On Languages

Formal vs Natural Languages

Natural languages do not have a well-defined syntax like formal languages (i.e., there isn’t a procedure to construct new words, some words are just arbitrarily adopted and assigned a loose meaning), nor a deductive system to infer statements (this is in part to the ambiguity of its semantics). Natural languages can then be thought of a means to express ideas (either about the physical world or the immaterial in our own mental faculties) via informal means. Thus, this naturally includes our human intuition and thought.

A formal system has a syntax (which is often referred to as a formal language) and semantics, as well as a deductive system to infer new theorems

Natural languages cannot define truth (at least not in an unambiguous universal way) as we can formulate the sentence “this sentence is false” as shown by Tarski’s undefinability theorem, this is the reason for formal languages and the Tarskian framework where truth is able to be defined in the metatheory (although we run into the problem of infinite regress and truth not being ‘grounded’). This does not result in a full semantic collapse as natural language does not have inference rules like formal languages in order to prove everything, rather there is ambiguity surrounding the meaning behind specific words and semantics become context-based. Thus, liar-type sentences are isolated ‘nonsensical’ statements that have no effect on the validity of other sentences (independent) and so natural language is an inconsistent-tolerant language. Natural language expresses different views on truth (see On Truth), hence for it not being well-defined.

The reason why a trivial language can define truth by allowing the liar’s paradox is because they are still a formal language (just ones w/ contradictory syntax and semantics) and so truth is formally defined via Tarski’s T-schema ( $T (⌈ ϕ ⌉) \leftrightarrow ϕ$ ), semantic collapse then follows from a language being able to define its own truth predicate. Meanwhile natural languages do not have an adequate definition of truth due to ambiguous and vague semantics (although it can still express the liar’s sentence).

Note: Metalanguages are not just used for formal languages, but also natural languages, although a full semantics for natural language cannot be fully defined due to containing contradictory and self-referential statements like the liar sentence. We could then define a construction akin to a Tarskian hierarchy (although not exactly due to the fundamental difference between natural vs formal languages). E.g., consider studying the language Spanish, the object language is Spanish and the metalanguage is English as we are translating a Spanish word to English to obtain its meaning.

Natural language (which contains things such as human intuition) acts as a sort of ‘ultimate’ metalanguage for formal languages or as a sort of irreducible ground for a hierarchy of formal languages. This is evident through logic being ultimately reliant on human intuition, no matter the infinite regression of metatheories justifying the lower level language (this is why logicians will merely consider a metatheory to formalize the object language and not bother w/ further levels and justification, as it is almost needless to do so if we can ‘jump’ to it being ultimately justified by natural language). Human intuition guides the construction of logic to what we believe logic should look like and how it should behave. Differing intuitions on how mathematics/logic should behave are reflected in varying Tarskian hierarchies and choices of object and meta-language. This aligns with the intuitionistic view of mathematics and logic.

It can also be argued that there is no formal definition of a ‘formal language’ as this is defined w/in the framework of natural language again via human intuition/thought and so there is metalogical vagueness in the what constitutes a formal language (i.e., there isn’t a universally agreed view on what a formal language is, this is evident by the existence of varying logics such as classical, intuitionistic, paraconsistent). One could think of complex ‘formal’ logical where the syntax, semantics, and proof system are modified, or perhaps even more complex logical architecture beyond that, in order to constitute some formal system. An imprecise language such as natural language can still define a precise language like formal language as we are effectively reducing a more complex system (natural language) to a simpler one (formal language). However, this is limits the type of statements that can be expressed by this ‘reduced’ language (universal imprecision to precision about a relative frame). Hence, imprecision is a result of complex (and expressive) systems while precision is a result of reduced, simpler systems.

A ‘logic’ is syntax + semantics, a full formal system is a logic with a proof theory (inference rules).

There are partial reductions of natural language to formal languages (i.e., Montague grammar), although a full reduction can never be done due to the ambiguity in meaning of expressions (i.e., context dependent) and can express the liar’s sentence (or analogues of it), while formal languages (which are not trivial) cannot by Tarski’s undefinability theorem. Truth is partially captured Tarskian hierarchies although this is ‘ungrounded’ as its truth is always dependent on being defined by a metatheory (see Tarski’s theory of truth). Montague grammar can be viewed as an attempt to create a framework where the metatheory is a formal language (which is the Montague grammar itself) and the object theory is natural language. This can be done, however since Montague grammar cannot fully formalize and so justify the semantics natural language, it is not a suitable metatheory (insufficient expressive power).

