In Quantified modal logic, "constancy’s defenders can point to certain powerful arguments in its favor.

Here’s a quick sketch of one such argument. First, the following seems to be a logical truth:

Ted = Ted

But it follows from this that: ∃y y = Ted

This latter formula, too, is therefore a logical truth. But if φ is a logical truth then so is □φ (recall the rule of necessitation from chapter 6). So we may infer that the following is a logical truth:

□∃y y = Ted

Now, nothing in this argument depended on any special features of me. We may therefore conclude that the reasoning holds good for every object; and so ∀x□∃y y = x is indeed a logical truth. Since, therefore, every object exists necessarily, it should come as no surprise that there are things that might have been ghosts, dragons, and so on— for if there had been a ghost, it would have necessarily existed, and thus must actually exist.

This and other related arguements have apparently wild conclusions, but they cannot be lightly dismissed,for it is hard to say exactly where they go wrong (if they go wrong at all!).8

8 On this topic see Prior (1967, 149-151); Plantinga (1983); Fine (1985); Linsky and Zalta (1994,1996);Williamson (1998, 2002)."

——Sider, Logic for Philosophers, Oxford, 2010, p307

  • But if we doesn't holds a view of free logic,it seems it should not be rejected. – AnduinWilde Feb 1 '20 at 12:19
  • And it based on what semantics we defined. – AnduinWilde Feb 1 '20 at 12:23
  • And if choose axioms for simple quantified modal logic(SQML),we may give a proof like this:1.a=a (RX) 2. ∀x∼ x=a → ∼a=a (PC1) 3. ∼∀x∼ x=a (1, 2, PL) 4. □∼∀x∼ x=a (3, NEC). the proof theory is comes from Sider2010,p315 and given Hughes and Cresswell have proved that SQML is sound(With a semantic theory without accessible relation),the syntactic consequence cones out a semantic consequence. – AnduinWilde Feb 1 '20 at 12:32
  • 3
    The problem only comes up when classical quantification rules are applied to systems with non-referring names, and then unrestricted necessitation rule is slapped on top of that. However, traditional classical systems do not allow non-referring names, and those that do do not allow unrestricted existential generalization. Either way, ∃y(y = Ted) will either not be derivable, or not be a logical truth to which the necessitation rule can be applied. – Conifold Feb 1 '20 at 22:30
  • e. j. Lowe on this:thiseven.blogspot.com/2013/11/blog-post_4436.html?m=1 – AnduinWilde Oct 7 '20 at 9:43

If I'm understanding correctly, I could use your logic to make the argument that unicorns exist as follows:

  1. unicorn = unicorn
  2. there exists some x = unicorn

therefore, unicorns exist

I agree that (1) holds. I also agree that (2) implies the conclusion. Step (2) strikes me as more problematic. Why is there some x = unicorn? If I assume that you are using (1), presumably it's because you can choose unicorn = x. But then you are assuming your conclusion and your reasoning is circular.

  • I find it quite implausible to treat "unicorn" as an individual constant, rather than as a predicate. Following your English translation "unicorns exist", "unicorn" should be a property, not a name that refers to a concrete individual. – lemontree Feb 5 '20 at 14:19
  • The notation is sloppy, but this is the right counterargument. By most notions of equality, any nonexistent thing would also be equal to itself. It shares absolutely every one of the zero properties it has, with just the same value. In fact, in some reckonings, by having no properties it would also share all of its properties' values with every other nonexistent thing. That does not mean it exists... The very first step in the main post is nonsense. – hide_in_plain_sight Feb 6 '20 at 2:43

You have two logical formulas here. Note that Ted, = and y are just symbols, they don't, in the logic, correspond to any actual objects. What we need is a model for that.

I am now proposing this model:

=(A,B) is always false The set of object in my universe is empty.

Under this model your formulas are both false.

Let's now take = to be what you expect, the identity relation. Still in a universe with no objects. The first relation is true but the second false.

Finally, a universe where there is some person called Ted that is equal with himself. Of course in that universe Ted exists.

=> Don't mix up syntax and semantics, you need both. You get strange answers if you forget that first-order logic makes no assumptions wrt. the model.

You can clearly see something is wrong when you come to the conclusion unicorns exist in our world, when in fact they don't. They exist in a model world if you say they do. But for the logic, there is no distinction between whether an object is a number, a unicorn, a grain of sand, or russels teapot.


The magician asks for a volunteer to offer their watch. He then takes the watch and puts it in a black cloth bag. He then smashes the smithereens out of the black bag, using a large metal hammer. He then opens the bag. You know that watch=watch, so you expect that there exists a watch in the black cloth bag.

But the magician is not a capable magician. He made a mistake.

So he hands you back a pile of formerly-parts-of-a-watch debris. The watch is no more. You sadly recall your trip to the watch factory, where you saw this very watch constructed. Never more will you know what time your favorite philosophy related TV show starts.

So it clearly is not true without a time component. You may then infer that, since there is one category of "it's not true" cases, there could well be more.

You can still say watch=watch and have it be true, even when the watch is not existent. In this case, the nature of the watch is "does not exist."

  • Would this count as a valid reasoning?

(1) For all x, x=x.

(2) the whole number between 4 and 5 = the whole number between 4 and 5.

(3) So there is some y such that y is the whole number between 4 and 5.

  • But in which way does it differ from Sider's reasoning?

  • I think that no name ( or description) should be substituted for x in sentence (1) unless this name actually refers to something.

In that case, the conclusion would be trivial in case of actually existent objects, and could not be arrived at in case there is no object the " name " refers to.


I don't think everything exists. It only exists if it can exist. But since dragons never existed just before we imagined them doesn't mean they can or ever will exist.

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