Observations on CIL

The individual concept of an individual is not a kind of infinitary logic conjunction of descriptions which denote this individual. For each one of such descriptions may be modally conditioned. Rather it is a special essence which describes what is unique, essential and invariant to the individual under any state-of-affairs or possible world.  But this essence perhaps does contain contingent definite descriptions suitable specified by the state-of-affair. In this way the description 'the tutor of Alexander' plus the data of the actuality of a certain possible state-of-affairs (world history or something of the sort) are enough to attach the sense of the description to the individual concept and thus to reference the individual.

For Frege the reference of propositions are truth-values, for Husserl they are states-of-affairs or a kind of equivalence class of such, situation-of-affairs.  CIL takes a kind of medium ground, although the extension function for propositions yields truth values, many equivalence classes on propositions can be defined which might accommodate Husserlian references.

The concrete world, the spatio-temporal world,  the world of generation and corruption, the realm of shadows and images, of change,  uncertainty, possibility, vagueness and indetermination. When we move from the realm of pure sense, of ideality, to this world, what must happen ?  

Physics has its method.  This concrete world is captured through the geometry of phase spaces.  In philosophy we should consider moduli spaces, spaces of deformations and variations of structures. What is is what can be defined in a moduli space.  Thus an individual concept is a logically determined form within a space of possibilities $\mathcal{S}$. This form is perhaps continuous, connected and compact in $\mathcal{S}$. Thus the individual concept of 'Aristotle' is a continuous spectrum of individuals spanning deformations of the current world sufficiently like it - its domain of definition.  Thus a definite description is like an indexical, you must input a world in a domain of definition. 

We have at last decided to let $D_{-1}$ consist of individual concepts $i$ for which $Hi = i$. Then we have a function

\[  desc: \mathcal{H} \times D_{-1} \rightarrow \mathcal{P}D_1 \]

which for a given state-of-affairs $H$ and individual concept $i$ yields the set of properties which characterize $i$ uniquely in $H$.  There are situations $H$ in which the individual concept $i$ does not have an existence or embodiment, this is translated by the fact that $desc(H,i) = \emptyset$.

Non-denoting strict proper names: another pseudo-problem

When we use the expression 'Pegasus' not only does it have a sense but it has a well-defined (though context-based) referent which is never the literal embodiment of the sense (the mythology).  The referent of Pegasus can be for instance determined as a figure in work of art or a character is a story (we exclude the case in which an actual horse has been given the name Pegasus).  In the real world uttered expression 'Walking home from work I came across Pegasus and gave him a carrot.'  the sense of Pegasus cannot be the same sense as Pegasus in the context of the pseudo-problem of non-denoting strict proper names.  Also observe the difference between 'Imagine if I could meet Kant' and 'Imagine if I could meet the Easter Bunny'. 

Are extensions denotations ? Then an empty set is a legitimate denotation which is not the same as not having a denotation. Thus the property of being a round square has a denotation, namely, the empty set.

Problems with tense and modality in Topics 145b

 Aristotle asks if  the statement: at the present it is not possible for $A$ to be $B$ entails or not that it is not possible for $A$ to $B$ at all times.  That is, is the modality involved a virtuality of the present or is it inherently temporal as well, wherein 'being not possible' means at all times it is never the case that $A$ is $B$. If we take $P$ to be the present operator and $U$ to be 'at all times', then we have $PU\phi \rightarrow \phi$.  However there are sentences like 'at the present it is not possible for me to understand this'  in which the embedded modality has no temporal implication. In this case $P\square\phi$ does not imply that $U\phi$ or $U\square\phi$.

Deontic logic and acts

How do we formalize acts, in particular in the context of deontic logic ? For instance that an agent $x$ wishes to do something $y$ to another agent $z$ ?  Furthermore, how do we formalize that we should do something ? How do we formalize the language of acts in CIL  which does not really distinguish between state-relations and act-relations ? How do we express attitudes with regards to acts, not abstractly considered, but as applied to a given agent ?  But how do we quantify over acts ? If it is not even clear how we could formalize 'Do unto others as you would have them do unto you'  (the Golden Rule) how can we hope to formalize natural language or set up an axiomatic philosophy ?

