Logic which depends on the perspective? (Semantic space of logic / perspectivism)

Question

I am not sure if this question is appropriate for MO, since I have only a basic understanding of Boolean logic, and am maybe not qualified to ask questions beyond that, but still I will try to write what I have in mind:

In everyday logic, we emphasize the perspective / point of view of someone who is trying to convince someone else with "his logic".

I tried to take this literally to marry vectors in some Hilbert space with Luwaksiewiecz logic in a short unpolished essay (or a recent version) which answers some questions that it raises:

The ideas here are to first start with a set of things and a positive semidefinite kernel on these things. The theory of RKHS / Moore-Aaronszajn theorem, gives us a Hilbert space $H$ where $k(x,y) = \left < \phi(x), \phi(y)\right>$ where $\phi: X \rightarrow H$ is an injective function, called 'feature mapping' in machine learning. Now suddenly we can do geometry on this set abstract set of things $X$. The conclusion of machine learning community is, that a 'feature vector' $\phi(x)$ of $x$ captures the 'semantic of the element $x$' relative to $k$.

Taking the perspective of everyday logic, we might want be able to give a semantic meaning to the phrase 'different point of views'. But this is very easy done in geometry, since if $x$ takes the perspective of $w$, then we might want to project the unit vector $\phi(x)$ to the unit vector $\phi(w)$, to get:

\begin{equation} \label{eq:pi_w} \pi_w(x) = \frac{k(w,x)}{|w|^2}w = k(w,x)w \end{equation}

For this last equation, we write:

\begin{equation} \label{eq:relative_to_w} x \text{ (relative to } w) := \pi_w(x) = k(w,x)w \end{equation}

and

\begin{equation} \label{eq:length_x_w} |x|_w := |\pi_w(x)| \end{equation}

which is the length of the projected vector.

Now we can imagine what the meaning of 'I have a different perspective' means. We simply associate the projection of the semantic vector of $x$ to some other perspective vector $w$. Then 'changing perspective' simply means to change the perspective vector from $w$ to say $\hat{w}$.

Let us define

\begin{equation} \label{eq:land_relative_to_w} x \land y \text{ (relative to } w) := \min(k(w,x),k(w,y)) \phi(w) \end{equation}

\begin{equation} \label{eq:lor_relative_to_w} x \lor y \text{ (relative to } w) := \max(k(w,x),k(w,y)) \phi(w) \end{equation}

\begin{equation} \label{eq:not_x} \lnot x := - \pi_w(x) \end{equation}

and following Lukasiewicz:

\begin{equation} \label{eq:implies_relative_to_w} x \rightarrow y \text{ (relative to } w) := \min(1,1+|y|_w-|x|_w) w \end{equation}

\begin{equation} \label{eq:equvi} x \leftrightarrow y \text{ (relative to } w) := x \equiv y := (x \rightarrow y) \land (y \rightarrow x) \end{equation}

We have the de Morgan rules, double negation, contraposition:

\begin{equation} \label{eq:de_morgan_1} (\lnot x) \land (\lnot y) = \lnot ( x \lor y ) \text{ (relative to } w) \end{equation}

\begin{equation} \label{eq:de_morgan_2} (\lnot x) \lor (\lnot y) = \lnot ( x \land y ) \text{ (relative to } w) \end{equation}

\begin{equation} \label{eq:double_neg} (\lnot (\lnot x)) = x \text{ (relative to w)} \end{equation}

\begin{equation} \label{eq:contraposition} (x \rightarrow y ) = ((\lnot y) \rightarrow (\lnot x)) \text{ (relative to w)} \end{equation}

Let us define three cases of truth values: $T,F,I$ which should be translated to True, False, Indeterminate.

\begin{equation} \label{eq:truth} \mu(x) = \begin{cases} T & \text{if } k(w,x) < 0, \\ I & \text{if } k(w,x) = 0, \\ F & \text{if } k(w,x) > 0. \end{cases} \end{equation}

