Comment About The 'Accessibility Relation'
Though useful in all kinds of philosophical applications, the key innovation is, as David Lewis states, that "old disputes give way to new. Instead of asking the baffling question whether whatever is actual is necessarily possible, we could simply try asking: is the relation symmetric?" (David Lewis, 1996)
In other words, the oddity of 'necessarily possible' can be remedied by simply asking if two statements, one about 'necessity' and one about 'possibility,' are symmetric logically. However, this doesn't really solve the issue at all. Any property characterizing whether 'transitivity,' 'symmetry,' and so on, is dependent upon the very concepts of 'possibility' and 'necessity.' But in turn, this dependency means that all properties of modal logic depend on 'possibility' and 'necessity,' and not fundamentally 'symmetry,' 'transitivity' and so on. For instance, the first principle says that if a modal statement is classified under this principle, then whenever a statement about 'necessity' is uttered, it is the same as another statement about 'possibility.' To say that 'It's necessary that the apple is red' is to say that 'It's not possible that it is not the case that the apple is red.' In other words, to say that it's 'necessary' that something is to say that it is not 'possible' that something. In fact, in the example, I essentially said that it's necessarily possible that 'the apple is red' because the way I explain what I mean by 'necessity' is through a more detailed sentence about 'possibility,' albeit the statement about 'possibility' just happens to be one framed negatively, where the 'possibility' is that it is not 'possible' that it's not the case that something. Pointing this out helps to show that although the sentence about 'necessity' is characterized by duality, that duality is dependent on the concepts of 'necessity' and 'possibility,' and as long as these concepts are used, they inevitably give rise to the strange paradox about how something can be both 'necessary' and 'possible.' So talking of 'duality' or any other logical property like 'symmetry' or 'transitivity' doesn't clear up this paradox that the language of 'possibility' and 'necessity' give rise to.
Kripke may argue that this objection is a failure to understand what is supposed to do. only uses the concepts of 'necessity' and 'possibility' to show that the seeming contradiction of a 'necessary possibility' is in fact not a contradiction but actually a relation between two propositions characterized in one of the four types, 'duality,' 'symmetry' and so on. In other words, he might say that it's not 'necessary' and 'possible' propositions that are primary, but that they only appear to be until the relation comes to light.
But notice that if this were true, the concepts of 'duality' and so on would lack substance. As such, they wouldn't be useful in some instances, like when examining the idea of time as linear, or plotting out different courses of action. In fact, is useful only when it is articulated as a characterization between propositions of 'necessity' and 'possibility.' So although the property of 'transitivity' can be understood readily (as for instance, a bushel of straw gradually decreasing in size from a handful to two few blades), its application is only meaningful when looking at the specifics of say, the application of moral principles or laws from one society to another or others. At this point, 'necessity' and 'possibility' will certainly come into play since 'transitivity' involves at what point something is 'possible' and something isn't, or 'necessary' for that matter. Put this way, deducing from general concepts of 'transivity' and so on will inevitably lead to propositions about (and thus concepts of) 'necessity' and 'possibility.' cannot be understood without 'necessity' and 'possibility,' though 'possibility' and 'necessity' can be understood without the relation . So rather than understanding Kripke's contribution as getting rid of the seemingly messiness of 'necessary' and 'possible' propositions, it's better to understand his relation as helping to clarify what's happening when using principles in modal logic, or modal logic in general since, according to logicians, all modal propositions can be characterized in as at least one type of modal axiom. Still, this doesn't clear up the oddity of the very concepts of 'necessity' and 'possibility.'
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