Philosophical miscellany

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 ?

6. 

Gustavo Augusto Fonseca Silva has written a very interesting monograph of Wittgenstein in the tradition of the following previous works.

 

Ernest Gellner, Words and Things: A Critical Account of Linguistic Philosophy and A Study in Ideology, with an introduction by Bertrand Russell, Beacon Press, 1960.

C. W. K. Mundle, A Critique of Linguistic Philosophy: with Second Thoughts - An Epilogue after Ten Years, foreward by P.L. Heath, 2nd edition, Glover and Blair, 1979.

Aaron Preston, Analytic Philosophy: The History of an Illusion, Continuum International, 2007.

Uwe Meixner, Defending Husserl: A Plea in the Case of Wittgenstein and Company Versus Phenomenology, De Gruyter, 2014.

Mark Steiner, Mathematical Knowledge, Cornell University Press, 1975.

J.N. Findlay, Wittgenstein, A Critique, Routledge and K. Paul, 1984.

Alain Badiou, Wittgenstein’s Antiphilosophy.

Laurence Goldstein, How Original a Work is the Tractatus Logico-Philosophicus ?

J.W. Cook, Wittgenstein’s Metaphysics, Cambridge, 1994.

J.W. Cook, Wittgenstein, Empiricism, and Language, Oxford, 1999.

Note on Tabak's Plato's Parmenides Reconsidered

M. Tabak's book Plato's Parmenides Reconsidered (Palgrave Macmillan, 2015) is a breath of fresh air in the ocean Platonic and specifically Parmenidean literature which is characterized not only by the insoluble difficulty of the subject matter, but by a certain propensity to a heavy philosophical hermeneutic bias which is ultimately more informative about the philosophical ideas of the author than about Plato's precise intention and method in writing this puzzling dialogue. The original nature of Tabak's thesis - that of the ironic, parody-like - even light-hearted - content of the second half of the dialogue, and the equation of Zeno's and Parmenides' argumentation to that of the sophists criticized in the earlier Platonic dialogues - as well as his according due importance to the briefer treatment of the same questions in the Sophist, certainly invites and a more neutral and lucid approach.

One of our interests in the second half of the dialogue involves the following question: is it possible to formalize in detail the arguments therein in a system of modern logic (including mereology) ? In particular, is this possible for the first part of the second of the eight arguments ? 

Tabak makes some very interesting observations on this last matter. Basically he says that if A is a part of B then since A is a part then A participates of unity and since A is something it participates of being. We can state this in the general case as

\[\phi x \rightarrow (Ux \& Bx) \tag{1}\]

It also clear that Plato is assuming that if $\phi x$ then $x$ participates of something (the form corresponding to $\phi$) and if $x$ participates of something then it has a (proper ?) part ($Pyx$).  Let us just state this very weakly as

\[ \phi x \rightarrow \exists y. PPyx  \tag{2}\]

where $PP$ denotes proper part which excludes the cases $PPxx$.

Now the hypothesis of the second argument is : if the one is.  Let $\odot$ denote the one and let us state this hypothesis as

\[ B\odot \tag{H2}\]

It seems we could use 1, 2 and a convenient supplementation principle to show that

\[ \exists uv. u\neq v \& PPu\odot\& PPv \odot \& \neg Au\& \neg Av \tag{T1} \]

and in particular that $\neg A\odot$.  But the problem here is that in 2 we have no guarantee that the parts of $x$ corresponding to two different predicates (either logically or syntactically or intensionally) will be in turn different.  Since $U$ and $O$ are both different and apparently co-extensional, this might be difficult to formulate.  We note that passages in the Parmenides as well as others in Aristotle's Topics already anticipate some of Frege's later distinctions.  One solution is to introduce a form-forming operation $[\phi]$ and an axiom scheme guaranteeing fine-grained distinction

\[ [\phi]\neq [\psi] \tag{Fg} \]

where $\phi$ and $\psi$ range over syntactically distinct formulas with one free variable. We replace 2 by

\[ \phi x \rightarrow  PP[\phi]x  \tag{2'}\]

