Priest does not think that the issue here is whether or not we can devise consistent formal theories. More important is whether or not our self-consistent formal theories are compatible with “the phenomenon we are trying to model: natural reasoning.” [Natural reasoning may require something different than a rigidly self-consistent theory, and thus] “It is disturbing to see how many logicians think that the problem has been solved once some formal construction, which is (putatively) consistent, has been given.” (9)
Priest divides the paradoxes of self-reference into two families: 1) the semantic variety and 2) the set theoretic ones. The semantic types include “the paradoxes of truth, denotation, predication, and so on (the liar, Grelling’s, Berry’s, Richard’s, Köenig’s, etc.),” while the set theoretical type includes “the paradoxes of membership, cardinality, etc. (Russell’s, Cantor’s, Burali-Forti’s, Mirimanoff’s etc.)” (9). Although for a long time this distinction seemed clear, it became “impossible to draw satisfactorily” – with the advent of mathematical semantics and Tarski’s truth definition – in a set theoretical metalanguage (9-10). * [The following section is a bit technical, but we will work through it using guesswork and material by Tarski and Gary Hardegree. Priest will say there is an isomorphism between the following two formulations:
x ∈ {y|α} ↔ α(y/x)
x satisfies α ↔ α(y/x)
(Priest, 10)
For a more thorough examination of the notion of isomorphism as it is understood in logic (and applied in artificial intelligence), please see this page. If we may summarize from that page, the isomorphism in this case at hand is perhaps the following. Consider certain substitutions that we may place into one formulation. The other formulation is of another nature and purpose, and so it may not be able to take the exact same values as we can give the first one. However, these values in the second formulation might have terms that correspond somehow to those in the first one, in a one-to-one fashion (like our example in the linked post: we may have one series of Roman numerals and another series of Arabic numerals. They are different series of different terms, but the one may be mapped onto the other in a one-to-one fashion. Also, the structural and logical relations between the one set are preserved in the other). So perhaps Priest is saying that so long as the both formulations give comparable ‘outputs’ for comparable ‘inputs’, they are isomorphic. His more basic point is that the first formulation is set theoretical and the second one is semantic. Normally with regard to paradoxes of self-reference we distinguish the one type from the other. But the isomorphism between these formulations calls into question that distinction.
So let us examine these formulations each in turn, then together, starting with the second one, the Tarski satisfaction scheme. Recall Tarski’s (T) scheme, which serves to provide a semantic definition of truth:
(T) X is true if, and only if, p.
For example,
“Snow is white” is true if, and only if, snow is white.
(Tarski, ‘Semantic Conception of Truth’, 334)
Tarski explains that the concept of ‘truth’ in this scheme can be understood in terms of satisfaction. But satisfaction would not apply if we begin with “snow is white”, because it has already been satisfied with the term ‘snow’. Other things, like angels and polite lies, can also be white. So the subject here is predicated by “is white”. We can introduce a variable as the subject that is being predicated, and we would get “y is white”. This is no longer a sentence but rather is a sentential function, since it is completed or fulfilled when we substitute an ‘input’ in for the variable, and the ‘output’ is one of two values, true or false. The input satisfies the function or predicate when it makes it true (this seems circular, but given the distinctions of meta- and object language, and the axiomatic recursive method that he uses to define satisfaction, it is perhaps in the end not problematic). So again, the beginning of our formulation is:
“y is white” is true
But since y is a variable, it is not yet true. We just have “y is white”. We then make substitutions, using symbols or names for objects. So the word “snow” when substituted in for y yields a true formulation, since snow itself is indeed white. So the object satisfies the formula when it can rightfully (correctly, truly) be substituted, and it does not satisfy the formula when it incorrectly or falsely is substituted. So let us replace ‘is true’ with ‘x satisfies’:
x satisfies “y is white” if and only if x can be rightly substituted for y in y is white
Now, let us replace ‘is white’ with a symbol for its formulation (as if it were a predicate or function symbol): α.
