Ramsification and Semantic Indeterminacy : What if the meaning is indeterminate? Can we do classical semantics as we traditionally want? The answer is Yes. Classical semantics = 'for evey number x, Prime (x) or not Prime(x)' is true It presupposes the existence of a uniquely determined intended interpretation. It is a meta-semantic presupposition. And here is where the problem lies, because for math, and scientific language there is no uniquely determined intended interpretation. So how come that classical semantics is still so successful? There is semantic indeterminacy concerning extensions of predicates but also in intentions of predicates. You can ramsify the extension or intentions. So according to Putnam, if you point at a tiger (d) and there is this other object (d') that is similar physically to the original tiger since d falls under the extension of the tiger then the d' does also. So here the idea is that we get from intentions to extensions. And the intended admissible interpretations are only one, that is what makes it classical. This is semantic determinacy, but in reality, it is wrong to do such presupposition since there is semantic indeterminacy. So what if the meaning is indeterminate? Vagueness is one example of semantic indeterminacy, like the predicate "Bald". Here there is more than one intended interpretation. Another set of indeterminacy without vagueness in the second-order language of arithmetics. And you are a structural-semantic. They claim that the interpretation of arithmetical symbols is only determined up until isomorphism. So the set of all admissible or intended interpretations of the language of arithmetic is the set of all interpretation mappings F that satisfy the second order Dedekind-Peano axioms for arithmetic. They proved that the second-order Peano axiom for arithmetic is categorical. So they pin down the intended interpretation of the arithmetical symbols up to isomorphism. And so if you collect all the interpretation matrix that satisfy these axioms you will get lots and lots of isomorphic of the second order language of arithmetic. And if you are a structuralist about arithmetics you want to say everything that's to be saved about arithmetic is structure anyway. The proper mathematical content of arithmetic is given by the structure of the natural numbers and nothing else. It is unimportant which sets of objects realize the structure, it's just the structure that counts, so the theoretic symbols meaning is only given up to isomorphism, there is no uniquely determined interpretation, but in fact, you can prove that there is infinite pairwise isomorphic set-theoretical interpretation. So they are semantically indetermined. So only their structural content is fixed but there are lots of set-theoretic ways of realizing that structure and all these interpretations that satisfy the structure (second order axioms), count as members of the set of admissible or intended interpretations. min 43 The 3rd example is about language in Newtonian mechanics. The term mass as being used by Newtonian physicists is semantically indeterminate if looked at from the modern relativistic point of view, as Field argued. So we have two meanings for the term mass and we do not know which one is the correct one. The intended interpretation of mass could either be relativistic mass, that is the total energy of a particle divided by the square of the speed of light. Or the 2nd interpretation coincided with proper mass, that is the non-kinetic energy of a particle divided by c square. Both interpretations save Newtonian mechanics and their experiments. So there is no fact of the matter which interpretation is correct, empirically speaking. Ramsey Semantics to the Rescue: It can preserve classical truth and logic. Accept everything that you find in classical semantics but do not assume there is a uniquely determined interpretation, meaning allow for semantic indeterminacy. Allow the cardinality of Adm (admissible/intended meaning), it might be one set or an infinite set of members, we leave it open. min 55 The idea now is to take those terms in the classical semantical package that you can view as theoretical terms, and those are the term's intended interpretation and the term the truth predicate. In the classical package, the intended interpretation is a member of adm, basically, it conforms to the way we use the term. Now we wanna ramsify them, which means that we replace theoretical terms with corresponding variables of the same type, and then the resulting open formula you existentially quantify ( which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃). What we replaced is the true predicate by a variable and the intended interpretation by another variable. 1:02:00 So the idea of ramsification is to get rid of semantic determinacy. Ramsification only presupposes that there exists a classical interpretation F that conforms to all existing metasemantic constraints. So all it is saying is that there is an intended meaning but maybe we do not know it. So it allows for semantic indeterminacy. We come back to our examples. 1-Newtonian mechanics, let Adm=(L1, L2) L1 (mass) coincides with relativistic mass (total energy/C square) L2 (mass) coincides with proper mass (non-kinetic energy/C square) L (the intended interpretation of mass)= L1 or L2 You can derive such truth using the Ramsey system without deriving both disjunct. So your semantics leaves open which one of them is the right intended interpretation because there is no fact to the matter. Imagine you are a scientific realist, and that there is continuity between old science L1 and new science L2, then it is consistent with my approach. And also it is consistent with antirealism. Carnap says, suppose we use temperature as a theoretical term, and now we specify the meaning of temperature by some postulates, ways to determine the temperature experimentally. This is in 1900, now suppose we are 1910, the new scientists specify the term temperature by further laws, and further ways to determine temperature. Then Carnap wants to say, they have specified the meaning of the term temperature in more ways, more specific about the term than they have been previously. So he is saying that scientific terms' meaning is open-ended, open to extension, and can be narrowed down during scientific progress. This is conceptual progress by finding more about the term in science. So for many fragments of natural, mathematical, and scientific language, there is no uniquely determined intended interpretation. End