We have abstract but complete acts, such as John's kissing of Mary, or , the kissing of Mary by John,  which are a kind of object in $D_0$ (i.e. sort 0 CIL terms).   We could distinguish these with a predicate $Act$.  Then there is the actuality or for the act, the state-of-affairs that the abstract act is actually occuring (now, or at a specified time). For this we use a further predicate $\mathcal{A}$.  Furthermore abstract acts can be objects of propositional attitudes, but it seems that it is the actual abstract act, or the actuality of the abstract act that is what is in most cases the object of the propositional attitude. Thus 'Mary wants to be kissed'  is not a relationship between Mary and the abstract complete act of Mary being kissed by somebody, but rather between Mary and the actuality of such an act. Can we consider an actuality operator $act$ which transforms the sense of an abstract act in this way ? 

Let us then try to formalize the Golden Rule, which we parametrize as a predicate $G(p)$ for a person $p$. If $x$ is incomplete abstract act (like 'kissing')  (determined by a predicate $iAct$) then $p.x.q$ denotes the complete abstract act obtained by specification to the agents $p$ and $q$.

\[ G(p) \equiv \forall x, q.  iAct(x) \rightarrow ( Wish (p, act( q.x.p)) \rightarrow Should(p, act(p.x.q)))\]

Of course our deontic 'Should' predicates needs to be clarified. What would it even mean if we said that $p$ should do an act in which $p$ is not even part of the act ? Perhaps we should write rather $Should(p, \lambda v. v.x.q)$.  It is a relationship between an agent $p$ and a semi-complete abstract act which expresses that $p$ is obliged to contribute to the actuality of the concrete act in which $p$ completes the act by becoming its agent. But how can an abstract complete act have a truth-value (as in CIL). Only the actuality of such can have a truth value.This suggests rather that the potentially actual complete act - bearing a truth value according to if such potentiality is actual or not - should be the default. The abstract complete act should be seen as the result of an operator which turns the previous $D_0$ element into something else which is no longer a bearer of truth-values. Cf. 'John is kissing Mary' and 'the kissing of Mary by John'.

Topics 148b

 Topics 148b deals with 'complex terms'. It appears a complex term can have a definition.  What is a complex term for Aristotle ?  The example presented, involving 'finite line' and 'straight line'  is obscure. But the idea seems to be something like if $\Sigma E = \Gamma \Delta\Delta'$ and $\Sigma = \Gamma \Delta$ then $E = \Delta'$, that is $\Delta'$ must be the definition of $E$. Aristotle speaks of 'compound notions' and of the number of elements in the definiens and the number of names and verbs in the definition. Most definitions must be given by phrases (i.e. not by simple terms). It is not in the least bit clear what Aristotle means be 'name', 'verb' and 'phrase'.

Topics 141a - 142b

In the previous post we left open the question: what does it mean for Aristotle to predicate something of something according to essence ?  And can a subject, as in modern mathematics, have more than one valid definition ?  Topics 141a provides a clue. Apparently a characteristic of essential predication is using 'anterior and better-known terms' (see 141b for an interesting discussion involving absolute anteriority and relative, according to us, better-known-ness).  Anyhow the uniqueness of definition is assumed as used by Aristotle as argument against definitions using terms which are not anterior and better-known.  For given such a definition we could replace the lesser-known term by a better-known term. The result would be a different definition. So there would be two definitions of the same subject which  contradicts the uniqueness postulate. Interestingly Aristotle speaks of the 'expression', the definition being equal to the 'expression'.  This is clearly not an expression in our modern syntactic sense. It must be its sense, its Fregean Sinn.  Aristotle rejects definitions using terms merely better known to us, rather than being in themselves anterior - or  absolutely better known. Such definitions would not express the essence of the subject.  It is curious how in this passage a point is seen as anterior to a line but a line better known to us than a point (this does not seem to be consistent with the later theory of the continuum in the Physics).  Aristotle states that if we know the subject then we know its genus and difference but not vice-versa. 142a is a strikingly insightful anticipation of Mates' puzzle, indeed the problem of the relativity of definitions. It shows the consequences if we take anteriority according only to a person or group of people. 

