Fundamental problem in the philosophy of logic

The fundamental problem in the philosophy of logic is understanding the nature and meaning of formal logic, that is,  so-called mathematical or symbolic logic.

The key notion involved is that of self-representation and self-reflection.

We have informal but rigorous proofs concerning abstract axiomatic systems. Then we have abstract axiomatic systems representing reasoning and proof concerned with abstract axiomatic systems. But then we must prove that a given structure is a proof of a proposition in the same way we prove a proposition in the object axiomatic system. And we require an abstract axiomatic system to reason about proofs in the deductive system - or to prove soundness and consistency.  But how do we prove that what we informally can prove we can also formally prove ?

In order to carry out deductions we must have the concepts of rule and what it means to apply a rule correctly. Likewise we must have the concepts of game and goal. The concept of rule is tied to logic and computability. 

The concept of game includes counting, computing and reasoning.

Kant's question: how is pure mathematics possible ? should not have gone the way of synthetic a priori intuitions but rather to the question: how is formal mathematical proof possible ? That is, how would Leibniz's characteristica be possible ?

Hilbert's treatment of geometry vs. Kant.

Another problem involves the countability of linguistic expressions vs. the possible uncountability of objects.  It follows that there are uncountably many indefinable objects which hence cannot be uniquely identified. Any property they have they must share with other such objects.

We find  the term 'sociologism' very apt to describe the 'linguistic turn'  (meaning-as-use, inferentialism) of Wittgenstein, Ryle, Austin and it continuation in Sellars, Brandom, etc. There is a strict parallelism with the earlier psychologism. It is likewise untenable. It is part of the physicalist assault against the mind, consciousness, individually accessible knowledge and truth (for example a priori moral, logical and mathematical truth) and moral conscience and freedom. It is a pseudo-scepticism and pseudo-relativism/conventionalism  and is ultimately nonsensical. It is reductionism (grabbed from neuroreductionism and functionalism) and is circular.  While sociology is a legitimate scientific discipline, sociologism is not based on science and is bad philosophy.

The idea that meaning of the term 'and' can be given by exhibiting a rule does not appear to be very cogent.

A: What does 'and' mean ?
B. That's simple. IF you postulate a sentence A as being true *AND* a sentence B as being true THEN you can postulate that the sentence "A and B" is true (and vice-versa).
A: I asked for you to define 'and' and you gave me an explanation that uses 'and', 'if...then', 'being true' and the concept of judgment. Sorry, that just won't do ! 

 It is also obvious that A may be possible to infer from B but that a person that accepts A is not sociologically obliged in anyway to state or defend B, for example, Fermat's last theorem before its proof by Wiles.  Any adequate language for fully describing the full range of sociological behavior, norms and practices is at least Turing complete.  So appeals to sociology cannot be used to furnish foundations for either logic or language.

Sociologism stands Frege on his head. It is a transposition to the social plane of the false dogma of functionalism and behaviourism.

Given a sentence S we can consider the recursively enumerable (but not recursive) set I(S) of all sentence which can be inferred from S in a system T.  Clearly I(S) cannot count as the meaning of S. Elementary number theory abounds in statements involving only elementary concepts the truth and inferentiability of which is not known.

Recommended reading: C. W. Mundle - A Critique of Linguistic Philosophy (Oxford, 1970).

Another strand of linguistic philosophy which seeks to undermine the certainty, clarity, objectivity and a priority of knowledge has roots in the later Wittgenstein's theories of polymorphism and his assault on definitions and meanings (but see the discussion in the Theatetus). In its current form it revolves around what we call 'the cult of vagueness'.

The cult of vagueness attempts to undermine the clarity, precision and non-ambiguity of language, and most importantly the language of philosophy, ethics, psychology - not to mention logic, mathematics and science.  Two of its sources are the  'paradoxes' and obvious peculiarities of certain natural language elements, specially the more homely and down-to-earth terms like 'bald' and 'cup' - there is nothing strange about certain adjectives having a trifold decomposition.  Of course to do this it has to assume a certain doctrine about language and its relation to the mind and the world.

The meaning of a property can be crystal clear and yet the application of the property can be difficult and uncertain. And it is only uncertain because the meaning is clear.

The cult of vagueness has its own peculiar rhetorical style which involves never stating one's assumptions clearly but only insinuating them.  

