Modal logic

3rdDecember 2020

Classical logic is a binary logic, it’s based on truth or falsity.
Nonetheless, not all propositional logic are binary.

It’s based on necessity, probability, possibilities. Not all statements are binary, indeed. In a broader sense:

  • epistemicsentences: they all start with “It is known that”. They have to do not with truth, but with what we know.
  • deonticstatements: they all start with “It ought to be the case that”, “It should be the case that”. They have to do with norms.
  • temporalstatements: at their premise, they have a temporal function, “It has been the case that”. The time factor here is what drives the reasoning.

An example of this type of logic dates back to Leibniz, who presented the theory of possible worlds.

+++ what is or is not necessarily the case, Aristotle

A good type of logic which could lead to an understanding of our reasoning method

Aristotelian modal logic

Of course, it’s applied in the context of syllogisms.

  • Plain or assertoric statements (A applies to every B)
  • +++

Aristotle’s work on modal logic was found to be incoherent and strongly criticized by most commentators. We have to wait until modern commentators (1930s) to give value to Aristotle’s modal logic approach

Problems with Aristotelian modalities

Examples:

  1. all lions are necessarily lions
  2. Everything lying down in a particular place is in fact a lion

In the first case this works universally and specifically, in the second case we could never have the certainty of what’s stated.

De dicto/de remodalities

  • de dicto: necessity of a property
  • de re: properties of the things of which the propositions assert something

Whatever is truede dictois not necessarily truede re.

ex. if x is

+++

Essentialism

It’s the essence of water to have the chemical structure H2O.

Natural necessity is not exactly logical necessity.

A different type of necessity, which may seem “weaker”, works only in specific instances. Truth of a statement is not valida priori, buta posteriori. Evaluations are contingent, strictly related to the essence of the object observed in its context.

If we claim some de re truth, some type of essentialism follows. Having some properties necessarily

Some essences can be related to the element contingently, and some others may concern the mode, the type, of the object in general


C.I. Lewis is considered to be the father of Modal Logic. He formulated this idea while he was studying material implication: he was in particular focused on its paradoxes. He used Modal Logic to overcome material implication issues.



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