We conclude that the better-known relation $\triangleleft$ is not something odd, an intrusion of sociological factors, but one of the deepest and most purely logical concepts in the Topics. We might call it the relation of logical precedence. In 142a we get the rules: $\sim A \triangleleft A^\circ$ and $A,B < C$ implies that $\sim A \triangleleft B$ and $\sim B \triangleleft A$.

The discussion in 142b is one of the most important in all the Topics. It deals with the non-circularity of definitions.  A definition is proposed $\Sigma = \Gamma\Delta$. But $\Delta$ involves a term $\Sigma'$ whose definition involves $\Sigma$.  Aristotle proposed that we substitute in the definition of $\Sigma$ such a term by its definition to detect the circularity.  But what if there is no circularity. Is the resulting expression still a definition ? Does this contradict the uniqueness of definitions ? This seems contradicted by 143a when Aristotle seems fine both with a definition involving proximate genus and difference and a definition involving a further genus and all the necessary differences. For instance if $\Sigma = \Gamma\Delta$ and $\Gamma = \Gamma'\Delta'$ then Aristotle apparently accepts also the definition $\Sigma = \Gamma'\Delta'\Delta$.

A problem in the interpretation of CIL

What is, linguistically,  $[B([A(x)])]_x$, that is,  $comb_{(1)} B^{(1)} A^{(1)}$ ? Prima facie the only real linguistic interpretation we can think of involves propositional attitudes. $[B([A(x)])]_x$ is the property that an $a$ satisfies when it is the case that $[B([A(a)])]$. We could read it as: the property of  $[A(x)]$ being $B$. But could this be written $\lambda x. [B([A(x)])]$ or $\lambda x. comb_{(0)}Bcomb_{(0)}Ax$ ? 

Consider the following system called $\lambda$CIL.  As in CIL each term that is not a variable has a unique sort.  For each sort we have a list of primitive terms. We have a set of variables which are terms. Given a sequence  $s$ of  $m$ distinct integers for $m \leq n$, $m$ terms $T_i$ and a term $T$ of sort $n$ we have that $comb_s T T_1...T_m$ is of sort $n- m$.  Given a term $T$ of sort $n$ we have that $\lambda x. T$ is a term of sort $n+1$. CIL can be represented in $\lambda$CIL and vice-versa. But $\lambda$CIL is not suitable for CIL models.

CIL must be extended to be able to talk about its own syntax. Expressions of the form 'an A is a B' or 'A = B' convey information or might be even seen as definitions. If A and B are both constants then A = B conveys the meaning that the expressions 'A' and 'B' have the same meaning. Given a CIL term T we can write T' for the term whose sense is the expression T itself (not the sense of T). This may seem odd. But consider the definite description 'the letter in the Roman alphabet after 'a''. The denotation is contained in some way in this sense.

Analysis of the superlative in Topics 152a

Some genera 'admit degree' and some do not. Take an adjective admitting degree like 'desirable'. Of course this must be specified to desirable to a certain kind.  It might seem that an adjective admitting degree particularized to a given kind can be represented as a set $X$ and relation $<$ expressing comparison. Then the superlative (if it exists) is unique (152a). For this to be the case we might need the anti-symmetry property $x < y\,\&\, y < x \rightarrow x = y$. We define a superlative of $X$ relative to $<$ , to be an element $s \in X$ such that $\forall x \in X. x\neq s \rightarrow x < s$.  The problem with Aristotle's discussion is the expression 'the superlative', or rather for instance 'the most desirable'. Does this definite article already imply that there is only one superlative (if at all), so that the mentioned argument of Xenocrates is just a case of the topic stated further ahead, that involving two things equal to a third being equal amongst themselves. The only way to make sense of this argument is to cast it in the form:  $x$ is a superlative and $y$ is a superlative (of $X$) therefore $x = y$.  Notice that in category theory we also use the expression 'the product of $A$ and $B$' when in reality we are speaking not of a unique object (and pair of arrows) but of an equivalence class. Anyhow we have that the superlative is a case of category-theoretic construction, it is the terminal object in the induced category of the pre-order $(X,<)$. 