x satisfies “α(y)” if and only if x can be rightly substituted for y in y is white
We will shorten this again by using the following notation. “x can be rightly substituted for y in y is white”as: α(y : y/x). (This is not standard notation. I need something with the meaning of: y with the property α where y is substituted with x.) Now we have:
x satisfies “α(y)” if and only if α(y : y/x)
[Again, excuse the poor notation. It is for the sake of the next step]
Now, we will embed y into the formula, such that the whole expression “y is white” or α(y) is now symbolized as α, but we keep in mind that y is hiding inside that formulation. So now we have:
x satisfies “α” if and only if α(y/x)
Now we change the quotations to underlining.
x satisfies α if and only if α(y/x)
And finally, we replace the text ‘if and only if’ with its symbol ↔, and we obtain the formulation as Priest writes it, which again is:
x satisfies α ↔ α(y/x)
Now let us turn to the first formulation, the set theoretical one.
x ∈ {y|α} ↔ α(y/x)
To arrive at this, we will draw from Gary Hardegree’s “Basic Set Theory”.
We begin with a set. Let us say it is {snow, polite lies, angels, ….} with the ‘ …’ meaning the list of all other white things. The curly brackets mean that all the contents between them form a set. But it is too impractical to actually list all white things. So let us again use the predicate symbol, beginning first with ℱ, and we will use variable symbol v to mean all items that can given in that list. So now we have:
{v : ℱ}
which means, the set of things that are white [the set of v’s such that ℱ(v)] . Now {v : ℱ} is defined as that one particular set of things that includes the members v if and only if ℱ(v). This can be written as:
{v : ℱ} =df ℩S∀v(v ∈ S ↔ ℱ)
But instead of the S for the name of the set, we can just use its curly bracket form.
∀v(v ∈ {v : ℱ} ↔ ℱ)
Now at this point I am not exactly sure how to inch closer to Priest’s formulation. But I propose the following. Let us think about ∀v (for all v) as indicating that many substitutions are possible, but this will hold for all cases of v in this formulation. So we are thinking in terms of substitutions, with x being a term that can substitute for v. So we now mean: x is included in the set of v things that are ℱ if and only if the v things are ℱ.
x ∈ {v : ℱ} ↔ ℱ
Let us also exchange the variable name v with y and formula name ℱ with α, and let us also change ‘ : ’ with ‘ | ‘, to get:
x ∈ {y | α} ↔ α
In Hardegree’s text, ℱ implied the variable was embedded in the formula symbol (see page 6). So like with the other formula, we can use α(y/x) to mean again: α(y) when y is substituted by x. Using our example, this could be something like: y is white when “snow” is substituted for “y”. Now we obtain the formula that Priest writes:
x ∈ {y|α} ↔ α(y/x)
But please read the following to interpret it for yourself.] Priest writes:
To discuss these issues, it will be convenient to divide the paradoxes into two families: the semantic and the set theoretic. The former comprises the paradoxes of truth, denotation, predication, and so on (the liar, Grelling’s, Berry’s, Richard’s, Köenig’s, etc.). The latter comprises the paradoxes of membership, cardinality, etc. (Russell’s, Cantor’s, Burali-Forti’s, Mirimanoff’s etc.). The received wisdom on the subject, dating back to Peano, is that the two families are quite distinct, the former belonging not to mathematics but to ‘‘linguistics’’. Since the advent of mathematical semantics, and of Tarski’s | definition of ‘truth’ in a set theoretic metalanguage, etc., this distinction has become virtually impossible to draw satisfactorily. There is also an obvious formal isomorphism between the abstraction scheme of set theory and the Tarski satisfaction scheme:
x ∈ {y|α} ↔ α(y/x)
x satisfies α ↔ α(y/x)
where a is a formula with one free variable, y, α(y/x) is a with all free occurrences of ‘y’ replaced by ‘x’ (with the usual precautions concerning clash of variables taken), and underlining is used for quotation. With a little ingenuity, we can extend the isomorphism to the case where α contains free variables other than y. Moreover, under the isomorphism, some of the semantic paradoxes transform into some of the set theoretic ones and vice versa. For example, Grelling’s paradox and Russell’s transform into each other. It is not surprising, therefore, that we have witnessed a number of papers resurrecting Russell’s original view that there is really only one family here.
