# Update of Language Definition

Note that I have removed the factorization requirement from the definition of a language in Fundamental Knowledge Part 1;  so we will just have $\mathcal{L}_{F,T,W}=F[T[W]].$  This will remove some triviality in examples of fuzzy logic systems in the upcoming post.  The original motivation behind the factorization was that traditionally compound terms are considered formulas, but terms themselves are not considered formulas.  I don’t really see why we can’t let terms be formulas;  let us assume “substitutions” have already been made.

I have also removed the requirement that $\varphi(\phi)=\varphi(\psi)$ for all $\phi,\psi$ in a theory $X$ where $\varphi$ is a logic system.  Instead I have defined a logic system that satisfies this condition as a normal logic system.

# Fundamental Knowledge-Part 2: Models

The next task is to absorb the traditional area of mathematical logic.  One key missing ingredient is a model.  Let us recall the traditional setup (taken from [1]).

Definition 2.1.  Let $S$ be a set (of symbols).  An $S$-structure is a pair $\mathfrak{A}=(A,\mathfrak{a})$ where $A$ is a nonempty set, called a universe, and $\mathfrak{a}$ is a map sending symbols to elements, functions, and relations of $A$.  An assignment of an $S$-structure $(A,\mathfrak{a})$ is a map $\beta:S\to A$.  An $S$-interpretation is a pair $\mathfrak{I}=(\mathfrak{A},\beta)$ where $\mathfrak{A}$ is an $S$-structure and $\beta$ is an assignment in $\mathfrak{A}$.

For shorthand notation, the convention (with some of my modifications) is to write:  $c^\mathfrak{A}=\beta(c)$, $(f(t_1,...,t_n))^\mathfrak{A}=\mathfrak{a}(f)(\beta(t_1),...,\beta(t_n))$, and $(xRy)^\mathfrak{A}=\beta(x)\mathfrak{a}(R)\beta(y)$.  These are the terms.  Formulas are then built from the terms using traditional (although this can be generalized) logical connectives.

The notion of a model is then defined via induction on formulas.

Definition 2.2.  Let $\mathfrak{I}=(\mathfrak{A},\beta)$ be an $S$-interpretation.  We say that $\mathfrak{I}$ satisfies a formula $\phi$ (or is a model of $\phi$), denoted $\mathfrak{I}\vDash\phi$, if $\phi^\mathfrak{A}$ holds, where $\phi^\mathfrak{A}$ is defined via its components and $\beta$ and $\mathfrak{a}$ where necessary.

Formal languages in convention are built up from the formulas mentioned above, which are nothing more than special cases of Alt Definition 1.3.  A model for a language is hence nothing more than an $A$-interpretation into a structure, where $A$ is an alphabet (provided it is equipped with a logic system).  This is precisely what I have constructed in Part 1;  the symbols of $W\subset A^*$ are mapped to the universe $\mathcal{L}_{F,T,W}$.  The next thing to establish is that every model is a language model.  This is trivial since a model by definition satisfies a set of formulas as well as compounds of them (i.e. it must satisfy a language).  Hence we have no need to trouble ourselves with interpretations and may simply stick to the algebra of Part 1.

While we have absorbed model theory, there are a few more critical topics to absorb from mathematical logic. We return to the language of Part 1 (no pun).  Let $X$ be a theory of $\mathcal{L}_{F,T,W}$ and $\varphi:F[X]\to V$ be a binary logic system.  A formula $\phi\in\mathcal{L}_{F,T,W}$ is derivable in $X$ if it is a proposition (i.e. is in $F[X]$).  We may write $X\vdash\phi$.  This definition is in complete agreement with the traditional definition (namely, there being a derivation, or finite number of steps, that begin with axioms and use inference rules);  it is nothing more than saying it is in $F[X]$.  Similarly $\phi\in\mathcal{L}_{F,T,W}$ is valid if $\varnothing\vdash\phi$, or equivalently, it is derivable in any theory.  In our setup this would imply $\phi\in F[\varnothing]=\varnothing$.  Hence no formula is valid.

Let $F$ have a unary operation $\lnot$ and $\varphi:F[X]\to V$ be a logic system on a theory $X$.

