[FOM] FOM: The Law of Excluded Middle

praatika@mappi.helsinki.fi praatika at mappi.helsinki.fi
Mon Oct 17 03:30:54 EDT 2005

There is now a relatively wide agreement, among those who are aware about 
the relevant differences between them (many are not), about the use 
of "LEM" and "PB":

The Law of the Excluded Middle (LEM) is a schema in the object language:  
A v -A. Each instance of it may be taken, according to the classical 
logician, as a logical axiom (of course, not every formalization of 
classical logic in practice does exactly that; but they all have something 
logically equivalent to this). 

The Principle of Bivalence (PB), on the other hand, is a single universal 
statement in the metalanguage: every statement (of the object language) is 
either true or false. 

There is certainly a close connection between LEM and PB, but they are not 
trivially equivalent. There are logical systems (e.g. certain many-valued 
logics and supervaluational languages) in which LEM is valid but PB does 
not hold, and vice versa. 

Brouwer, for example, did not distinguish LEM and PB, but given that he 
could not be aware of the later developments, it would be uncharitable to 
accuse him for that. Lukasiewicz emphasized the need to distinguish 
between them. Also Dummett, in the intuitionistic camp, has underlined 
their difference. But even today, it seems that their difference and exact 
relation is unclear to the great majority. 

Best, Panu

Panu Raatikainen

Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
Department of Philosophy
P.O. Box 9
FIN-00014 University of Helsinki
E-mail: panu.raatikainen at helsinki.fi

> Q1 Is there any generally accepted meaning of 'the law of excluded
> middle' or related expression as it occurs in foundational debates?
> Q2 Does any of the contributors to the FOM exchange have a specific
> preferred sense for 'the law of excluded middle' or related expression
> as it occurs in foundational debates?
> Q3 Is there a place in the literature that discusses the range of
> meanings that this expression has been made to carry over its long
> history?
> Q4. Do all of the contributors agree as to what this law is about:
> strings of characters, people's thoughts, people's assertions, abstract
> propositions in the sense of Alonzo Church 1956, "meaningful sentences"
> in the sense of Tarski 1956 or something else?
> Q5 Has any contributor to this debate prefaced his or her contribution
> with remarks sufficient to clarify the intended meaning of this
> expression?

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