Now Aristotle's furnishes us with an interesting discussion of superlatives applied to pluralities (which might be seen to answer the objection above as well as present an early example of brilliant linguistic analysis, pointing out the misuse of the definite article and the assumptions implied). For instance the 'Spartans' and 'Peloponnesians' are both said to be the bravest of the Greeks.  What do such plural superlatives mean ?  How are we to make sense of Aristotle's argument that the Spartans being the bravest and the Peloponnesians being the bravest only means that one term is contained in the other ? The definition of $A$ being the supremum in $X$, for $A \subset X$ (let us write this as a predicate $Sup_X(A)$) seems to be stated explicitly to be $\forall x \in X. \forall y \in A. x\notin A \rightarrow y > x$. S

Suppose that $Sup_X(A)$ and $Sup_X(B)$ and that $A$ and $B$ were not contained in each other. Then there is an $a\in A$ such that $a \notin B$ and a $b\in B$ such that $b \notin A$. But then by definition $a > b$ and $b < a$ from which we derive a contradiction. Note the rather complex logical steps and rules for dealing with multiple implicitly  required in Aristotle's argument. But if for instance $A \subset B$ there is no contradiction. It only follows that the $A$ are the best within $B$.

Question: what exactly does Aristotle mean by a predicate 'expressing the essence' of a subject ? And is it possible for Aristotle for there to be two different valid definitions of the same subject ? 

In 153b we get an anomalous genus-difference constructions:  justice is the virtue of the soul, where 'of the soul' is the difference.  The context is as follows. Given a definition $\Sigma = \Gamma\Delta$ how does this relate to the definition of $\Sigma^\circ$ ? How do $\Gamma^\circ$ and $\Delta^\circ$ come in ? Sometimes the same $\Delta$ is used.  Justice and injustice are contraries, yet their definitions are virtue of the soul and vice of the soul. So it is $\Gamma^\circ$ that enters. Aristotle concludes that if we now the definition of $\Sigma^\circ = \Gamma' \Delta'$ then we know the definition of $\Sigma = \Gamma\Delta$, there being three cases. i) $\Gamma = \Gamma'$ and $\Delta' = \Delta^\circ$, ii)  $\Gamma' = \Gamma^\circ$ and $\Delta' = \Delta$ and iii)  $\Gamma' = \Gamma^\circ$ and $\Delta' = \Delta^\circ$. Here we assume that $X^{\circ\circ} = X$.

It seems that modifiers such a difference in the Topics are perhaps not quite correctly expressioned as conjunctions of properties.  In 155a Aristotle gives some basic quantifier rules.  Harder to refute a definition through reasoning that to construct one !  Property is closest to definition. Accident is just plain predication. Book VII seems to be part of an independent treatise.

Ancient relational logic in Scotus Erigena

Consider the general four-fold division of all things in book IV of the Periphyseon.  The relation is $C(x,y)$, $x$ creates $y$.  Then derived notions are 'creates (something)',  $C_1(x) \equiv \exists y. C(x,y)$, 'being created (by something)', $C_2(x) \equiv \exists y. C(y,x)$. Then the four categories are \[A_1(x)\equiv C_1(x)\& \sim C_2(x)\] \[A_2(x)\equiv C_1(x)\&C_2(x)\] \[A_3(x)\equiv \sim C_1(x)\& C_2(x)\] \[A_4(x)\equiv \sim C_1(x)\& \sim C_2(x)\]

In CIL we would have that $C_1$ is $log^\exists_{(0,1)} C$ and $C_2$ is $log^\exists_{(0,1)}per_{(1,2)} C$.

Adjectives and adverbs unveiled

We must not confuse grammatical, logical and semantic aspects of language. This is a fatal error.  Take 'big'. This can be an attribute as in 'X is big' or  a modifier as in  'a big X'.  As an attribute it cannot be represented by a primitive unary predicate $big(x)$.  Because a semantic-phenomenological analysis reveals that: for X to be 'big' means that X is larger than the typical object Y in the kind K to which X belongs. Thus 'big'  does not correspond to a primitive unary predicate $big(x)$.  'Big' is dependent on the kind of the object to which it is applied and on the relation of size-comparison.  Thus to say something is big I implicitly posit:

1) That the object to which I am applying this attribute has a kind.

2) That this kind has a typical object.

3) That I can compare the size of the object with the size of the typical object of the object's kind.

4) That the above comparison shows that the object is larger.