If we assume $\lnot$ to be idempotent ($\lnot\lnot\phi=\phi$), then since $\varphi$ is a homomorphism, we have $\varphi(\phi)=\varphi(\lnot\lnot\phi)=\lnot\lnot\varphi(\phi)$.  That is, the corresponding unary operation in $V$ must also be idempotent on $ran(\varphi)$.

Definition 2.3.  A unary operation $\lnot$ (not necessarily idempotent) is consistent in $\varphi$ if for all $\phi\in F[X]$, $\varphi(\phi)\neq\varphi(\lnot\phi)$.

If we assume $\lnot$ is consistent in $\varphi$ and that $\varphi$ is a binary logic system, then the corresponding $\lnot$ in $V$ is idempotent since

$\varphi(\phi)=0\Rightarrow\varphi(\lnot\phi)=1\Rightarrow\varphi(\lnot\lnot\phi)=\lnot\lnot\varphi(\phi)=0$.

Again, proofs in a binary system are independent of the choice of valence.   If we assume consistency and idempotency, then we have a nonidentity negation which is idempotent on the range.  The case for assuming binary system and idempotency yields either a trivial mapping of propositions (all to $0$ or all to $1$), or that $\lnot$ is consistent and idempotent on $V$.  And lastly if we assume all three (idempotency and consistency of $\lnot$ together on a binary system), we obtain a surjective assignment with idempotent negation in $V$.

Let $\varphi:F[X]\to V$ be a binary logic system where $V$ is a boolean algebra.  Then the completeness and compactness theorems are trivial.  Recall these statements:

Completeness Theorem.  For all formulas $\phi$ and models $\mathfrak{I}$,

$\mathfrak{I}\vDash\phi\Rightarrow X\vdash\phi$

where $\mathfrak{I}\vDash X$.

Compactness Theorem.  For all formulas $\phi$ and models $\mathfrak{I}$,

$X\vdash\phi\Rightarrow\mathfrak{I}\vDash\phi$

where $\mathfrak{I}\vDash X$.

Traditionally these apply to, what we would call, a binary logic system $\varphi:F[X]\to V$ where $V$ is a boolean algebra (hence $F$ has a consistent, idempotent negation) under traditional operations, and in particular this fixes the operational/relational structures of $F, T$, and $W$ , but $X$ is arbitrary.  In this setup, all “formulas” (or what we would hence call propositions since they are generated by a theory) are trivially satisfiable since they have a language model.  Hence Compactness is true.  Moreover since they are propositions in a binary logic system, they are in some $F[X]$ for a theory $X$ and are hence derivable; so we have Completeness.

Lastly we wish to address Godel’s Second Incompleteness Theorem;  recall its statement:

Godel’s Second Incompleteness Theorem.  A theory contains a statement of its own consistency if and only if it is inconsistent.

We have only defined what it means for a unary operation in a logical system to be consistent.  Hence we can say that a binary logic system with a unary operation is consistent if its unary operation is consistent.  But all of these traditional theorems of mathematical logic are assuming a binary logic system where $V$ is a boolean algebra , $\lnot$ is idempotent, and the map $\varphi:F[X]\to V$ is surjective.  Hence $\lnot$ is consistent (from above discussion), and the consequence in the theorem is false.

The weakest possible violation of the antecedent of Godel’s theorem is to use a structure to create itself (i.e. that it is self-swallowing), which makes no sense, let alone using it to create a larger structure within which is a statement about the initial structure.  That a binary logic system with unary operation could contain a statement of its own consistency is itself a contradiction, since the theory itself, together with the statement $\phi$, are in a metalanguage.  It is like saying that one need only the English language to describe the algebraic structure of the English language.  As we previously said at the end of Part 1, one can get arbitrarily close to doing this–using English to construct some degenerate form of English, but you can never have multiple instances of a single language in a language loop.  Another example would be having the class of all sets, then attempting to prove, using only the sets and operations of them, that there is a class containing them.

Hence the antecedent is also false.  So both implications are true.

[1]  Ebbinghaus, H.-D., J. Flum, and W. Thomas.  Mathematical Logic.  Second Edition.  Undergraduate Texts in Mathematics.  New York: Springer-Verlag.  1994.

