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My understanding of many valued logic is that it is used in cases where an aspect of a proposition is vague or not well defined. The example Wikipedia gives is:

'This apple is red.' Upon observation, the apple is an undetermined color between yellow and red, or it is motled both colors. Thus the color falls into neither category " red " nor " yellow ", but these are the only categories available to us as we sort the apples. We might say it is "50% red". This could be rephrased: it is 50% true that the apple is red. Therefore, P is 50% true, and 50% false. " https://en.wikipedia.org/wiki/Principle_of_bivalence#Vagueness

It seems to me that this is only the case if 'red' is not well defined. If it were well defined as, for example, 'such that at least 90% of the apple's surface area were to reflect light at a wavelength of 650nm (give or take 25nm) when viewed under white light' then it would seem that there is, in fact, a sharp line between true and false, such that the proposition, 'the apple is red', is true if and only if the apple fits the criteria given in the definition of 'red'. This would seem to me to indicate that many valued logics are not useful in cases where a proposition is well defined, however, people like Graham Priest seem to hold the position that many valued logics can hold even in well defined cases, i.e., 'some real contradictions are true'.

My question is, is it true that many valued logics can hold in unambiguous cases, and if so, how?

  • There's also Zadeh's fuzzy logic en.wikipedia.org/wiki/Fuzzy_logic where truth values can take on any real value between 0.0 and 1.0 inclusive. So your 50%,90%,etc red are easily and quantitatively accommodated. – John Forkosh May 11 '17 at 23:05
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I'm not sure to correctly interpret your question...

The relevant discussion is about the difference between truth-value gaps and truth-value guts: see G.Priest, An Introduction to Non-Classical Logic: From If to Is, 2nd ed., page 127-on.

The three-values logic of Kleene and Lukasiewicz treat the third truth value i as neither true nor false (a truth-value gap), while other logics (like the LP logic, proposed by Priest himself) treat i as both true and false (a truth-value glut).

According to Priest, the concept pf truth-value glut is necessary to treat e.g. the Liar Paradox; in this case, the "liar" sentence: ‘this sentence is false’, is both true and false.

In this example we have no vagueness, but a case of a 'real contradictions that is true'.

  • I would tend to view the liar sentence as somewhat vague, or, if not, non-referral. by that I mean, it is not obviously clear what 'this sentence' refers to, if it refers solely to the words 'this sentence' then the paradox doesnt make sense, and if it refers to 'this sentence is false' then you get "(this sentence is false) is false" which seems to lead to an infinite regress, this is Steve Patterson's view I think. – Kiah May 11 '17 at 21:07

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