We could ask Aristotle: can you use 'big' as a genus' difference in a  definition  ? It seems that to know big we already have to know the genus.

The same questions can be asked for adjectives. 'John is moving fast'.  'Fast' is not a modifier of 'moving' generating a new state  'moving fast' which is applied to John.  Fast, like for adjectives, implies that John has as kind (human being) and that this kind has a typical embodiment of motion to which John's motion can be compared.

Quantification of free variables in CIL

 Let $T(X)$ be a CIL term in which the variable primitive $X$ occurs (as an argument to a $comb_s$ substerm).  What is then the analogue of $\exists X. T(X)$ ? This must be a term of the form $log^\exists_s T'$ where $X$ does not occur in $T'$. The problem is the determine $T'$ given $T(X)$. Take the case of $T(X) := comb_{(0)} S^{(1)}\,X$. Then $T'$ is $S$ itself and we get $log^\exists_{(1)} S$. In general given the concept-graph of $T(X)$ we take the $X$-nodes and pull them to the bottom of the graph as open nodes then link them together. Then we apply $log^\exists$.

Meaning and proof

The meaning of a statement is not one of its proofs, because there might be others.  Not all of its proofs because we might not know all of these. And a proof consists in turn of statements which must in turn have meaning leading to an infinite regress.  If the statements of a proof do not have meaning, how can it be a proof ? And  proofs also contains relations between statements. For a proof is not just a bag of statements.  Then what are these relations and what is their meaning ?  

Axioms have meaning, but they are accepted as not needing proof. Proofs depend on there being axioms. So proofs are not meaning. 

Who could deny that Gödel's sentence still has meaning in PA without it having a proof ?

Do we need a proof that a certain thing is a proof ? Do proofs themselves have meaning and does this meaning not have to be in turn a proof ? And if it does not, then  meaning is not proof. And if proofs do not have meaning how can they constitute the meaning of a statement ?

A term as a mere string has no meaning, cannot be a meaning, and is not a proof. To interpret this string as a proof, that is, as a term in the $\lambda$-calculus, as an algorithm, we need theory. We cannot invoke proofs-as-terms without dependence on the meaning of mathematical statements. We just pushed the whole matter upstairs as we can ask meaningful questions about the theory of terms which in turn require proof.

Proof-theoretic semantics is false and untenable. Meaning is constituted by meaning-constructions applied to primitive meanings but the total meaning is seized as a whole. Proofs themselves have meaning which can be grasped as a whole. Linear logic, intuitionistic logic and classical logic show a profound convergence and inter-relatedness which counts as a devastating argument against logical pluralism rather than for it. Incompleteness does not imply lack of categoricity.

Nor are the meanings of connectives and quantifiers given by rules. Rules in which they appear ? Rules in in which they exclusively appear ? And what if there is more than one rule, is the meaning  the whole collection or is the whole meaning in each part  or is the meaning fragmented into the various rules so each rule only gives the partial meaning of the connective ? What about equivalent systems of rules ? Why are the rules accepted in the first place if not because their are judged to conform to the meaning of the connective ? And what is a rule in general but a partial recursive functions on sequences of strings. These can be given through Boolean circuits and other models of recursion. So we just pushed back the problem to studying the meaning of recursion theory. And what about embeddings into higher-order logic which allows connectives to be expressed in terms of a fewer number of connectives.  Is the meaning then dissolved ?

Let us look at the natural deduction rule for $\&$ (the same argument applies for the sequence calculus).  Suppose we wanted to teach somebody this rule, that we wanted to describe this rule.  Can we do so without using the word 'and' ? Indeed, can we describe anything or teach anybody how to do anything without the concepts mediated by the basic connectives (and specially intuitionistic connectives) as well as universals (senses, intensions, meanings, abstract ideas), that is, quantifiers not reducible to extensional, possibly infinitary disjunctions or conjunctions ? 

And how do we recognize the equivalence between different types of deductive system involving the 'same' connective ? We already need logic to describe and know the relationship between rule systems. Why do we accept that the cartesian product corresponds to conjunction and the disjoint union to disjunction (considered from the point of view of category theory) ?  Product and coproduct are presented differently than deduction or sequent rules. Yet we recognize the same concept and idea.