Natural languages have a ‘self-extending’ syntax as they often adopt new words, although this does not mean it is not syntactically-closed (i.e., cannot talk about its own formation rules and method of adding new words, especially since natural language syntax is not well defined) and so doesn’t undergo syntactic collapse. They also have a ‘self-extending’ semantics as new meanings may be attached to a pre-existing word depending on context, further enforcing the ambiguity of natural language.

Syntactically-closed languages can (internally) expand their syntax to be non-recursive

It was shown by Noam Chomsky that formalized natural language grammars (based off of English) is at least recursively-enumerable.

There is no formal definition of what constitutes a natural language (it could be anything which can be used to express concepts and discuss the world,…etc. which is not a formal language), this means we can have particular languages w/ a non-recursive syntax or even those which can infinitely-long sentences (via embedding infinitary logic into natural language by some methods, or allowing infinitary logic to be the metatheory). Natural language can essentially talk about anything and so in this sense is maximally expressive. Thus, (complete) natural language cannot be ‘pinned down’ by a formal construction, whether we combine modal logic, many-valued,…,etc. there are some expressions that cannot be characterized (i.e., analogues of the liar paradox) that natural language can express. Nothing short of semantically-closed formal languages can be used to emulate natural language as new syntax can be introduced (see Trivialism). However this ultimately fails as we now have a complete semantic collapse (triviality) rather than just vagueness and so nothing actually is being expressed.

Natural languages are infinitely complex in that it would require an unbounded, complex set of syntactic and semantic rules in formal language to fully emulate natural language and so is an impossible/infinite task (cannot completely formalize natural language). This is because we can express more and more of natural language by adding and modifying syntactic & semantic rules, however this process is asymptotic in that we’ll never fully express every statement in natural language w/in formal language (hence why an infinitely complex set of formal rules). This is particular evident via formal language never being able to fully escape the liar, we can always construct new liar analogues that destroy the semantics of a formal system.

Underlying Logic

Attempting to completely ground truth requires a metaphysical notion of an ‘underlying logic’ that is external to language (both formal and natural). This is because such an entity presupposes language itself as the metaphysical backdrop against which logical truth and semantics is derived (it must as well define truth about itself which languages cannot do), and as such can be considered as the ‘ultimate metasystem’ (not a ‘ultimate metalanguage’). Such an idea depends on philosophical belief (as it inherently would not be able to be proven by logic or w/in ‘reality’). This is unlike human intuition and expression of thought through natural language, which is proven.

A metaphysical realist— who believes there is an underlying logic and notion of truth to the world (‘ultimate metasystem’) — would think that formal languages and natural languages are mere linguistic representations of this underlying logic which (attempt to) convey it (however not completely). Hence, the underlying logic is independent of language. $\to$ Recall: Natural language can convey logic (as said by other comments) and there is such a thing as natural language arguments, however ambiguity may be present when talking about informal notions.

Since this ‘underlying logic’ is external to language, it cannot be fully described or characterized as such by languages. Hence, this underlying logic is ‘ineffable’ (“The limits of my language are the limits of my world” - Wittgenstein, see Philosophy 2) and so we don’t exactly know how it operates.

The underlying logic being an ineffable metaphysical structure that underlies reality supports the idea of some kind of coherence theory of truth (coheres semantics). However, where ‘truth’ is more complex than simple truth-conditional semantics or truth corresponding to state of affairs as described by Wittgenstein in the Tractatus (see Philosophy 2). This is evident through reality not operating on bivariant truth/falsity, but has additional complexity via context-dependance, ambiguity, degrees of truth (what exactly this notion of truth is not exactly known due to the underlying logic being ‘ineffable’). We only know that ‘reality’ is the result of the underlying logic in operation and so ‘reality’ ‘shows’ (like a picture) how the underlying logic behaves (it cannot be ‘pinned down’ using merely logical/linguistic notions).

Note: The coherence theory of truth itself supports logical holism, where truth of an individual proposition cannot be (fully) understood in isolation but rather only (fully) understood through how it relates to every other proposition.

Attempting to describe the underlying logic (w/in language) would instead be describing a mere shadow or projection of the underlying logic (almost in a apophatic sense). If these metaphysical truths are interpreted by these languages, this may yield ‘illogical’ results (by the standards of formal logic) like contradictions or paradoxes. However, as mentioned above, these interpretations are mere shadows of the underlying logic. This avoids the issue of the ineffability paradox (an entity being ‘ineffable’ is a description and so that entity becomes effable, contradictorily) via accepting this is as an expressive limit of language.

The shadow of the underlying logic is akin to the object language (in this case, any language in the Tarskian hierarchy with the underlying logic being the ‘ultimate metasystem’) internalizing the metatheory, however in a different sense as internalization is a relation between languages (which the underlying logic itself is not). This relation of ‘internalizing’ is external to language itself by consequence of the underlying logic (a constituent of the internalizing relation in this case) being ineffable. Hence, these shadows are mere approximations of how the underlying logic would behave as a result of being interpreted w/in language.