We have a version of 'Modus ponens':

\begin{equation} \label{eq:mp} \text{ If } \mu(x)=T \text{ and } \mu( x \rightarrow y ) = T \text{ then } \mu( (x \land ( x \rightarrow y)) \rightarrow y) ) = T \end{equation}

The Liar Paradox in Semantic Spaces

The Liar Paradox is a classic illustration of self-reference and inconsistency in logic, often summarized by the statement, "This statement is false." If we assume the statement is true, then it must be false as it claims. Conversely, if we assume it is false, then it would paradoxically be true. This paradox highlights the difficulties in dealing with self-referential statements in formal systems. It is well-known that using multi-valued logic, one can give the statement "This statement is false" an indeterminate logical value, different from true or false. Here, we aim to illustrate that it is possible to find a framework where both the classical paradox and the new interpretation are logically possible by altering the perspective.

To illustrate how semantic perspectives can resolve or interpret this paradox differently, let us consider an example involving vectors.

We will be interpreting the Liar Paradox like this: We are searching for a sentence—or better yet, let us call it formula $ L $—whose negation $ \lnot L $ is equal to $ L $: $$ \lnot L = L $$

The basic idea is to change the perspective or point of view $ w $, so that the truth value or even the formula itself satisfies: $$ \lnot L = L \quad \text{(first interpretation)} $$ or in terms of truth values $$ \mu(\lnot L) = \mu(L) \quad \text{(second interpretation)} $$

Let $ X := \{ e_1, -e_1, e_2, -e_2 \} $, where $ e_1 = (0, 1) $, $ e_2 = (1, 0)$ are standard basis vectors in $\mathbb{R}^2$ with the usual inner product. We choose $ L = (0, 1) = e_1 $ and $ w = (1, 0) = e_2 $. With this choice, the negation of $ L $ relative to $ w $, denoted $ \lnot L \operatorname{rel} w $, is given by $ -L \operatorname{rel} w $. With this choice, $ w $ is perpendicular to $ L $ and $ -L $, hence this perspective might be suitable for "resolving" the paradox, as it is not in the same subspace spanned by $ L $ and $ -L $.

We compute: $$ k(w, L) = 0 \implies \mu(L \operatorname{rel} w) = I $$ and also $$ k(w, \lnot L) = -k(w, L) = 0 \implies \mu(\lnot L \operatorname{rel} w) = I $$ Hence $$ L \operatorname{rel} w = \pi_w(L) = k(w, L)\phi(w) = 0 = -k(w, L)\phi(w) = k(w, \lnot L)\phi(w) = \pi_w(\lnot L) = (\lnot L) \operatorname{rel} w $$ and the first interpretation of the Liar Paradox is satisfied, and also $$ \mu(L \operatorname{rel} w) = I = \mu(\lnot L \operatorname{rel} w) $$ so the second interpretation of the Liar Paradox is also satisfied.

If we change the perspective $ w $ to $ w := L = (0, 1) $ or $ w := -L = (0, -1) $, then in the first case we get: $$ k(w, L) = 1 \implies k(w, -L) = -1 \implies \mu(L \operatorname{rel} w) = T \neq F = \mu(\lnot L \operatorname{rel} w) $$ hence the second interpretation of the Liar Paradox cannot be satisfied, and consequently, the first interpretation also cannot be satisfied.

In the case of $ w := -L $, we again find that the Liar Paradox cannot be satisfied: $$ k(w, L) = -1 \implies k(w, -L) = 1 \implies \mu(L \operatorname{rel} w) = F \neq T = \mu(\lnot L \operatorname{rel} w) $$

Comment:

It appears that different logical meanings can be attributed to the Liar Paradox through a change in perspective. If one insists that the perspective must lie within the subspace generated by $ L $ and $ -L $, then the Liar Paradox cannot be satisfied, as shown above. However, if one allows a perspective perpendicular to the subspace generated by $ L $ and $ -L $, then the Liar Paradox can be satisfied in some sense. This is analogous to a common situation in mathematics: while some equations may not be solvable in a certain set or mathematical structure (such as a ring or field), they can be solved in a larger space that encompasses the original set as a substructure. For example, $ x^2 = -1 $ is not solvable in $ \mathbb{R} $, but it is solvable in $ \mathbb{C} $, which contains $ \mathbb{R} $.