Practical philosophy

Primum vivere...Can practical philosophy lead us to the goal of philosophy itself, something that an abstracted compartmentalized philosophy cannot ? Sure, to refute popular philosophies, philosophies entrenched in institutions and upheld not by reason but by prestige and power,  is good and liberating.  But such Pyrrhonism does not suffice and is not enough to constitute a solid path, a modus vivendi. Language and the mind. These are indeed the great objects of investigation. But we need the right radical attitude, a praxis both practical and theoretical, a humility with regards to the elementary, lowly and the concrete which at the same time does not divorce  theory from  art, from ability. Why is it  so few can see logical analysis is lost without linguistic analysis ? Clarity of expression, clarity of meaning, would this not save us endless foggy labyrinths of pseudo-logical argument ? Linguistic analysis means doing linguistic analysis, it means living according to linguistic analysis in our everyday life-world. It means understanding the full significance of grammar  and lexicology,  including what relationship, attitude and work we ought to have, adopt and engage in.  Grammar itself requires an epokhe and a discarding of heaps of dogmatic theory - including undue focus on certain languages and on modern language at the expense of ancient language. A curious idea: a linguistic phenomenology in which we force meaning to manifest by trying to define - or tell stories about - concepts using only a fixed set of semantic primitives. The work of Anna Wierzbicka is very relevant here.

 As for the mind, we are quite lost. To approach the mind we must knock on other doors of temples in far-away places and times. We don't even know what it means to approach the mind and how theory is meaningless without practice.

On definition

Definitions, though logically and semantically constrained, are established for logical purposes. Many terms in natural language are attached to different senses by different people, including logically equivalent senses.  Thus the word 'circle' can have different senses for different people. In mathematics the same concept can be defined in different ways, the  equivalency between these different ways being often non-trivial. For instance 'Noetherian ring' in terms of i) satisfying the ascending chain condition or  ii) any ideal being finitely generated. A mathematician may well not know the equivalence between the two. So if we take the sense of Noetherian ring to be the sense of one of these definition we are lead to the 'paradox of analysis'. For instance from  '$A$ knows that a Noetherian ring satisfies the ACC' we could use Leibniz' rule for $=_I$ to obtain a falsehood.  This suggests that definitions should involve $=_N$ not $=_I$.  We just have to be careful with the introduction of new constants via $=_I$. For often not all people agree on the sense-attribution of terms. The paradox of analysis reveals itself to be a pseudo-problem. But note that it can pop up again in indirect discourse where there is a certain ambiguity between verbatim reported speech and looser reported speech. 

Problems with definitions are quite distinct from  substitutivity pseudo-problems such as Mates' puzzle which are merely about linguistic incompetency or individual divergence. Different agents may attach different senses to the same expression or no sense at all.  In CIL terms stand for senses, hence senses which a given agent attaches to CIL expressions.  We might even introduce a 'non-sense'  sense.  Groundhog $=_I$ Woodchuck means that the agent attaches the same sense to the expressions 'Groundhog' and 'Woodchuck'.  Groundhog $=_N$ Woodchuck means that even if the attached senses are different the reference is the same in all possible states-of-affairs. Other pseudo-problems could, as we saw above, arise from reported speech which hovers been sense and expression. He said 'here is a woodchuck !'  vs.  he said that there was a woodchuck there.  Obviously there is a perspective in which we cannot substitute 'woodchuck' for 'groundhog'.  CIL needs syntactic abstraction operations $[A]$ on terms  for which the meaning of $[A]$ is to be seen as the expression $A$ itself. But how can  an expression be a meaning ? But consider 'the letter coming after 'a''. This has both sense and reference.  So $[A]$ might be seen as the individual concept of the expression 'A'.

Dispelling epistemological confusions

Modern convoluted treatments of the concepts of analytic, synthetic, necessary, contingent, a priori and a posteriori, a mixture of epistemology, logic and metaphysics,  seem to be hopelessly muddled and should not be taken too seriously.

Rather we put forward the following division of knowledge. Logical knowledge consists in statements of the form: in axiomatic-deductive system A expression B can be derived.  Evidential knowledge takes the form:  A consists in evidence for accepting axiomatic-deductive system B (or even just a sentence D) as valid in domain C. These are quite distinct kinds of knowledge.  The first kind is absolutely certain, verifiable, necessary and non-trivial. The second kind is prone to error but equally important, non-trivial and both  'investigable' and epistemically 'enhanceable'.  Leibniz's approach wins.