# Fundamental Knowledge-Part 1: The Language Loop

We must start with a language.  A language can be defined in two ways.  First let us begin with the axioms of pairing, union, and powerset and schema of separation.  This gives us a cartesian product of sets, and hence functions.

Definition 1.1.  An $n$ary operation on a set $X$ is a map $O:X^n\to X$.  A structure is a set $X$ together with an $n$-ary operation.  The signature of a structure $X$ is the sequence $(n_1,...,n_k,...)$ where $n_k$ is the number of $k$-ary operations.

Definition 1.2.  Let $X$ and $Y$ be structures with the same signature such that each $k$-ary operation of $X$ is assigned to a $k$-ary operation of $Y$ (i.e. $f(O_i)=O^i$ where $O_i$ is the $i$th $k$-ary operation of $X$).  A homomorphism between structures $X$ and $Y$ is a map $\varphi:X\to Y$ such that

$\displaystyle \varphi(O_i(x_1,...,x_n))=O^i(\varphi(x_1),...,\varphi(x_n))$.

Note that a nullary operation on $X$ is a map $O:\varnothing\to X$.  That is, it is simply an element of $X$.  Now let $A$ be a set which we will call an alphabet, and its elements will be called letters.  A monoid $X$ has a nullary operation, $1\in X$ called a space, and a binary operation, which will simply be denoted by concatenation.  We define the free monoid on $A$ as the monoid $A^*$ consisting of all strings of elements in $A$.  We now have two definitions of a language, of which the first is traditional and the second is mine:

Definition 1.3.  A language is a subset of $A^*$.

Alt Definition 1.3.  Let $W\subset A^*$, $T$ be a relational structure (a set together with an $n$-ary relation), and $F$ be a structure.  The language $\mathcal{L}_{F,T,W}$ is defined as $F[T[W]]$ where $X[Y]$ is the free $X$-structure on $Y$.  In particular elements of $W$ are called words, elements of $T[W]$ are called terms, and elements of $\mathcal{L}_{F,T,W}$ are called formulas.

Definition 1.4.  A theory of $\mathcal{L}_{F,T,W}$ is a subset $X\subset\mathcal{L}_{F,T,W}$.  Elements of a theory are called axioms.  Elements of $F[X]$ are called propositions.  A theory $X$ of $\mathcal{L}_{F,T,W}$ is called a reduced theory if for all $\phi,\psi\in X$, $\psi\neq O(\phi,x_1,...,x_{n-1})$ for all $n$-ary operations of $F$ and all placements of $\phi$ in evaluation of the operation.  (That is, the theory is reduced if no axiom is in the orbit of another).

For example, the theory $\mathcal{L}_{F,T,W}$ is called the trivial theory.  The theory $\varnothing$ is called the empty (or agnostic) theory.

Definition 1.5.  An $n$ary logic system on a theory $X$ is a homomorphism $\varphi:F[X]\to V$ where $F$ and $V$ have the same signature and $V$ has cardinality $n.$  We may also say the logic system is normal if $\varphi(\phi)=\varphi(\psi)$ for all $\phi,\psi\in X.$

In traditional logic $V$ is a two element boolean algebra.  Traditional logic also has a special kind of function on its language.

Definition 1.6.  A quantifier on $\mathcal{L}_{F,T,W}$ is a function $\exists:T[W]\times\mathcal{L}_{F,T,W}\to\mathcal{L}_{F,T,W}$.  We may write:

$\exists(x\in X,\phi)=(\exists x\in X)\phi$.

In particular it is a pseudo operation, and gives the language a pseudo structure.  This is similar to modules, where in this case a product of a term and a formula are sent to a formula.

Hence our initial assumption of four axioms (as well as the ability to understand the English language), have in turn given us the ability to create a notion of a language of which a degenerate English can be construed as a special case.  This is certainly circular in some sense, but in foundations we must appeal to some cyclic process.  One subtlety worth noting is that the secondary language created will always be “strictly bounded above” by the initial language;  they aren’t truly equivalent.  (In fact this last statement is similar to the antecedent of Godel’s Second Incompleteness theorem).