What is the difference between Law of Excluded Middle and Principle of Bivalence?

Question

Law of Excluded Middle:

In logic, the law of excluded middle (or the principle of excluded middle) is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is. The principle should not be confused with the principle of bivalence, which states that every proposition is either true or false, and has only a semantical formulation.

Source: http://en.wikipedia.org/wiki/Law_of_excluded_middle

Principle of Bivalence:

In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false. A logic satisfying this principle is called a two-valued logic or bivalent logic. In formal logic, the principle of bivalence becomes a property that a semantics may or may not possess. It is not the same as the law of excluded middle, however, and a semantics may satisfy that law without being bivalent.

The principle of bivalence is related to the law of excluded middle though the latter is a syntactic expression of the language of a logic of the form "P ∨ ¬P".The difference between the principle and the law is important because there are logics which validate the law but which do not validate the principle.

Source: http://en.wikipedia.org/wiki/Principle_of_bivalence

I'm not quite sure I get the difference. It seems that 'excluded middle' is a syntatic problem and 'bivalence' would be a semantic one. Is this correct? Also, it seem that in the realm of bivalence, stating that "P" is false, doesn't necessarily mean "non-P" is true, which would be the case with the principle of the excluded middle. Is this correct?

I don't understand precisely in which situations one or the other principle are at play, it seems that they may appear together, but not necessarily. Can someone give me examples and help me clarify the differences?

Answer 1

OK, I think I've got it now:

• A certain logic validates the Law of the Excluded Middle (LEM) if the following is a theorem in the logic: p v not p

• A certain logic abides by the Principle of Bivalence (PB) if every well formed expression according to the logic has exactly one truth value: true or false

Some semantics may make it the case that LEM is true and PB is not true. Consider the following supervaluationist treatment of vague predicates. A statement such as

Schiphol is bald

will be supertrue (superfalse) iff under all (no) acceptable precisifications of the predicate "bald", the sentence comes out true. A precisification will have the form "... has n hairs", where, e.g., n = 0 is acceptable, but n = 10^6 is not. Sadly, the sentence above is supertrue -- which is the supervaluationist criterion for accepting it as true.

Luckier folk, such as, say, Andy, might come out bald according to some precisifications and not bald according to others. Thus,

Andy is bald

is neither supertrue nor superfalse: it lacks truth value, according to supervaluationism. PB, therefore, is false: that sentence is neither true, nor false. Now, what happens with a sentence of the form [p v not p], such as

Andy is bald or Andy is not bald

Well, such sentences will be true for all precisifications, because either Andy has n hairs or he doesn't, for all n. Therefore, the sentence comes out supertrue -- this is the supervaluationist for accepting it as true. Its negation ("it's not the case that Andy is bald or Andy is not bald"), by the same token, comes out superfalse.

The same will happen with every other vague sentence: the supervaluationist semantics validates LEM. Supervaluationism is a semantics that validates LEM but not PB.

Answer 2

It may help to have an example of a logic where the excluded middle doesn't hold. Probably the most well known one is Intuitionistic Logic, also known as Constructive Logic. It was formulated in the early part of the 20C in reaction to certain (mathematical) existence proofs where certain mathematical objects were shown to exist but no construction given, this was traced to use of the excluded middle. The intuitionists insisted on being given a construction.

It is correct here to say that not true=false. But there are other truth values. So the bivalence law doesn't hold.

It isn't correct to say that something can be true and false simultaneously. So the non-contradiction law does hold.

aside: Whereas classical logic is associated with Boolean algebras and standard set theory, intuitionistic logic has an associated Heyting algebra and categorical set theory (topos).

Answer 3

This is the initial thread to the discussion:

In logic, the law of excluded middle (or the principle of excluded middle) is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is. The principle should not be confused with the principle of bivalence, which states that every proposition is either true or false, and has only a semantical formulation.

^{Source: http://en.wikipedia.org/wiki/Law_of_excluded_middle}

This sloppy formulation of the law of excluded middle (for propositions) is slightly inaccurate (i.e., is mistaken) - although the cause of the inaccuracy (mistake) is very natural.

The law of excluded middle for propositions should instead read: Given any proposition, either it's true or it is not true. Or, alternatively, [given a two-valued logic where the two values are true and false] Given any proposition, either it's false or it's not false. More abstractly, but more precisely, it can be expressed as follows: Given any proposition, either it has property P or it doesn't have property P.

One law of excluded middle for natural numbers is: Given any natural number, either it is even or it's not even. One law of excluded middle for animals is: Given any animal, either it's a vertebrate or it's not a vertebrate.

Truth is not the point here - nor is falsity. Instead, the point is the logical exclusivity that (necessarily) holds between IS and ISN'T.

At this juncture, it might be helpful to state the law of excluded middle for properties, which is a second-order logical truth: Given any property and given any individual, either the individual has that property or it does not have that property. [Please note that it doesn't matter what the property is or what the individual is.]

