1. Arguments for logical pluralism are at times not very impressive. It is the idea that logic is an arbitrary collection of rules which for some reason were static for milennia until the revolutionary discovery that these are arbitrary and conventional and that now everybody can have their own preferred logic...and at the same time it is claimed (rather incongruently) that one's choice of logic is not metaphysically neutral. The pet example is the 'law of the excluded middle'. But the fact of the matter is that this particular example completely fails as an argument for logical pluralism/conventionalism. The relationship between classical and intuitionistic logic exhibits abundantly many characteristics of surprising pre-established harmony, mutual interpretability, conceptual refinement (or subsumption if you prefer) and the sharing of common structural-conceptual spaces (i.e. topos semantics) which would not be in the least expected if classical logic where just an arbitrary collection of rules in which one rule was arbitrarily changed. And the same goes for minimal logic which jettisons the negation rule(s) completely. To put it in other terms: classical and intuitionistic logic are brother and sister, not complete strangers or a pair consisting of a human and an alien. Intuitionistic logic is of immense mathematical interest and used in a wide range of applications in computer science. What other tinkerings or changes of classical logic - or any deviant logic for that matter - can even remotely compare to this situation ?
2. What are meaning-as-use theories but attempts to ground logic and language in sociology ? That logic and language factor in importantly in any cogent sociological theory has always been patent. That the study of the social and functional aspects of logic and language is legitimate and important nobody would doubt. However all this is a far cry from a claim of setting up a sociological and behavioristic reductionism as a standpoint to level arguments against logical realism or other theories of logic. This stance is circular from the start because in order for us to be clear about sociological matters we need forcefully to deploy a wide range of sophisticated formal concepts and axioms (for instance those pertaining to general systems theory) which can only be couched in formal logic and mathematics. And if we stick to ordinary vague natural language terms to attempt to describe complex sociological behaviors, changes, interactions and emergent structure then what do we have but a caricature of medieval or Aristotelian science ? It is no use saying that word A is explained by language-game B if we lack the formal conceptual and axiomatic apparatus to analyze and classify the language-game. If we lack a rigorous way of formulating descriptions of language-games then how can we ever hope to be able to test our theories or confront them with the empirical facts of human behavior ?
3. Suppose we put forward the theory that the 'meaning' of a proposition P in a system T is the set M of proofs of P in T. So is T- 'meaning' the whole set M itself or is each proof in M a possible T-meaning of P among others ? And given a proof p what criteria do we have that it is in fact a T-proof of P, a member of M ? What is the meaning of the statement 'p is a proof of P in system T' ? This is generally accepted in virtue of some kind of intuitive-conceptual process rather than by producing an elaborate meta-logical proof in some meta-system Z. So if the proof-as-meaning view fails at the meta-level why accept it at all at the basic level ? The meaning-as-proof theory does not seem prima facie more desirable or economical than rival views.
4. What is mathematical logic ? It is the mathematical treatment of formal systems, but mostly those formal systems of interest to the foundations of mathematics itself. Mathematics was done with a high degree of logical accuracy before formalism. The full force of logic, encompassing multiple generality, higher-orderness and even set theory was deployed in mathematical proof long before the advent of modern formalism, for example in Gauss' Disquisitiones Arithmeticae. So is mathematical logic the mathematical study of formal systems serving as a foundation of mathematics conducted in a pre-formal or semi-formal mathematical way ? That is, a reflection-into-self of mathematics ? The mathematical logician assumes various structural induction principles and sneaks arithmetic, combinatorics and even ordinal arithmetic under the table. The use of the term 'finitary' is questionable. Key theorems in first-order logic such as $\forall x. A(x) \& B(x) \rightarrow \forall x.A(x)~ \& ~\forall x. B(x) $ are actually theorem-schemes which could only be stated in second-order logic, and this argument might be repeated. Or can mathematical logic itself be conducted within a formal metasystem but discarding knowledge-claims to any significant properties of this metasystem ? A mathematician when thinking of a proof often skips many logical steps according to a formal system which could formalize such a proof. In fact whatever formal system we choose we will find the mathematician skipping steps. And yet the mathematician does not invent new rules as he goes along. Does he process the skipped rules very quickly at an 'unconscious level' ? There is no evidence for this. Is he referring by memory and analogy to case were a similar step was in fact gone through in detail ? Metamathematical theorems about intuitionism or nonstandard models depend on their conception and proof on classical arithmetic, combinatorics and computation.
5. One dogma reads as follows: for every natural language $L$ there is exists a set $\Sigma$ of symbols and a subset $M \subset \Sigma^*$ together with a pair $(T, S)$ ,where $T$ takes expressions of $L$ into elements of $M, $ such that any meaning that can be expressed by an expression $E$ in $L$ can be given as the assignment under $S$ of a unique expression expression $E' \in M$. In other words: natural language can be disambiguated. Furthermore whenever natural language is used such a disambiguiation is actually somehow effected internally at the level of expressions rather than meaning, even if this last is not registered phonetically or graphically. We propose that each speaker may have their own particular disambiguated language. While meaning is objective and extra-linguistic whenever a sequence of signs is presented as part of a language then we must relativize to a given speaker, perhaps through a proper name or some sort of description or indexical device. We have degrees of variability of languages from individual to individual and we can hope for close or rough correspondences for members of the same socio-linguistic groups. The fact that we learn other languages not by syntactic transformation but by direct attributions of meaning (or transference of meaning) to a new system of signs is of great philosophical importance. We propose the question: how can two different speakers determine if they are using a given collection of signs in the same way in terms of meaning-attribution ? The question of identical reference (for instance for proper names) is even more difficult. What is exactly that which Peter means when he states that : Pierre means the same thing by 'poisson' as I mean by 'fish' ? In general we could expect definite descriptions to be still liable of being conditioned by the different culturally determined semantic webs of the two speakers. Is modern formal-axiomatic mathematics arare example of a 'universal language' ? How could the logical pluralist account for the clarity and universality of mathematical language if one's mathematical concepts and understanding rested on one's particular logic ?
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