The Constitution of the Russian Federation says, in Chapter 1, Article 1:

The Russian Federation - Russia is a democratic federal law-bound State with a republican form of government.

The names "Russian Federation" and "Russia" shall be equal.

Why is this argument I came up with wrong?

  1. Russia = Russian Federation [says so in the Constitution]
  2. Russian Federation ≠ Russian Empire
  3. Russia ≠ Russian Empire [by substitution]
  4. Russian Empire ≠ Russia [by reflexivity of equality]

I am curious what you guys think.

  • Are you interested in the logic, and willing to substitute abstractions for the words you used, or are you interested in the relation between the natural-language monickers 'russia' and 'russian empire'?
    – bukwyrm
    Dec 5 '18 at 22:27
  • where did #2 come from?
    – user4894
    Dec 5 '18 at 22:35
  • Because they are different states?
    – alex811
    Dec 5 '18 at 22:37
  • I think it's problematic to equate "-" with "=". This could lead to type errors that explain the unintuitiv nature of the argument. Consider following example: John - a man is... could lead to a deduction among the lines of "John is a (the only) man therefore every man is John therefore John = man." If we accept your premisses the argument is correct. However reconsidering the "-" relation and it's usage in english might be helpfull but is not my expertise.
    – CaZaNOx
    Dec 6 '18 at 0:15
  • [Russian Empire ≠ Russia (of this time)] but [Russia = Russian Empire (of the Russian Empire's time)] therefore [Russia now ≠ Russia of before] - Welcome to Philosophy SE!
    – christo183
    Dec 6 '18 at 5:35

The argument you present is correct given the first two statements as premises. From those two statements one can derive using equality elimination and introduction both the third and fourth statement.

To see this, symbolize the names as follows:

  • Let a be "Russia".
  • Let b be "Russian Federation".
  • Let c be "Russian Empire".

The following uses Klement's proof checker and natural deduction using rules for identity found in forallx, section 27.4:

enter image description here

Lines 1 and 2 symbolize the first two lines viewed as premises:

  1. Russia = Russian Federation [says so in the Constitution]
  2. Russian Federation ≠ Russian Empire

Line 3 results from taking the assumed identity in line 1 and substituting in line 2 b for a. This is the symbolization of the third statement:

  1. Russia ≠ Russian Empire [by substitution]

Line 8 after going through a subproof shows that the fourth statement using the symbolization above can be derived.

  1. Russian Empire ≠ Russia [by reflexivity of equality]

The subproof may require some explanation. On line 5 the identity a = a is introduced. Such a line may be introduced anywhere without reference to prior lines. Using the assumed identity on line 4, I can substitute the second a in line 5 with c to get line 6. This contradicts line 3 which allows me to introduce a contradiction on line 8 which is the desired result.


Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/

  • What a wonderful and detailed post, sir! Thank you. My only worry is that Russian Empire is not Russia. This seems counterintuitive. What do you think about this?
    – alex811
    Dec 5 '18 at 22:25
  • @alex811 Just as names your argument is correct. However, there may be hidden assumptions that make the premises unreliable. That would be for a subject matter expert to determine, not a logician. But if we assume the premises and the symbolization is correct, then the derivation should follow. Dec 5 '18 at 22:28
  • I was more interested in semantics rathen than syntactics.
    – alex811
    Dec 5 '18 at 22:29
  • @alex811 I assumed so. I am just addressing the syntactics. There may be a relevant stack exchange on international politics or Russia that deals with the semantics. Dec 5 '18 at 22:31

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