(9-10)
However, Priest will still keep this distinction, because a) some set theoretic paradoxes have no equivalent in semantics, and vise versa, and b) at least in the eyes of mathematical logicians, set theoretical paradoxes have solutions while the semantic ones do not. (10)
__________________________________________
Note: The original version of the blog post was revised after I worked more on Tarski and basic set theory concepts. The asterisk above marks the place where the text below was deleted from the original version:
* [The following section is technical and difficult for me to grasp. Let us work through it slowly, and I invite your corrections for improving our grasp. I first will quote from Wilfrid Hodges’ Stanford Encyclopedia article “Tarski’s Truth Definitions”.
The two standard truth definitions are at first glance not definitions of truth at all, but definitions of a more complicated relation involving assignments a of objects to variables:
a satisfies the formula F
(where the symbol ‘F’ is a placeholder for a name of a particular formula of the object language). In fact satisfaction reduces to truth in this sense: a satisfies the formula F if and only if taking each free variable in F as a name of the object assigned to it by a makes the formula F into a true sentence.
(Hodges)
So let us now look at the Tarski formulation in the material that we will quote below.
x satisfies α ↔ α(y/x)
Keeping with Hodges’ explanation, this would seem to mean something like the following. We begin with a sentence in our object language, for example, y is red. We want to know if x is red is true. We would know that if all our substitutions of x in for y are true. But we are dealing with an object language and a metalanguage. α is supposed to be alpha with quotes, “α”, which would mean it is in the object language. So consider if our sentence is “y is false”, and y can be some sentence. Then,
x satisfies “y is false” if and only if all substitutions into the object language of x for y are true.
A problem might arise if we want to refer to that very sentence itself. Then we would have “this sentence is false”. Perhaps because we are substituting x into the object language itself, we then have that sentence refer to its very self.
Priest says that this Tarski satisfaction scheme is isomorphic with the set theoretical abstraction scheme. I suppose this means that the two formulations behave identically in the sense that given equivalent ‘inputs’ with equivalent relations, the outputs share the same structure of parts and relations. In our situation here, that would seem to mean that the same sorts of problems can result from both formulations when given equivalent inputs. The other formulation reads:
x ∈ {y|α} ↔ α(y/x)
I am guessing this might mean that x is a valid part of sentence α (which has the free variable y), if and only if we can substitute x in for y. I request a better explanation. But what I gather is that because these formulations are isomorphic, the same problems given a semantic expression can also be found when given a set theoretical expression. Please read the following to interpret it for yourself.]
Citations from:
Priest, Graham. In Contradiction: A Study of the Transconsistent. Oxford/New York: Clarendon/Oxford University, 2006 [first published 1987].
Or as otherwise noted from:
Hodges, Wilfrid. “Tarski’s Truth Definitions.” In The Stanford Encyclopedia of Philosophy.
http://plato.stanford.edu/entries/tarski-truth/
Tarski, Alfred. “The Semantic Conception of Truth and the Foundations of Semantics”. In The Nature of Truth: Classic and Contemporary Perspectives. Michael P. Lynch, ed. Cambridge, Massachusetts / London: MIT, 2001, pp.331-363.
A hyperlinked online version can be found here:
http://www.ditext.com/tarski/tarski.html
Hardegree, Gary. “Basic Set Theory”. A course text for his class “Philosophy 595 - Formal Semantics”.
http://people.umass.edu/gmhwww/595/text.htm
http://people.umass.edu/gmhwww/595/pdf/set%20theory/Set-Theory-Chap0.pdf