I could not find something similar for reference in Google so I am asking here, if this point of view has been explored in the literature before?

Thanks for your help!

Answer 1

I will try to answer my own question, by showing that Boolean algebras can be constructed using semantic spaces of logic and that we can use finite groups to construct finite semantic spaces of logic:

Separable Semantic Space

A finite semantic space $S = (X,k)$ with labels $y_i = \pm 1$ for each $x_i in X, 1 \le i \le n$ is called separable by the $y_i$ if there exist a $w$ such that $\hat{X} := X \cup \{w\}$, $\hat{k}:\hat{X}\times\hat{X}\rightarrow \mathbb{R}$ is a positive semi-definite function on $\hat{X}$ which extends $k$, that is: $\hat{k}(x,y) = k(x,y) \forall x,y \in X$ and such that $\hat{S} := (\hat{X},\hat{k}$) is a semantic space, such that the following holds:

$$\forall 1 \le i \le n : \operatorname{sign}(\hat{k}(x_i,w)) = y_i in \{\pm 1\}$$

If $S=(X,k)$ is separable by $Y= \{y_i| 1\le i \le n\}$, then there exists a $w$ such that:

$$\forall 1 \le i \le n : \mu( x_i \operatorname{rel} w ) = T, \text{ if } y_i > 0, \text{ else } F$$

The proof, consists in the definition of $\mu$ and because the $\operatorname{sign}$ can, by separability assumption, take only values $\pm 1$.

From Separable Semantic Space to Boolean Algebras

First we begin with a few properties which are easy to prove: Let $S = ( X,k)$ be a semantic space and $\phi(w) \neq 0$ be a perspective vector. Then the following are immediate from the definitions:

• $\pi_w(w) = \frac{k(w,w)}{|\phi(w)|^2} = \frac{1}{1} \phi(w) = \phi(w)$
• $w \operatorname{rel} w = \phi(w)$
• $(w \land x) \operatorname{rel} w = x \operatorname{rel} w$
• $(w \lor x ) \operatorname{rel} w = w \operatorname{rel} w$
• $(\lnot w) \operatorname{rel} w = - \pi_w(w) = - \phi(w) $
• $((\lnot w) \land x) \operatorname{rel} w = (\lnot w) \operatorname{rel} w$
• $((\lnot w) \lor x ) \operatorname{rel} w = x \operatorname{rel} w$

For instance with $\mathbf{1} \operatorname{rel} w:= w \operatorname{rel} w$ and $ \mathbf{0} \operatorname{rel} w := (\lnot w) \operatorname{rel} w$, we have:

$$ (\mathbf{1} \land x) \operatorname{rel} w = x \operatorname{rel} w $$

because,

$$ (\mathbf{1} \land x) \operatorname{rel} w = (w \land x) \operatorname{rel} w = $$ $$ = \min( k(w,w), k(w,x)) \phi(x) = \min(1,k(w,x)) \phi(w) = $$ $$ = k(w,x)\phi(w) = x \operatorname{rel} w $$

and similarly

$$ (\mathbf{0} \lor x) \operatorname{rel} w = w \operatorname{rel} w $$

because,

$$ (\mathbf{0} \lor x) \operatorname{rel} w = ((\lnot w) \lor x) \operatorname{rel} w = $$ $$ = \max( k(w,\lnot w), k(w,x)) \phi(x) = \max(-k(w,w),k(w,x)) \phi(w) = $$ $$ = \max(-1,k(w,x))\phi(w) = k(w,x) \phi(w) = x \operatorname{rel} w $$

Let now $S = (X,k)$ be a separable space by $Y := \{ y_i | 1 \le i \le |X| = n \}$ and let $\phi(w)$ be a perspective vector such that:

$$\forall 1 \le i \le n : \operatorname{sign}(\left < \phi(x_i),\phi(w) \right> = \operatorname{sign}(k(x_i,w)) = \pm 1 =^! y_i$$

Then the semantic space $\hat{S} := (\hat{X},\hat{k}$) with $\hat{X} := X \cup \{w, \lnot w\}$ give rise to a Boolean algebra $A = (\hat{X}, \mathbf{1}= w, \mathbf{0} = \lnot w, \land,\lor)$.

