cases, it seems as if the identity operator plays an important role and still therefore seems central to the failure substitutivity in an intensional context.

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    There is no "lack of clarity". The identity operator denotes coreferentiality whereas what you introduced doesn't. The lack of substitivity in intensional contexts doesn't challenge this notion of identity, just poses some challenges about semantics of these statements. Commented Mar 27 at 23:52
  • @abracadabra If "the identity operator denotes coreferentiality" and nothing further, then it seems that there is an inherent ambiguity to the operator... it does not make clear the scope of properties the two notions have in common (as a second notion is introduced and simply said to be coreferent with the initial notion).
    – Max Maxman
    Commented Mar 28 at 0:00
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    Adding on..... it seems that such an appeal merely pushes the question back a single step, that being, "Does coreferentiality entail that all intrinsic and extrinsic properties are shared between (a) and (b)?"; if you reject the initial characterization of (a) and (b) as two notions, then it merely comes down to a question of labels and naming conventions. If so, the syllogisms would be valid prima facia, in which case, invalidity becomes the case only if the form is changed from "Lois Lane believes Superman can fly" to "If Lois Lane is asked 'can Superman fly' she would answer yes".
    – Max Maxman
    Commented Mar 28 at 0:24
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    The intrinsic/extrinsic distinction is highly suspect beyond colloquial uses, see SEP, and is not very helpful in understanding why substitutivity fails. Indeed, we can use substitutivity in turn to make the distinction, so this just trades the frying pan for the fire. You can see how this plays out with a similar de re/de dicto distinction, which is traditionally used to analyze substitutivity in opaque contexts.
    – Conifold
    Commented Mar 28 at 0:32
  • @MaxMaxman Well, sure, Quine proposed that all intensional/referentially opaque contexts are treated like quotes, i.e. by embedding the sentence, as, for example, answer to a question. But this way you cannot make any sense of quantification in modal contexts. And this is terrible news for many perfectly valid parts of scientific and ordinary discourse, ex. counterfactual conditionals, ascription of beliefs etc. The appropriate solution is to allow for various ways in which expressions can refer to objects. And yes, this can involve something like the distinction between essential... [1/2] Commented Mar 28 at 0:40

1 Answer 1

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No. The identity operator indicates that two names or definite descriptions refer to the same object; it does not directly say anything about properties, intrinsic or extrinsic, although from A=B one can infer that A and B have the same properties.

A sentence like "Lois Lane believes Superman can fly" does not express a property of Superman, but a property of Lois Lane, namely what she believes. And what she believes is not a state of affairs but a proposition: "that Superman can fly". Superman himself does not directly figure into this proposition, only the concept of Superman or the meaning of the word "Superman" does.

One way to think about it is to ask what Lois Lane would say if you asked her, "Do you believe Superman/Clark Kent can fly?" When viewed this way, you can see that the meanings that Lois associates with the words are relevant to how she will answer. That is where the context sensitivity comes in; not in the use of identity, but in the nature of "believes" and how it is used in English.

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    +1 Explained in a concise and lucid manner as usual; you responded while I was in the middle of writing my addendum, but I touched in the addendum the interpretation that you seem to be using here. Under such an interpretation the identity operator can reasonably be understood as such whilst taking “belief” here to mean Lois’ response to a particular proposition (I initially felt as if this deviates quite far, but on second thought it holds quite a bit of merit, though I’m not sure if it’s applicable to examples where modality is involved such as the planet example).
    – Max Maxman
    Commented Mar 28 at 5:29
  • You might reword Lois lanes belief as "Lois believes the person she recognises as superman can fly." Then it becomes apparent that you can't swap in Clark Kent for "the person she recognises as superman" - but crucially, once she finds out they're the same person, suddenly you CAN swap it in.
    – TKoL
    Commented Mar 28 at 7:33

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