A description of the shadow — for example, the question “How do you define the behavior of the underlying logic if it is ineffable?” — is a meta‑linguistic claim about our representation, not a direct description of the ‘underlying logic’ itself. That meta‑claim must itself be described in a higher metalanguage, which in turn produces its own shadow requiring further description at a higher meta-level. Hence, we infinitely defer such claims about the shadow to be answered in a higher metalanguage (in fact, talking about the ‘underlying logic’ right now invokes this infinite deferral). These infinitely-many metalanguages are between the object language and underlying logic. Consequently, complete knowledge of the underlying logic cannot be achieved by any purely linguistic or logical procedure; as this is something that is beyond what logic can capture. e.g., “how are you defining how the underlying logic behaves if it is ineffable” may be answered by “we are not talking about the actual underlying logic as we have mentioned it w/in language.” However, the issue persists at the next metalevel as “how are describing this now?”

Both the platonic and intuitionistic theory of logic hold that there is such a thing as intuition. The platonic theory of logic argues that intuition is a (mental) faculty that behaves in alignment or accordance w/ the underlying logic of reality (i.e., perception of abstract truth). This would also explain why intuitive truths may be contrary to formal languages (linguistic apparatuses of interpreting the underlying logic). On the contrary, the intuitionistic theory of logic asserts that ‘logic’ is something that has been abstracted from perceived patterns in reality (both material and immaterial) (see Philosophy) and so is embedded into language, not external from it.

It can be argued that ‘reality’ cannot be (fully) mapped out and the boundaries of it cannot be fully defined (i.e., what it encompasses and what it doesn’t) as humans can only have a partial knowledge about reality. This relates to the ineffability of the underlying logic as the metaphysical structure that underlies reality.

Additional Notes

Formal language (syntax) comprises of a set of symbols called an ‘alphabet’ (in the context of mathematics, specifically model theory, this is instead called a vocabulary), formation rules to forms WFF’s, and a set of WFF’s (which is a subset of all possible strings that can be formed).

Naive set theory is formulated w/in natural language (evidently) due to being ridden w/ paradoxes such as the Russell’s paradox. The issue precisely lies w/ totalized predicates which were interpreted linguistically (in natural language) and not formally (e.g., the unrestricted comprehension schema: For any property $P (x)$ there exists a set of all $x$ , there is ambiguity concerning what exactly a ‘property’ is).

The exact structural order of these languages depends on the philosophical background of the logician:

For a intuition-based view of logic where one believes truth cannot ultimately be grounded: Natural language $\to$ Tarskian hierarchy
For a metaphysical view of logic where one believes truth can be ultimately grounded:
- Underlying logic $\to$ natural language $\to$ Tarskian hierarchy
- Underlying logic $\to$ Tarskian hierarchy (formal languages) if one holds a formalist or constructive view of logic

Classification of Formal Grammars

Defining Formal Grammars

Formally, a grammar is defined as the following: Grammar: A grammar is a 4-tuple $G = (N, Σ, P, S)$ , where - $N$ is the set of non-terminating symbols, symbols which cannot be appear in the formal language produced by a formal grammar. - $Σ$ is the set of terminal symbols - $P$ is the set of production rules (functions map a string of symbols, which could bee terminating or non-terminating, to another string of symbols). - $S$ is the starting symbol

E.g., Suppose we have the production rules: $S \to a S b$ $S \to ε$ (the empty string) then, the following strings can be outputted $ab$ $aabb$ $aaabbb$ …etc.

Notes:

The starting symbol is always a non-terminal symbol.
Each production rules is of the form $(Σ \cup N)^{*} N (Σ \cup N)^{*} \to (Σ \cup N)^{*}$ , where ' $*$ ' is the Kleene star operator.

Chomsky Hierarchy

Type 3: Regular language; recognized by a finite state automaton Type 2: Context-free Language; recognized by a non-deterministic pushdown automaton Type 1: Context-sensitive Language; recognized by a linear bounded automaton Type 0: Recursively-enumerable; recognized by a Turing machine

Note: Formal grammars do not have a semantics, only a formal syntax which has been generated. Meanwhile, something like Montague grammar has both a syntax and semantics.

Montague Grammar

Montague grammar (also called Montague semantics) is a formalization of (a fragment of) natural language. This is equivalent to the metalanguage being a formal language (the Montague grammar) and the object language is natural language.

References

Tarski’s theory of truth On Truth Trivialism Proof Theory Logics, Structures & Models Philosophy Formal Grammar

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