Proof: First we recall the definition of Boolean algebra:

A Boolean algebra is a set $ A $, equipped with two binary operations $ \land $ (called "meet" or "and"), $ \lor $ (called "join" or "or"), a unary operation $ \neg $ (called "complement" or "not") and two elements $ 0 $ and $ 1 $ in $ A $ (called "bottom" and "top", or "least" and "greatest" element, also denoted by the symbols $ \bot $ and $ \top $, respectively), such that for all elements $ a $, $ b $, and $ c $ of $ A $, the following axioms hold:

Property	Meet	Join	Name
Associativity	$a \land (b \land c) = (a \land b) \land c$	$a \lor (b \lor c) = (a \lor b) \lor c$	Associativity
Commutativity	$a \land b = b \land a$	$a \lor b = b \lor a$	Commutativity
Absorption	$a \land (a \lor b) = a$	$a \lor (a \land b) = a$	Absorption
Identity	$a \land 1 = a$	$a \lor 0 = a$	Identity
Distributivity	$a \land (b \lor c) = (a \land b) \lor (a \land c)$	$a \lor (b \land c) = (a \lor b) \land (a \lor c)$	Distributivity
Complements	$a \land \neg a = 0$	$a \lor \neg a = 1$	Complements

Then we proceed: Commutativity, associativity are true because $\min,\max$ are commutative and associative. Absorption and distributivity are true, because $\{ \min, \max \}$ over the real numbers is a distributive lattice. We prove the complements, as the identity has been proven for all semantic spaces:

$$(x \lor (\lnot x)) \operatorname{rel} w = \max( k(w,x) -k(w,x))\phi(w) =$$ $$|k(w,x)| \phi(w) =^{\text{ separable}} |\pm 1| \phi(w) = $$ $$\phi(w) = w \operatorname{rel} w = \mathbf{1} \operatorname{rel} w$$

Similarly:

$$(x \land (\lnot x)) \operatorname{rel} w = \min( k(w,x) -k(w,x))\phi(w) =$$ $$-|k(w,x)| \phi(w) =^{\text{ separable}} -|\pm 1| \phi(w) = $$ $$-\phi(w) = (\lnot w ) \operatorname{rel} w = \mathbf{0} \operatorname{rel} w$$

Construction of Separable Semantic Spaces

Idea: Given a finite set $X$ with a positive definite kernel on $X$, such that $S = (X,k)$ is a semantic space, then the Gram matrix is positive definite and so invertible. Let now $Y = \{ y_i=\pm 1| 1 \le i \le |X| = n \}$ be 'any' labeling of the $x_i$. Then we can find an extension $\hat{k} : \hat{X} \times \hat{X} \rightarrow \mathbb{R}$ of $k$, where $\hat{X} := X \cup {w}, w \notin X$ such that, there exist $c_i in \mathbb{R}$ with:

$$ \forall 1 \le i \le n : y_i = \operatorname{sign}(\left [ \sum_{j=1}^n c_j y_j k(x_i,x_j) \right ]) $$

This would allow us to define:

$$\hat{k}(w,x_i) := \sum_{j=1}^n c_j y_j k(x_i,x_j)$$

and so we would get a separable semantic space:

$$ \forall 1 \le i \le n : y_i = \operatorname{sign}(\hat{k}(w,x_i))$$

But because the Gram matrix $G=(k(x_i,x_j))_{i,j}$ is invertible, the first equation with the $c_j$ can be solved by setting:

$$c := G^{-1} y$$

where $c = (c_1,\cdots,c_n), y=(y_1,\cdots,y_n$. The same strategy has been used to show, that 'primes are linearly separable': Are primes linearly separable?

