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(2nd version to make explicit my implicit assumptions about A, B and C, and the definitions of the non-logical constants "⊂" and "≡".)

Intuitively, the following modal argument seems valid to me and to other people. However, can it be formally proved valid? Or proved not valid?

Definition: "◇p" means "We don't know that not p".

Definition: "◻p" means "We know that p".

∃A

∃B

∃C

∀x, ∀y, ∀z, ((x ⊂ y) ∧ (y ⊂ z) → (x ⊂ z))

∀x, ∀y, ((x ⊂ y) ∧ (y ⊂ x) → (x ≡ y))

◇(A ⊂ C)

◻(B ⊂ C)

∴ ◇(A ≡ B)

Or, in ordinary language, where "⊂" means set inclusion and "≡" means identity:

A, B and C exist

A may be some part of C

B is some part of C

Therefore, A and B may be the same part of C

I'm researching expressiveness of modal logic, i.e. to what extent modal logic can express the kind of assertions people make using some ordinary, informal language.

The particular relation identified in the argument here, which I think we can best describe as one of vagueness, seems to polarise the population surveyed into (mainly) two groups, and broadly equal in size: those who accept the argument as valid, and those who do not. However, those who do not accept validity appear for the moment, as I see it, unable to articulate a conclusive rationale in favour of invalidity.

The only substantial rationale for invalidity offered so far is to exhibit an interpretation of A, B and C which keeps both premises true but makes the assertoric version of the conclusion, i.e. "Therefore, A is the same as B", false. This is sometimes loosely qualified as a form of "undistributed term".

However, I take this to be inconclusive as it doesn't preclude other interpretations of A, B and C that make true the same assertoric conclusion "Therefore, A is the same as B". So, A and B may be the same. QED.

As I see it, there is nothing in the premises that implies that A and B are necessarily different. This in turn implies that they are possibly the same, which is the conclusion of the argument. QED.

Isn't that good enough?

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How about just a blanket statement with no assumptions:

A and B may be the same part of C

This would seem to be valid by the same reasoning! However it is only so if "may" ranges over all models of the language. If, as is usual, "may" only ranges over the possible worlds in a single model, it is not valid.

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  • $\begingroup$ My argument, like all arguments, is a statement, but not all statements are arguments. Your statement doesn't have premises nor a conclusion, and therefore is not an argument, and cannot therefore be said to be valid or invalid, only that it is either true or false. My question is about the logical validity of my argument, not about whether the conclusion is true. $\endgroup$ – Speakpigeon Jan 29 '19 at 7:41
  • $\begingroup$ Sorry, I don't understand talk of ranging "over all models of the language" or over "the possible worlds in a single model". I talk myself of the possible interpretation of the variables A, B and C, some of which seem to falsify the conclusion even though the argument seems valid and, in these interpretations, the premises are true. This may or may not correspond to your notion of "model" and "possible worlds". What do you think? $\endgroup$ – Speakpigeon Jan 29 '19 at 7:47
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    $\begingroup$ @Speakpigeon If your modal language is not interpreted in the (usual) "possible worlds" semantics, then what is the exact meaning of your modalities? $\endgroup$ – mrp Feb 4 '19 at 11:21
  • $\begingroup$ @mrp Sorry, yes, I define "◇p" as "We don't know that not p" and "◻p" as "we know that p". $\endgroup$ – Speakpigeon Feb 4 '19 at 17:58

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