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Adam I. Gerard
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Non-Eternalist Logic

From: 2018.

Logic was traditionally conceived as:

  1. A priori - known or intuited through reason alone
  2. Eternal - existing outside of time or being unchanged
  3. Classical - two-valued

Many systems that deny 3 have been studied. Putnam argued that 1 is incorrect - our discovery of logic is at least partly empirical (a posteriori) and might need to be recast according to Quantum Mechanics (and the logic thereof).

I will attend to 2 - logics that self-modify, that change.

This class of logics is not be confused with well-known formal systems that pertain to belief revision (like AGM) or logic systems that model temporal relationships (like modal-based temporal logics). Instead, these logics change.

Regarding Classical Logic

Classical Logic was and is most often still held to be:

  1. Eternal - timeless
  2. Static - fixed and unchanging
  3. Invariant across time and space - even delimiting what's possible within spacetime. What governs all things in spacetime
  4. Abstract - existing outside space and time even - a calculus of abstract propositions (on some views)
  5. A Priori - known or intuited through reason alone

Non-Eternalist Logic is not explicitly non-Classical (in that every change could still be Boolean - e.g. as a superset of Classical Logic) but it is a repudiation of these often implicit, dogmatic, assumptions about Classical Logic that permeate the Philosophy of Logic (to echo Graham Priest's criticism elsewhere).

Non-Eternalist Logics are logics that change. There are not to be confused with logics that model change (but which are themselves static and fixed) nor logics that are time- or context-sensitive (also fixed and static in terms of grammar, axioms, truth-conditions, etc.).

How Can Logics Change?

In what ways can a logic change? Some ideas:

  1. Axiom sets that change - axiom sets that are expanded, reduced, or modified over time
  2. Semantics that change - truth assignments change (the truth-tables or definitions themselves and not the physical truth-conditions or domain)
  3. Grammatical changes - the underlying grammar of the logic changes

Such that a specific wff may change in its truth-assignments (e.g. - T or F in Classical Logic).

Precursors

This is not so radical:

  1. We can for instance (and often do) introduce temporal predicates to model time-dependent statements (“it’s raining now”).
  2. This approach models sentences or propositions as being time and/or context-sensitive. But the semantic and/or grammatical rules of the formal apparatus do not themselves change only the real-world conditions the statements are about.
  3. Other systems attempt to append a time-dependent operator to each statement. In such models, the statement does not change in its truth assignment. p @ t1 = T, p @ t2 = F, etc.
  4. Changes to the axioms mean that propositions or their truth-evaluations are themselves altered.
  5. Naturally, we can consider techniques from group theory or view such changes as transformations / rotations - invariance between transformations is precisely captured by propositional stability.

When two logics comprise a dual pair they can be modeled as cross-logics and the specific operations between them in terms of computing their propositional truth-values can be modeled as transactions (e.g. - discrete operations - though these could be understood as financial operations - perhaps a natural evolution to information and credibility markets).

I believe that this is less a philosophical proposal to radically replace some system of thinking with another. Instead, it's an attempt to more accurately model what’s already being done, in human life, in our ordinary day-to-day lives.

Epistemic Logics

We can replace at least some epistemic notions (particularly those that use logical operators or predicates) with the idea of an interface (and add some other notions) while reducing some of the grammatical baggage (trading an enlarged semantics for a smaller overall language footprint).

Here, the temptation is to treat two so-interfaced logics as one (continuous but semantically or at least proof-theoretically richer) logic - each self-consistent and contained, but sprawling and overlapping.

The potential scope of application includes epistemic modeling of belief change (like AGM) and logic revision (axiom revision like Euclidean to Hyperbolic geometry).

We can take an alternative approach here too - shedding the epistemic wrapper to review changes to the raw underlying logic itself.

Logical Module

See: logical module

Applied to two logics as an interface and allowing for:

  1. Opposing or contradictory wff’s to be believed simultaneously without contradiction in the logic (like considering two alternatives).
  2. Hegel’s critics unfairly criticized the Science of Logic as permitting outright contradictions (which renders a logic useless for science which minimally requires falsification/falsifiability)
  3. The module also permits reasoning about epistemic transitions between logics (or theses - theory change).
  4. Hegel’s critics were unfair to his views and we see how the simple addition of an epistemic abstraction layer removes some of the perceived wonkiness of his view - many early thinkers read the dialectic as ~(P & ~P) > (P & ~P).
  5. A stepping stone to a better way of analyzing these considerations (which propositional stability captures).

Tracking Changing Thinking Patterns

Classical Logic is well-known to be limited in at least the following two ways:

  1. It's monotonic - information change does not impact changes in truth-determination - e.g. (P > Q) does not entail (~P > ~Q) ((A > C) & (A & B > D)) > (A & B > C & D) - Classical Logic is best-suited for perfect idealizations - perfect knowledge, lack of ambiguity, or epistemic states rather than real-life scenarios where information is changing, dynamic, and our knowledge limited.

i. Note for instance the distinction the imperative logic condition IF p DO x (used in programming) and the material conditional IF p THEN q (in Classical Logic).

ii. Control-flow programming is also trivially a mixed-logic scenario - in that it combines Boolean as well as imperative logics to perform operations.

  1. It's bivalent - depending on one's view about truth-values, this is a bad thing - propositions are either true or they are false. They are not, for example, indeterminate (as under a Kleene 3-Value Logic).

One other potential advantage of this approach is to more-effectively model changes in the logic of one's thoughts rather than just the thoughts or beliefs themselves (which AGM, for example, captures).

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