Using H2 and 1,2' we can now derive T1 directly as well as stronger results closer to Plato's conclusion.  The problem with this solution is that Fg is already postulating an infinite number of distinct entities so the Platonic conclusion of $\odot$ having an infinite branching tree of proper parts is no longer too surprising. An even more serious problem is that the same form will be a part of completely distinct entities so that there will be universal overlap, $\forall xy. Oxy$. One solution would be to introduce a function symbol $pxy$ which yields the participated mode of form $y$ for $x$. In other words, we replace 2' by

\[ \phi x \rightarrow  PP(px[\phi])x  \tag{2''}\]

and replace Fg by

 \[ px[\phi]\neq py[\psi] \tag{Fg'} \]

 where $\phi$ and $\psi$ range over syntactically distinct formulas with one free variable.

Pyrrhonian strategy in Rorty's Mirror of Nature

Regarding Rorty let us quote from J.N. Mohanty's The possibility of transcendental philosophy (1985) p.59 :

Impressive as he is in his scholarship, he has given very few arguments of his own. He uses Sellars' arguments against the given and Quine's against meaning, as though they cannot be answered, but he has done little to show they cannot be. He plays one philosopher against the other, and would have one or both dismissed, according as it suits his predelineated moral. These are rhetorically effective but argumentatively poor techniques. What does it matter if Sellars rejects the concept of the given - one may equally rhetorically ask - if there are other good philosophers who accept the viability of that concept? There is also an implied historicist, argument that has little cutting edge. If the Cartesian concept of the mental had a historical genesis (who in fact ever wanted to say that any philosophical concept or philosophy itself did not have one?) whatever and however that origin may be, that fact is taken to imply  that there is something wrong about the concept.

On Susanne Bobzien’s groundbreaking discovery in Frege and Prantl

https://handlingideas.blog/2021/02/05/the-stoic-foundations-of-analytic-philosophy-on-susanne-bobziens-groundbreaking-discovery-in-frege-and-prantl/

Why we need formal ontologies

 Consider the sentence I wish I had been born earlier. This implies that we are considering possibilities in which the same individual $A$ had a different life-history in a different possible world. By life-history we mean the total sequence of states, actions and events involving $A$ (all a part of a world $W$) during a certain time interval L from the time of birth to the time of death. 

 For a world $W$ can the life-history of an individual $A$ be deduced or defined ? Or is the identity of an individual $I$ fixed  a priori ? Why do two life-histories corresponding to two different worlds correspond to the same individual ? And even for the actual world how could we interpret prexistence or reincarnation or survival in some different form all in the same time-line of the actual world ? What is that which guarantees the identity and continuity of such different embodiments ? 

 Let each world $W$ is endowed with a partition according  to world-time $T$ (of which the $L$ of an individual is a subinterval). Thus we speak of the world $W_t$ at time $t$. The relationship between possible worlds will reflect a branching structure for $T$.  Given a possible world $W$ and time $t\in T$, we can consider a set of possible worlds $W^t_1,...W^t_n$ which coincide with $W$ up to $t$ and then begin to differ.  

This same construction carries over to life-histories of individuals.  It gives reasonable sense to sentences of the type I wish I had not made that decision. Of even statements of an individual $I_1$ regarding an individual $I_2$: if $I_2$ hadn't died young he might have become famous.  This is expressed by the branching structure on possible life-histories in possible worlds which coincide up to a time $t$.

But there is a great asymmetry involved in If he had been born earlier he would have been able to have met Bertrand Russell. Without a previous theory of individuation and identity we are forced to conclude that such an expression is meaningless.  But a theory of individuation and identity means some kind of more or less sophisticated formal ontology and even a  'general systems theory'. We consider the idea of a 'rigid designator' and thought experiments such as the 'twin earth'  problematic.

Mereology is also connected to such a formal ontology. Here are some older notes and sketches illustrating what a formal ontology or general systems theory might look like.