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From: Mild Shock <janburse@fastmail.fm>
Newsgroups: comp.theory,sci.logic
Subject: =?UTF-8?Q?Re:_G=c3=b6del's_Basic_Logic_Course_at_Notre_Dame_=28Was:?=
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Date: Tue, 23 Jul 2024 22:52:31 +0200
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In Montague Semantics the bivalance principle
is expressed by the type "prop". You find that in
the intial definition of Montague:

 > 2 Intensional Logic
 > By ME_a is understood rhe set of meaningful expresions of type a
 > 1. Every variable and constant of type a is ME_a.
https://www.cs.rhul.ac.uk/~zhaohui/montague73.pdf

If I am not mistaken then Montague simply uses
the letter "t" for the type "prop".

Mild Shock schrieb:
> Of course you can restrict yourself to
> only so called "decidable" sentences A,
> 
> i.e. sentences A where:
> 
> True(L,A) v True(L,~A)
> 
> But this doesn't mean that all sentences
> are decidable, if the language allows for
> example at least one propositional variables p,
> 
> then you have aleady an example of an
> undecidable sentences, you even don't
> need anything Gödel, Russell, or who knows
> 
> what, all you need is bivalence, which was
> already postualated by Aristoteles.
> 
> Principle of bivalence
> https://en.wikipedia.org/wiki/Principle_of_bivalence
> 
> if you assume that a propostional variable
> is "variably", meaning it can take different truth
> values depending on different possible worlds,
> 
> or state of affairs, or valuations, or how ever
> you want to call it. Then a propositional variable
> is the prime example of an undecided sentence.
> 
> Mild Shock schrieb:
>> Thats a little bit odd to abolish incompletness.
>> Take p, an arbitrary propositional variable.
>> Its neither the case that:
>>
>> True(L,p)
>>
>> Nor is ihe case that:
>>
>> True(L,~p)
>>
>> Because there are always at least two possible worlds.
>> One possible world where p is false, making True(L,p)
>> impossible, and one possible world where p is true,
>>
>> making True(L,~p) impossible.