Ramblings 2

There is no merit in writing obscure, elliptic, terse, sinuous, cryptic philosophical prose. It may seem that you are showing that you are more intelligent and quicker than the reader but you are in reality just lacking courtesy and expressing a need to hide the substance of your arguments. 

When we translate a text in one language into another language, what exactly are we doing ? Is translation linguistic activity ? What do we say or who do we talk to when we write or translate ?

Formal systems represent linguistic expressions. We should be able to transform a linguistic expression into a formal expression, apply the formal rules and then translate back and obtain something acceptable. Formal semantics represent what linguistic expressions mean.  Formal semantics must consist of mathematical structures. Syntactic constructions can be mirrored in operations on such structures. The structures capture how meaning is built up.

A formal semantics must have reflection-into-self, intensionality. You can't escape the fact that we can talk about senses and references, hence senses can themselves be references. That the reference of sentences are states of affairs or equivalence classes of such.

While there are some good philosophical points in Bealer's Quality and Concept  our approach and views are quite distinct from those of Bealer.  Rather they follow the tradition of Leibniz, Bolzano, Frege, Husserl and more recently the work of Guillermo Rosado Haddock and Claire O. Hill.  The work of Edward Zalta exemplifies what a project of an axiomatic philosophy can look like.

Recursive axiomatic-deductive systems and their limits

The limitations, semantic and epistemic, of recursive-axiomatic deductive systems,  are often harped on.  But are not such limitations, after all, almost a truism ? What is important is not the 'axiomatic' or 'deductive'  therein: it is the  'recursive' .  The general theory of computatibility is as neglected as it is fundamental.  Theorems are $\Sigma^0$.  But arguments like those used for the Halting problem show that there are sets of arithmetical truths which are not in $\Sigma^0$. Thus recursive axiomatic-deductive systems are epistemically and semantically limited, incomplete. This is obvious.  Much more interesting is the philosophical elucidation of the concept of computability and of the whole arithmetical hierarchy (and hierarchies beyond).  And more interesting is the elucidation how we can apparently know and talk about sets whose elements we a priori cannot know or distinguish individually such as the set of numbers which codify arithmetical truths. If this is not a conclusive argument against extensionalism, then what is ? The algorithm is the intension,  extension is a computational question about the intension. Also there is a tendency to confuse a formal system and its philosophical agenda.  It is a happy truth that we can interpret ZFC however we want as long as we stick to the axioms and rules.

If incompleteness is a 'negative' trait of formal systems, there is also a surprising, significant, powerful  'positive' trait as well. This is the ability of formal systems to reflect, mirror, interpret and be embedded in each other. For one system to mirror within itself the meta-theory and semantics of another.  And that this mirroring and interpretation is itself mirrored in a certain encompassing system. Incompleteness itself was established by Gödel to be a consequence of certain formal systems mirroring aspects of their own meta-theory. Mathematical logic is the mathematical treatment of the formal logic used to represent mathematics itself. It is a reflection-into-self of mathematics.

It is not so much formal systems that are 'limited' and 'incomplete' as our own attitude and science if we do not start studying axiomatic-deductive systems with proof systems in all levels of the arithmetical hierarchy and also with possibly  infinitely long expressions.  Even physics suggests this.

Formal systems are supposed to talk about things. When those used in mathematics are set up it is implicit that the expressions are meant to relate to objects, not mental contents of certain people at certain times. What will a philosophical formal system talk about ? The objects of science such as physics and biology ? Mental states and consciousness ? The semantic, ontological and epistemic problems of logic and mathematics themselves ? Or abstractly all of the above ?

But notice how many philosophical questions are framed: what is X ? how is Y possible ? For instance what are mathematical objects ? How is knowledge possible ? Questions about finding the nature or definition of X are paradoxical because if we do not know what X is how can the question even be asked meaningfully ? For instance: what is a snark ? We start from a situation of social linguistic usage and desire to go to the realm of essence.  We also ask; what are numbers ? We use numbers and think we know what numbers are and yet we cannot say clearly and explicitly what they are (cf. Augustine's si nemo a me quaerat scio, etc.). This duality, division,  shadow, in our cognition and consciousness does not seem to be approachable by formal systems.  When we understand language there does not seem to be any conscious computation going on.

