Existential Import paper by Corcoran and Massoud

  • #1
nomadreid
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Main Question or Discussion Point

In the paper

John Corcoran & Hassan Masoud (2014): Existential Import Today: New
Metatheorems; Historical, Philosophical, and Pedagogical Misconceptions, History and Philosophy of
Logic, http://dx.doi.org/10.1080/01445340.2014.952947,

already in the introduction it says, as self-evident, that "The universalized conditional
∀x(x = 0 → x = (x + x)) implies the corresponding existentialized conjunction
∃x(x = 0&x = (x + x)). And ∃x(x = 0) is tautological (in the broad sense, i.e. logically true)."

There are two assertions here, and I have difficulty with each one. In the first assertion I do not see why a model in which there is no 0 (e.g., ZFC* obtained by negating all the axioms of ZFC), so that the conclusion of the implication is false and the premise would be (vacuously) true, would not be a counter-example.

In the second assertion, I don't see why the statement is tautological (since it is not satisfied by all models, such as ZFC*).
 
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  • #2
Stephen Tashi
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I can only see the abstract of the article. What is the definition of "0" in this context?
 
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Stephen Tashi
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I attach the full article in pdf. "0" is the symbol in the theory that would be interpreted by the empty set in the model.
I don't understand "model theory", but it seems to me that the crux of the matter is whether (for a particular x) "##x=0##" is a statement (be it true or false) or whether "##x=0##" is just meaningless notation.

If we grant that "##\forall x( x=0 \rightarrow x = x+x)##" is a statement (be it true or false) the we must grant that (for any particular x), "##x=0##" is a statement as opposed to being meaningless notation. To evaluate the truth or falsity of "##x=0##" we must have a definition of the relation "##=##" that applies to whatever things that "##x##" may represent and whatever thing "##0##" is.

If we want to say "##0##" "does not exist" then we are free to say "##x=0##" is meaningless notation as opposed to a statement whose truth value can be determined.

In the free-for-all logic used in mathematics might be possible to interpret the notation "##x R \theta##" as a compound statement meaning 1) ##x## exists and 2) ##\theta## exists and 3) the relation ##R## exists (as a set of ordered pairs of elements) and 4) ##(x,\theta) \in R##. This is the type of interpretation we need if we wish to say that it is false that "13 is greater than the largest number" instead of limiting ourselves to the observation that "13 is greater than the largest number" is a meaningless phrase that cannot be assigned a truth value.

However, insofar as I understand the paper, it limits itself to particular kinds of logic and I don't know how these types of logic handle the concept of "exists". In particular I don't know how the notation "##x=0##" is interpreted as a statement vis-a-vis the existence of the things it mentions.
 
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  • #5
nomadreid
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Thanks for your analysis, Stephen Tashi. Analyzing the paper further, I believe that you are right that the crux lies (a) in which type of logic is permitted, (b) how 0 is interpreted, and (c) how the quantifiers are used. In the first matter, the paper was not explicit but rather following a common convention that I should have taken into account. In the second one I was following a common convention that I should not have. That is, the paper follows the convention to exclude a "free logic" which allows an empty universe (domain), so that the concept for "vacuously true" is limited. Secondly, it does not use "0" in the usual set theoretic way of the symbol interpreted by the empty set, but rather as a symbol which could be interpreted as anything whatsoever, as long as the interpretation remained constant in that model. I can accept these two, successfully unblocking two hurdles to understanding the paper. The third one still presents difficulties. You are right that the use of "=" is a key point, but here the authors seem to stick to the convention that the "=" sign between two constant symbols is always interpreted as meaning that the constant symbols were interpreted by the same individual in the model, and the sentence ∃x (x=c) would be interpreted that there is a constant in the universe to which the constant symbol is assigned. As you point out, if one uses a symbol in the theory, then one necessarily has something to assign it to: one is not allowed, in this way, to use a symbol that has no interpretation. Fine. But now to turn to the third hurdle, primarily the existence quantifier. In saying that ∃x (x=0) is a tautology, the authors are assuming that it is also true in theories which do not have ∃x (x=0) as an axiom. Also they discuss the existence of a unicorn, allowing that perhaps unicorns don't exist. That is, despite the fact that "unicorn" has a meaning in the theory, then in some interpretations "unicorn=horned horse" and "horned horse→⊥", so ~∃x (x=unicorn), so ∃x (x=unicorn) is not a tautology. I do not see how, if "0" is allowed free rein as a symbol (and not the standard set-theoretic interpretation), how one can treat "0" and "unicorn" as different. (Once one accepts ∃x (x=0) as a tautology, then the implication between the ∀ and ∃ sentences becomes clear.) I must be missing something here. Any further insights would be appreciated.
 
  • #6
Stephen Tashi
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Also they discuss the existence of a unicorn, allowing that perhaps unicorns don't exist. That is, despite the fact that "unicorn" has a meaning in the theory, then in some interpretations "unicorn=horned horse" and "horned horse→⊥", so ~∃x (x=unicorn), so ∃x (x=unicorn) is not a tautology.
Perhaps we must make a distinction between the concept of "is a unicorn" versus the concept of "has the property of unicorn-ness". When we say "A horse is a four legged animal" we don't actually mean "horse = four legged animal" as if "horse" and "four legged animal" define concepts that are the same in every respect.

Using that approach, "x is a unicorn" would not be denoted "##x = u##". It would denoted as a propositional function ##U(x)##. In set theory ##U(x)## could be "x is an element of the set U of unicorns", which would leave open the possibility that U is the null set.

By contrast, if we assume "##x = u_1##" is a statement where ##u_1## denotes a particular unicorn we must interpret ##x=u_1## in some way that a truth value can be assigned to it. So we are compelled to have a ##u_1## that can be used to compare ##x## to.

I don't understand how the paper distinguishes between concepts that are discussed by the logic versus meta-concepts that are concepts about the logic. For example, a meta-interpretation of "##=##" is "When p = q, every place you see a p, you can replace it by a q". This is a statement about manipulating strings that are used to express the logic. It is not a definition that applies within the subject matter that the logic discusses - unless you allow the logic to be self-referential.
 
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  • #7
nomadreid
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Thanks again, Stephen Tashi. Your analysis about the existential quantifier makes perfect sense, and seems to be the solution to this hurdle of my understanding of the article. Other difficulties will of course arise, partly because of the problem that he does seem to mix meta-mathematics with mathematics, and the fact that he seems to leave his axioms, rules of inference and definitions implicit, leaving them to be extracted from the discussion. As far as equality, I assume that he is using first-order logic with equality, so that the axioms for equality are reflexivity and the substitution schemata (so equality becomes a case of an equivalence relation).
 

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