On 10/21/2011 7:23 AM, Steve Vickers wrote:
What Vaughan called the "clean-shaven gods" argument is, I think, the normal logicians' excuse for barring empty carriers.
That's my alternative to St. Augustine's proof of the existence of god, with "clean-shaven" thrown in to draw attention away from the real fallacy (one would no more portray God without a beard than Santa Claus). What I don't understand is why the emphasis on existence to the exclusion of uniqueness, given that monotheism is the intellectual descendant not of atheism but polytheism. The only explanation I've been able to come up with is that uniqueness is an equational property (g = g') and therefore admissible as an axiom, say as part of the definition of god. Postulating existence in a definition is an obvious cheat since the definition might be inconsistent. A minimal requirement for an existence proof is a consistency proof, which are tough to begin with and get increasingly tougher with increasing scientific understanding.
The logical deduction
(all) x. P(x) ------------- P(a) ------------- (exists) x. P(x)
Even simpler: P(a) --> (exists) x. P(x) is a theorem of the pure predicate calculus, which is "clearly" false in the empty universe when P is taken to be the identically true predicate.
The fix was discovered by Mostowski (I believe). To use a free variable such as a is to hypothesize a denotation for it in the carrier. Therefore in each deduction one should be explicit about the context of free variables that may be used. This deals cleanly with empty carriers.
If you use cylindric algebras B as the subalgebras of the Boolean algebra 2^D^V of all V-ary relations on a domain D, there is nothing to fix. Given a first order language L of wffs over a set X of variables, for each subset V of X, let L_V denote those wffs of L using variables from V, and for each D and each B as above let I_B: L_V --> B interpret the wffs of L_V in B subject only to requirement T0 below. Define the theory T(L) to consist of those wffs P of L such that every I_B having P in its domain map P to the top element of B. Requirement T0 is that the intersection of T(L) and L_0 consist of the propositional tautologies. (Every reasonable notion of classical pure predicate calculus must surely satisfy these extremely mild conditions, which shouldn't be too hard to phrase appropriately for intuitionist logic as well.) Claim. T(L) is independent of whether D = 0 is forbidden. Proof. Allowing D = 0 can only decrease T(L). There are two cases. 1. V > 0. In this case D^V = 0 whence B has only one element and so I_B maps *every* wff to the top element of B. This cannot decrease T(L). 2. V = 0. This is propositional calculus, for which requirement T0 makes the choice of D irrelevant. QED I noticed this property of cylindric algebras when I first ran across them in mid-1979 and assumed it must be well-known since Tarski and Henkin had been playing around with them since the 1950s and could hardly have overlooked such an elementary fact. A month later I attended the sixth International Congress of Logic, Methodology, and Philosophy of Science in Hannover where I met a number of algebraic logicians for the first time: Routley, Dunn, Nemeti, Andreka, Maksimova, etc. When Nemeti and Andreka told me they were working with Henkin, Monk and Tarski on Vol. 2 of Cylindric Algebras, I asked them if they were aware that cylindric algebra semantics solved the logic-of-the-empty-universe problem. They said no so I showed them the argument. Since they were much better connected to the cylindric community than I, I figured they'd pass it on, relieving me of any obligation to write up what seemed to me little more than a passing remark. I was hoping it might appear in Vol. 2 but I don't see it there. Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Vaughan, - On 21 Oct 2011, at 23:06, Vaughan Pratt wrote:
The logical deduction
(all) x. P(x) ------------- P(a) ------------- (exists) x. P(x)
Even simpler: P(a) --> (exists) x. P(x) is a theorem of the pure predicate calculus, which is "clearly" false in the empty universe when P is taken to be the identically true predicate.
No, that's wrong. Your formula P(a) --> (exists) x. P(x) is valid in the empty carrier, because it is (vacuously) true under every interpretation of the free variable a. To avoid the vacuity and get a falsehood you have to quantify out the free variable, as ((all) a. P(a)) --> ((exists) x. P(x)) Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 10/22/2011 6:03 AM, Steve Vickers wrote:
Even simpler: P(a) --> (exists) x. P(x) is a theorem of the pure predicate calculus, which is "clearly" false in the empty universe when P is taken to be the identically true predicate.
No, that's wrong.
You may have overlooked the scare quotes.
Your formula P(a) --> (exists) x. P(x) is valid in the empty carrier,
I agree, but I said "false," not "invalid."
because it is (vacuously) true under every interpretation of the free variable a.
This would only make sense if you identify truth and validity. The formula is vacuously *valid* under every interpretation.