The law of excluded middle for properties is a logical truth, not merely a logical law of classical two-valued logic. [It's very important to realize that not every logical law is a logical truth.]

The principle of bivalence - although a law of classical (two-valued) logic - is NOT a logical truth, because it has the same logical form as some (i.e., at least one) falsehood. The principle of bivalence is that Every proposition is either true or false.

This proposition (call it a principle, if you like) has the same logical form as the known falsehood Every number is either odd or prime. In sharp contrast, every proposition that has the same logical form as the proposition that Every proposition is either true or it isn't (i.e., Every proposition is either true or it is not true) is a logical truth.

The distinction at issue here is well known by experts, but it's a rather technical (though quite important) distinction. The author of the Wikipedia article appears to be admirably informed, but not an expert. [The entry for the principle of bivalence (that directly follows the initial thread) is also goofed up in several respects.]

By the way, there are a great many other issues that very frequently cause confusion concerning such topics as this one. In particular, it's necessary to know/learn the difference between a proposition and a sentence. For example, the declarative sentence I am female expresses a truth when my girlfriend utters it, but it expresses a falsehood when I utter it. And yet this is not a good reason for claiming that some proposition is both true and false.

Answer 4

The difference between Excluded Middle and Bivalence:

Excluded Middle says every proposition of the form P v ~P is true

Bivalence says every proposition is true or it is false

and that's all she wrote

(forget all the technical jousting)

Answer 5

Principle of Excluded Middle: "A proposition p and its negation ~p cannot be false together."

Principle of Non-Contradiction: "A proposition p and its negation ~p cannot be true together.

Principle of Bi-Valence: "A proposition is either true or false."

PEM and PNC forbid a proposition and its negation having the same truth value.

PB forbids a proposition being both true and false or neither true nor false.

Answer 6

I think this is not quite right, or at least doesn't quite bottom out the issues. I am no great expert but as I see it...

The PBV is not (afaik) part of the laws of A's logic.

The LEM would be a stipulation for true contradictory pairs that must be met for the dialectic process to work properly and to decide between contradictory propositions. That is to say, the LEM will hold wherever the proposition to be tested meets A's rule for contradictory pairs (RCP), which is that it must be one of a pair of which one must be true and the other false. This rule would be inviolable.

None of this would imply anything for the world itself, about which statements may take on various truth-values, even be half-true and half-false.

Thus, say, when Heraclitus states, 'We are and are not' this would violate the PBV but not the LEM. It would not violate the LEM because Heraclitus is not suggesting that either half of his statement is true or false but, rather, that the truth lies elsewhere. His statement does not meet the requirement of the RCP so the LNC/LEM would not be relevant.

This is how it seems to me for now. This would be important because it allows us to use A's logic as the basis for a logic of contradictory complementarity and thus reconcile this logic with the world-view of Heraclitus and his like. If we see the LEM and rule for contradictory pairs as more than a formal device then we will be limiting our world-view.

Answer 7

It is called Excluded Middle because there is nothing in-between those two values: F, and V. In Fuzzy Logic, for instance, there is something in-between: T would be 1, F would be 0, and there is an infinity of values in-between 0, and 1 (0.1, 0.11, 0.23, and so on). Excluding the middle means taking away whatever could be a moderate position, so it is always Yes or No to any question you may have; never a 'more or less' or a 'so so': are you black? Yes. Are you happy? No. If someone asked you, are you rich, and you answered, so so, they would say: No! That is not an ACCEPTABLE answer, mate. In life, it is either an ABSOLUTE YES or an ABSOLUTE NO... That is the World of Classical Logic, or the World of the EXCLUDED MIDDLE... Bi-valence means two values, so that it could refer to any two arbitrary values. If we say Principle of Bivalence in Classical Logic, then it refers to False, and True or 0, and 1. You could not have the middle, and still have three values, so say 0 0.5 1, but no 0.3 or 0.6. Yet, in Classical Logic, you only have two, and that is why we say that, in that world, bivalence is a principle. Notice that it is Law of the EM, but 'principle' of bivalence. That probably means we are surer about not having anything in the middle than we would be of having only two values to hang on to... I have just read about the Principle of Non-contradiction, which is different from the Ex-Falso, which is a consequence of the laws, and principles of CL. In my interpretation, bivalence does not imply exclusion of paraconsistency, so that we could still have 2 values at the same time, or 'the door is open' is true, and 'the door is open' is false at the same point in time, and all else (Ceteris Paribus). That is why we should need a principle to say 'no contradictions accepted', or Principle of Non-contradiction. With this one, we would exclude paraconsistency, so that if 'the door is open' is true, 'the door is open' cannot be false in Ceteris Paribus Worlds: it is either one or, without concomitance, the other.