To make things more concrete, here is a semantic space with $n$ elements, which is separable for every $Y$:

• $X = \{1,\cdots,n\}$
• $k(i,j) = \frac{\gcd(i,j)^2}{ij}$
• Let $Y = \{ y_i = \pm 1 | 1 \le i \le n\}$ be any set on $n$ elements consisting of (\pm 1).

Then set

$$c := G^{-1} y$$

and put

$$\hat{k}_0(w,i) := \sum_{j=1}^n c_j y_j k(i,j)$$ $$\hat{k_0(w,w) := \sum_{1 \le i,j \le n} c_i y_j c_j y_j k(i,j)}$$

$$\hat{k_0(i,j) := k(i,j) \text{ }\forall 1 \le i,j \le n}$$

where $G = (k(i,j))_{1 \le i,j \le n}$ is the Gram matrix, which is shown below, to have non-zero determinant. We have to normalize the kernel:

$$\hat{k}(w,i) := \frac{\hat{k}_0(w,i)}{\sqrt{\hat{k}_0(w,w) k(i,i)}}$$ $$\hat{k}(w,w) := 1$$

By the discussion above this method produces a semantic space $S$ which given $Y$ is separable for $Y$. This concludes also the proof, that Boolean algebras can be constructed using semantic spaces.

Construction of semantic spaces of logic from finite groups

It is possible to construct to a finite group $G$ a semantic space $S := ( G,k )$ where the elements of $X=G$ are the group elements and $k$ is a positive semidefinite kernel on the group. Here is the construction of the kernel:

Let $G$ be a finite group with $n = |G|$ elements. By Cayley's theorem for finite groups, we have an injective homomorphism of groups:

\begin{equation}\label{eq1} \pi : G \rightarrow S_n, g \mapsto \pi(g) \end{equation}

where each group element $g$ is mapped to the permutation of the symmetric group $S_n$ on $n=|G|$ elements, wich it generates by left multiplication:

$$\pi(g): G \rightarrow G, x \mapsto g \cdot x$$

We want to associate to each group element $g$ a matrix and then use the Frobenius inner product to define a positive semi-definite function $k$ on the group $G$ as follows:

It is known , see for instance "The Kendall and Mallows Kernels for Permutations" by Yunlong Jiao and Jean-Philippe Vert, that the Kendall-tau function can be made to a positive semi-definite kernel $k$ for permutations.

Let $\mathbf{1}_{\{x\}}$ be the indicator function, which is $=1$ if the boolean variable $x$ is true and $0$ if the boolean variable $x$ is false, and let:

\begin{equation}\label{eq2} \phi: S_n \rightarrow M_n(\mathbb{R}), \sigma \mapsto (\mathbf{1}_{\{\sigma(i)>\sigma(j)\}}-\mathbf{1}_{\{\sigma(i)<\sigma(j)\}})_{1\le i,j \le n} \end{equation}

where $M_n(\mathbb{R})$ denotes $n \times n$ matrices over $\mathbb{R}$.

The embedding from the finite group $G$ to $M_n(\mathbb{R})$ is then given by:

\begin{equation}\label{eq3} \psi: G \rightarrow M_n(\mathbb{R}), g \mapsto \frac{1}{\sqrt{n(n-1)}}\cdot \phi(\pi(g)) \end{equation}

We use the Frobenius inner product on $M_n(\mathbb{R})$, which is given by:

\begin{equation}\label{eq4} \left \langle A, B \right \rangle := \operatorname{tr}(A \cdot B^T) \end{equation}

to define a positive semi-definite, symmetric function $k$ on $G$:

\begin{equation}\label{eq5} k: G \times G \rightarrow \mathbb{R}, (g,h) \mapsto \operatorname{tr}(\psi(g) \cdot \psi(h)^T) \end{equation}

The kernel $k$ can be normalized to take value between $-1$ and $1$ and since $k(g,g)$ is constant for all $g \in G$, we can normalize $k$ to take values $k(g,g) = 1$ and so we get a semantic space of logic given a finite group $G$.

It remains in each case to specify a perspective vector $w$.