Erroneous theory of 'semantic relations' including 'speech acts' like 'whispering'.   What do they mean by act (and old Aristotelean metaphysical concept)  ? And whispering is a quality of speech not a semantic relation. For instance 'Mary whispered the nonsense spell she read in the book' has no semantic component. 

Anna Wierzbicka's distinction between folk and scientific concept demolishes the cult of vagueness.  Our low level concepts do not have definitions in the technical sense, they have stories. They are also dynamic and socio-specific.  Thus it is a category mistake to concoct arguments which ignore this distinction.

Linguistics depends on psychology and the philosophy of mind but these last depend on language.

Most adjectives and many nouns are not analogous to mathematical properties such as 'prime number'.  Negation functions differently. Often the adjective property has a tripartite structure, for instance 'tall', 'short' and 'medium height'.  Thus is somebody is not tall is does not mean they are short.  These folk concepts (having the possibility of a fair range of adjectival and adverbial degree modifiers) can give place to scientific ones which generally will involve scale, a measure.  Temperature is measured by different instruments. There is a limit of precision and variations across measurements by different instruments or the same instrument at different times.  But this does not make the concept of temperature vague or ambiguous. In fact statistical concepts are not vague even if as properties they cannot describe the state of a system in a unique way.

We can transpose Gödel's arguments to Zalta's Object Logic.  Instead of numerical coding of formulas we use the encoding relation for properties and objects.  We can thus define predicates for an object codifying only a certain property, only a certain sentence, and only a proof of a certain sentence $Proof(p,a)$ where $p$ is to be seen as codifying a sequence of sentences.  Then we can define $Diag(a,b)$ iff $a$ encodes the proposition $Bb$ where $b$ encodes only property $B$.  Then we can construct the Gödel sentence by taking the formula $G$ (property) $\lambda z.\mathrm{\neg }\mathrm{\exists }x,yProof(x,y)\mathrm{\&}Diag(z,y)$ which is encoded by $g$ to construct the Gödel sentence $Gg$.

Consider a reference relation between expressions and objects. Suppose that there were uncountably infinitely many objects.  Then:

i) either there are objects which cannot be referred to by any definite description

ii) or there are objects which share all their properties with infinitely other objects (indiscernability)

Or infinitely many objects with one binary relation. There are uncountably infinitely many possible states of affairs which cannot thus be referred to in a unique way. The same argument applies.  And of course arguments involving categoricity.

"Speech acts", the vagueness of ordinary terms...this is already found in Husserl's Logical Investigation (see for instance vol II, Book I). And previously in Benno Erdmann. 

Meaning and psychology: the great question.  Consciousness is so much more than the lower sphere of (mainly audio-visual) fantasy and imagination processes.  When we think of the concept of prime number or the concept of 'meaningless sentence'...and of course there is the Fregean view.

Multiplicity of psychological experience in the meaning phenomenon. But we can abstract a type, a species of what is invariable. Husserl is lead from here to ideal objects à la Frege, the space of pure meanings. But in the first Logical Investigations when Husserl discusses the psychological content of abstract expressions, how these are very poor, fluctuating and even totally non-existent and hence cannot be identified with meanings. But Husserl mentions the hypothesis of a rich subconscious psychological content involved. What is going on really when we think of "prime number" ? Do we have a subconscious web of experience reaching back to when we first learnt the concept ? And could not all this ultimately correspond to a kind of formal rule such as : if a divides p then a is 1 or p,  or if a is not 1 or p then a does not divide p ? There is nothing social here or only in the most vague and general way. An extended and rectified Hilbertian view can be seen as depth phenomenology perhaps, specially in light of modern formal mathematics projects.

A priority, certainty, as well as intersubjective agreement - all this depends on recursion theory and arithmetic or its 'deep logic'. Logos is a web of relations which is not relative. 

Meinong's Hume Studies: Part I: Meinong's Nominalism

Meinong's Hume Studies: Part II. Meinong's Analysis of Relations

The deep meaning of Gödel's incompleteness theorem is the mutual inclusion of the triad: logic, arithmetic and recursion theory. 

Gödel's rotating universe.  Individual subjective time that parametrizes a path need to have any simple correspondence with cosmic time which implies a global foliation by hypersurfaces.