I wrote those notes in response to an essay of a classmates of the class on Philosophy of Logic by A. Marra during the3rd_Semester at theLMU. Many of the following passages are related to The Silent Assumption in Tarskian Semantics, an introduction on the distinction of object and meta language is present in the Mathematics First Thesis Concept.
Definability Issues with “PREM” and “SOLO”
Here I present an argument of mine against both “PREM” and “SOLO” argument by Russel. What I claim to hold for “PREM”, I claim it for “SOLO” too.
In this section I argue that even if one succeeds in finding one informal proposition of the kind of “PREM”, it would only prove a logic to be incomplete other than inconsistent in respect to natural deduction.
The “PREM” statement, sounds to me like defining an object in mathematics such that, if it is sunny and if it’s cloudy. Against the existence of such an object, or, more precisely, against the existence of the function (the interpretation in the “PREM” case) that maps to either or , I would argue that there is no proposition in (language of mathematics, namely of for more see Wikipedia such that it is true if it is sunny and false if it is cloudy.
Also, I do not see why one should start the research for “PREM” within the natural language instead of the formal system itself. I believe that the core assumption here is that (I): any argument valid in the natural language shall be interpreted (hence be either valid or invalid) one formal logic. Though, I am more tempted to doubt this assumption than the one that there is at least one correct logic. In fact if one is to say that (I) does not hold and (II): there is one statement, call it “PREM” that is true iff. it is a premise, then it may still be the case that it does not get mapped to any formal statement. On the other hand, I could think of several arguments that many hold for true in some informal ways but that, on a formal level, lose any of their validity (not that they are invalid, but simply cannot be written in the language). Considering the former example, I could informally correctly state that if then it is sunny, though the formal value of this statement would be the same as the one of random characters. On the other hand I see that one may wish to have at least one logic such that (I) holds, namely a “complete logic”.
Now, it might be the case that the author agrees that the refusal of (I) is equivalent to logical nihilism, if that were the case, it would seem to me to be a natural conclusion. On the other hand it seems to me far weaker than the attempts of the “PREM” argument, since coming to the conclusion that there is one false argument in any logic, like instantiated on “PREM”, would prove any logic to be false (i.e. inconsistent with informal deduction) instead of simply being incomplete in respect to informal deduction. And, arguably, one prefers incompleteness over inconsistency (the same choice one has in the Gödelian Incompleteness Theorem).
I believe that the argument I exposed in this section is highly influenced by the Tarskian Semantics and mostly by the distinction of meta- and object-language, I would therefore claim the informal statements to be metalinguistic and the formal ones to be in the object language, and I find it totally normal to have some arguments that are true at the meta level but not expressible in the object-language. For instance, “this is a false statement” is such a statement, it cannot be phrased in the object language but can be defined only through a deeper analysis of the truth predicate (which takes place in the meta level).
02.06.2024: After reasoning on the paper and discussing it, I got the impression that I didn’t really understand it.