We present a procedure which allows us to recover classical and nonclassical logical structures as \emph{concrete logics} associated with physical theories expressed by means of classical languages. This procedure consists in choosing, for a given theory $\mathcal T$ and classical language $\mathcal L$ expressing $\mathcal T$ , an observative sublanguage $L$ of $\mathcal L$ with a notion of truth as correspondence, introducing in $L$ a derived and theory-dependent notion of \emph{C-truth} (\emph{true with certainty}), defining a \emph{physical preorder} induced by C-truth, and finally selecting a set of sentences that are \emph{verifiable} (or \emph{testable}) according to $\mathcal T$ , on which a \emph{weak complementation} is induced by $\mathcal T$ . The triple consisting of the set of verifiable sentences, physical order and weak complementation is then the desired concrete logic. By applying our procedure we recover a classical logic as the concrete logic associated with classical mechanics and standard quantum logic as the concrete logic associated with quantum mechanics. We also show that our alternative view of standard quantum logic, which can be constructed in a purely formal way, can be provided with a physical meaning by adopting a recent interpretation of quantum mechanics that reinterprets quantum probabilities as conditional on detection rather than absolute. Our results then show that some nonstandard logics can be obtained as mathematical structures formalizing the properties of different notions of verifiability in different physical theories. More generally, they strongly support the idea that many nonclassical logics can coexist without conflicting with classical logic (\emph{global pluralism}), for they formalize metalinguistic notions that do not coincide with the notion of truth (described by Tarski's truth theory).We present a procedure which allows us to recover classical and nonclassical logical structures as \emph{concrete logics} associated with physical theories expressed by means of classical languages. This procedure consists in choosing, for a given theory $\mathcal T$ and classical language $\mathcal L$ expressing $\mathcal T$, an observative sublanguage $L$ of $\mathcal L$ with a notion of truth as correspondence, introducing in $L$ a derived and theory-dependent notion of \emph{C-truth} (\emph{true with certainty}), defining a \emph{physical preorder} induced by C-truth, and finally selecting a set of sentences that are \emph{verifiable} (or \emph{testable}) according to $\mathcal T$, on which a \emph{weak complementation} is induced by $\mathcal T$. The triple consisting of the set of verifiable sentences, physical order and weak complementation is then the desired concrete logic. By applying our procedure we recover a classical logic as the concrete logic associated with classical mechanics and standard quantum logic as the concrete logic associated with quantum mechanics. We also show that our alternative view of standard quantum logic, which can be constructed in a purely formal way, can be provided with a physical meaning by adopting a recent interpretation of quantum mechanics that reinterprets quantum probabilities as conditional on detection rather than absolute. Our results then show that some nonstandard logics can be obtained as mathematical structures formalizing the properties of different notions of verifiability in different physical theories. More generally, they strongly support the idea that many nonclassical logics can coexist without conflicting with classical logic (\emph{global pluralism}), for they formalize metalinguistic notions that do not coincide with the notion of truth (described by Tarski's truth theory).

Jan 1, 20139 citations

The Conditional in Quantum Logic

Gary M. Hardegree

Springer eBooksJan 1, 197675 citations

The conditional in quantum logic

Gary M. Hardegree

SyntheseDec 1, 197428 citations

On the concreteness of quantum logics

Pavel Pták, J. D. Maitland Wright

It is shown that for any quantum logic $L$ one can find a concrete logic $K$ and a surjective homomorphism $f$ from $K$ onto $L$ such that $f$ maps the centre of $K$ onto the centre of $L$ . Moreover, one can ensure that each finite set of compatible elements in $L$ is the image of a compatible subset of $K$ . This result is "best possible" - let a logic $L$ be the homomorphic image of a concrete logic under a homomorphism such that, if $F$ is a finite subset of the pre-image of a compatible subset of $L$ , then $F$ is compatible. Then $L$ must be concrete. In the second part one considers embeddings into concrete logics. It is shown that any concrete logic can be embedded into a concrete logic with preassigned centre and an abundance of two-valued measures. Finally, one proves that an arbitrary logic can be mapped into a concrete logic by a centrally additive mapping which preserves the ordering and complementation.It is shown that for any quantum logic $L$ one can find a concrete logic $K$ and a surjective homomorphism $f$ from $K$ onto $L$ such that $f$ maps the centre of $K$ onto the centre of $L$. Moreover, one can ensure that each finite set of compatible elements in $L$ is the image of a compatible subset of $K$. This result is "best possible" - let a logic $L$ be the homomorphic image of a concrete logic under a homomorphism such that, if $F$ is a finite subset of the pre-image of a compatible subset of $L$, then $F$ is compatible. Then $L$ must be concrete. In the second part one considers embeddings into concrete logics. It is shown that any concrete logic can be embedded into a concrete logic with preassigned centre and an abundance of two-valued measures. Finally, one proves that an arbitrary logic can be mapped into a concrete logic by a centrally additive mapping which preserves the ordering and complementation.

Applications of MathematicsJan 1, 19858 citations

Assumptions and Interpretations of Quantum Logic

Bas C. van Fraassen

Springer eBooksJan 1, 19814 citations

\mathcal T

and classical language

\mathcal L

expressing

\mathcal T

, an observative sublanguage

L

\mathcal L

with a notion of truth as correspondence, introducing in

L

a derived and theory-dependent notion of \emph{C-truth} (\emph{true with certainty}), defining a \emph{physical preorder} induced by C-truth, and finally selecting a set of sentences that are \emph{verifiable} (or \emph{testable}) according to

\mathcal T

, on which a \emph{weak complementation} is induced by

\mathcal T

. The triple consisting of the set of verifiable sentences, physical order and weak complementation is then the desired concrete logic. By applying our procedure we recover a classical logic as the concrete logic associated with classical mechanics and standard quantum logic as the concrete logic associated with quantum mechanics. We also show that our alternative view of standard quantum logic, which can be constructed in a purely formal way, can be provided with a physical meaning by adopting a recent interpretation of quantum mechanics that reinterprets quantum probabilities as conditional on detection rather than absolute. Our results then show that some nonstandard logics can be obtained as mathematical structures formalizing the properties of different notions of verifiability in different physical theories. More generally, they strongly support the idea that many nonclassical logics can coexist without conflicting with classical logic (\emph{global pluralism}), for they formalize metalinguistic notions that do not coincide with the notion of truth (described by Tarski's truth theory).

It is shown that for any quantum logic

L

one can find a concrete logic

K

and a surjective homomorphism

f

from

K

onto

L

such that

f

maps the centre of

K

onto the centre of

L

. Moreover, one can ensure that each finite set of compatible elements in

L

is the image of a compatible subset of

K

. This result is "best possible" - let a logic

L

be the homomorphic image of a concrete logic under a homomorphism such that, if

F

is a finite subset of the pre-image of a compatible subset of

L

, then

F

is compatible. Then

L

must be concrete. In the second part one considers embeddings into concrete logics. It is shown that any concrete logic can be embedded into a concrete logic with preassigned centre and an abundance of two-valued measures. Finally, one proves that an arbitrary logic can be mapped into a concrete logic by a centrally additive mapping which preserves the ordering and complementation.