Another angle is to consider a mathematical approach to natural language. We are given a large data set of linguistic data plus descriptions of their social context. Our task is to give formal structure (morphological, syntactic, semantic, pragmatic, etc) to linguistic units so as to largely predict or explain computationally or deductively the massive data set - and extrapolate new data. For originally we needed both to explain and deduce known mathematical truths as well as deduce new ones.  But we mentioned 'description of social context' : this itself is carried out within natural language. Also, how can we talk about mental states or the mind ? How can we examine or compare mental states ?

We wrote 'predict or explain computationally or deductively'.  But in itself the computational complexity may be too great for our resources. Imagine if to prove anything arithmetically significant  in ZFC or the Peano Axioms we actually needed billions of lines of proof.  What difference would this be from just having a handy list of arithmetic validities which we could query ? In some cases looking for counter-examples might be more efficient than looking for proofs (the proposition or its negation).  So measures of computational complexity are important, the way the length of the proof depends on the length of the theorem to be proven.

If we wish to focus on a formal analysis of natural language (which itself is highly problematic) then we are forced to restrict ourselves to a limited domain of language and linguistic practice; for either we are too general and simplistic or else we must be able to account for the philosophical use of language too, even in dealing with language itself. For instance, what does the word  'language' mean ' ? We are looking at the painting which we are inside in.

Notes on truth

There are multiple theories of truth,  that is, solutions to the challenge of giving a precise definition of truth, of saying what truth is.  But for a given theory we can object that it is still not precise enough. For instance the 'correspondence theory' of truth.  We make the strict distinction between expressions in a given formal language and what they mean or express.  An expression is  a sentence if it expresses a proposition, which is an extra-linguistic entity like the Fregean Gedanke.  Truth is a property of propositions. I question that we can even speak meaningfully of sentences in isolation being true or true-in-some-language.  That such contradicts linguistic and psychological evidence. For instance:  'Il neige is true in French' can only mean; 'What Il neige means is true and Il neige is a French expression'. Sentences are said to be true only in virtue of, relative to, what they mean or express. Tarski's notion of true-in-L only make sense in the context of a completely artificial set-theoretic construction.

Let us say, for the moment, that there is a realm of proposition-thoughts and a realm of conditions, states-of-affairs. And that there is a correspondence between the two so that a proposition is true if its corresponding condition obtains. We do not see yet how to formalize this.

Technical modal logic is not circular because it is all carried out in set theory and validity for modal sentences is defined purely set-theoretically. But this is not the same as defining necessity or possibility, for instance in terms of 'possible worlds'. A necessary proposition is what correspond to a necessary condition, a condition that must obtain. Thus our burden is to give a non-circular definition of necessary condition.  In our model there is (up to $=_N$) only one necessary condition $\mathfrak{N}$. Thus we define a condition $\phi$ to be necessary iff $[\phi] =_N \mathfrak{N}$.  There is a subset (finite, perhaps) of primitives $P_i$ in each $D_i$, that is $P_i$ is finite and $P_i \subset D_i$. It follows that there is only one $=_N$ equivalence class in $D_0$ containing a $p_0$ which is always true. Bealer wishes to reduce the equivalence classes of primitives to singletons. Thus there are no plural aspects of the primitive phenomenological quality 'being green', $g \in D_1$. 'Being green and big' is a different concept from 'Being big and green' (one may not know the commutative law for conjunction) but nevertheless corresponds to the same quality.  Even at the primitive level we have singletons in general qualities and conditions are to be given a $=_N$ equivalence classes. The problem is that we must pay attention to a primitive being fined or coarse grained.  Suppose 'knows' is a primitive $k$ in $D_1$. Them $comb_{(1)} k a$ may  not define a $=_N$ equivalence class. That is, $comb_{(1)} k b$ can have different truth-values for different $b \approx_N a$. The correspondence (as in the correspondence theory of truth) between thoughts and conditions is thus the taking of equivalence classes !