To avoid the vacuity and get a falsehood you have to quantify out the free variable, as
((all) a. P(a)) --> ((exists) x. P(x))
Your "have to" here brings to mind Dan Dennett's chapter "Qualia Disqualified" in his 1991 book "Consciousness Explained." Concerning "if a tree falls in the forest and there is no one to hear it, does it make a sound?", Dennett says "The answer is left as an exercise for the reader." Dennett appears to reject the possibility of two answers depending on whether "sound" is considered a physioacoustic or psychoacoustic phenomenon on the ground that it can't be the latter since (if I understand his argument on pages 370-411) there is no coherent notion of the latter. (How could anyone doubt this after 40 pages?) Dennett's argument in turn brings to mind the argument that, since we can't pin down climate sensitivity to better than a range of 1.5 to 5 degrees per doubling of CO2, global warming can't be real. What we have here is the logical counterpart of "no one to hear the sound" as "no interpretation to witness the truth." The distinction between physioacoustic and psychoacoustic becomes that between physio-logical and psycho-logical, truth vs. validity. But did the tree actually fall? Is "P(a) --> (exists) x. P(x)" (context should make clear that those are not intended as scare quotes) actually false as I claim? Enter the scare quotes on "clearly," and enter Tarski pointing out that truth can't be defined. For this sentence it's not impossible to define truth, it's just that there's no canonical definition, since in the empty universe the truth of P(a) becomes an incoherent notion, leaving one at liberty to define it however you wish. Correct me if I've misunderstood but you (following Mostowski?) appear to have chosen to assign "true" to every non-closed formula, which stops the bleeding, a victory for first aid. CCA, concrete cylindric algebras, namely subalgebras of the Boolean algebra 2^D^V, handle all this automatically when D = 0. When the formula contains no variables at all, free or bound (i.e. propositional calculus, V=0), D^V = 0^0 = 1 and B = 2 as appropriate for propositional calculus. That is, absent both individuals *and* variables, CCA automatically becomes ordinary propositional calculus. But if V is nonempty then D^V = 0 and B = 2^0 = 1, the inconsistent Boolean algebra. *CCA automatically recognizes the inconsistency of trying to define truth in the empty universe when the formula contains variables.* With CCA there is no need to staunch the bleeding because the patient was not injured in the first place. With other semantics, which invariably disallow B = 1, YMMV as they say. On 10/22/2011 3:36 PM, Dusko Pavlovic wrote:
maybe it's time that we all recall lawvere's "Adjunctions in foundations", that appeared some 35 years ago. in its best times, categorical logic offered powerful tools to expand our logical intuitions.
The version of my proof (that CCA handles this problem in stride) that I told to Andreka and Nemeti assumed a fixed V, per the usual convention among cylindric algebraists (whatever their radius). Knowing how strongly Bill feels about cylindric algebra, and anticipating that he would surely object to my argument in that form, I tweaked it slightly by introducing a master set X of variables so as to be completely consistent not only with Bill's nice way of handling quantifiers, which requires allowing V to vary, but even the most egregious ways including those in the Cylindric Algebra volumes. You will hopefully have noticed that the only property of interpretations I_B that I required was that they interpret propositional formulas (V = 0) so as to make T(L) agree with standard propositional logic. The proof does not depend on how quantification is defined, and is just as sound for Bill's approach as for anyone else's. Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
variable a. To avoid the vacuity and get a falsehood you have to quantify out the free variable, as
((all) a. P(a)) --> ((exists) x. P(x))
here is a proof of this apparent falsehood: (all) a. P(a) --> (all) b. P(b) (exists) b. P(b) --> (exists) x. P(x) -------------------------------- ------------------------------------- (all) a. P(a) --> P(b) P(b) --> (exists) x. P(x) --------------------------------------------------------------------------------------- (all) a. P(a) --> (exists) x. P(x) (it's is just adjunctions in foundations again: we compose the counit of the adjunction of one quantifier with the unit of the other one. note that it is valid constructively. i am not sure, but i think gentzen gave a different proof in an example.) -- dusko [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Even simpler: P(a) --> (exists) x. P(x) is a theorem of the pure predicate calculus, which is "clearly" false in the empty universe when P is taken to be the identically true predicate.
i assume that even this pure predicate calculus defines its predicates over a set A as the maps A --> Prop, where Prop is the set truth values. is that right? then there is just one predicate P:A-->Prop if A=0 (empty). you may call this predicate identically true, or identically false if you like. do we also agree (with Gentzen) that a sequent (exists x). Q(x,y)=>R(y) gives a sequent Q(x,y)=>R(y)? (tacitly, R gets a dummy variable x in this second sequent.) if we agree with this, then for the predicate P:0-->Prop the identity sequent (exists) a. P(a) => (exists) x. P(x) gives the sequent P(a)=> (exists) x. P(x) this is not so unintuitive if you realize that (exists) x. P(x) in this sequent must be a predicate over 0, in order to be compared with the truth value of the predicate P(a) over 0. but there is just one predicate over 0, so these two predicates must be equal. maybe it's time that we all recall lawvere's "Adjunctions in foundations", that appeared some 35 years ago. in its best times, categorical logic offered powerful tools to expand our logical intuitions. -- dusko [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
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Dusko Pavlovic -
Steve Vickers -
Vaughan Pratt