Excluded Middle, Bivalence, and Tertium Non Datur

Dave Gudeman comments:

I was surprised to see you distinguishing between bivalence and the LEM. As far as I can tell, in the traditional and most common formulations, they are identical.

Here is the way I understand it.  They are not identical.  Excluded Middle is a law of logic, whereas Bivalence is a semantic principle. (See Michael Dummett, Truth and Other Enigmas, Harvard UP, 2nd ed. , 1980, p. xix; Paul Horwich, Truth, Oxford UP, 2nd ed., 1998, p. 79) If 'p' is a place-holder for a proposition, any proposition, then Excluded Middle is:

LEM. p v ~p.

If 'p' is a propositional variable, and we quantify over propositions, then we have the universal quantification

LEM*. For all p, p v ~p.

It is understood that the wedge in the above formulae signifies exclusive disjunction. Why is that understood? Because both p and not-p is excluded by the Law of Non-Contradiction:

LNC. ~(p & ~p).  

If I may be permitted parenthetically to wax poetic in these aseptic precincts, (LNC) possesses a 'dignity' in excess of that possessed by (LEM). What I mean is that there are some fairly plausible counterexamples to (LEM), but none that are very plausible to (LNC).  Few philosophers are dialetheists; many more accept truth-value gaps.

The laws of logic are purely formal: they abstract from content or meaning. They are syntactic principles. Bivalence, by contrast, is a semantic principle. It goes like this:

BV. Every proposition is either true or false.

Tertium non datur means that a third is not given: there is no third truth value.  (TND) is also a semantic principle:

TND. No proposition is neither true nor false.

So the difference between (LEM) and (BV) is that the first is a syntactic principle and the second a semantic principle. But is this a difference that makes a difference? Is there a conceivable case where (LEM) is true but (BV) false?  I don't know the answers to these questions. Either that or I forgot them.

But if you conflate the two principles,  then you are in good company. W. V. O. Quine, Mathematical Logic, Harvard UP, 8th ed., 1976, p. 51: ". . . the law of excluded middle, which is commonly phrased as saying that every statement is either true or false . . . ."

Comments

[The Bad Ostrich]

Depends what is meant by 'false'. I always read it is 'not true', in which case LEM and BV coincide (I think).

I believe Peirce introduced the concept of truth value (and the term?).

I suggested in a separate comment, or may have been by email, that we can define ‘not’ in such a way that LEM is true. Take all the possible states P of the world where p is true. If p is contingent, P is a proper subset of the set W of all possible states whatever of the world. Let Q be the complement of P, i.e. all possible states in W which do not include members of P. And let q be the proposition that is true for all states in Q, and no states in P. Then it is necessary that p or q, since P and Q exhaust all possible states of the world. Hence for any proposition p we can define a proposition q such that necessarily one or the other is true. Let there be an operator which converts the name ‘p’ to an name of q, say ‘not-p’. Then under that definition of ‘not’, LEM holds.

[BV]

Peirce introduced the concept of truth value only if he introduced the notion of propositional function. Did he?

[The Bad Ostrich]

According to Church, in Peirce, C.S., 1885, “On the Algebra of Logic: A Contribution to the Philosophy of Notation”, American Journal of Mathematics, 7(2): 180–202.

But I have not 'verified' this.

[The Bad Ostrich]

But what of my proposal that we can define negation in such a way that LEM is true?

[John]

The law of excluded middle and the principle of bivalence are very clearly distinguished in the literature on vagueness. Supervaluationist approaches to vagueness are precisely those approaches that accept the law of excluded middle while denying bivalence. (Basically, supervaluationists believe that the law of excluded middle holds for each precisification of a proposition containing a vague term, but deny the principle of bivalence, since - typically - supervaluationists believe that a vague proposition is true if and only if every precisification is true. They call this 'supertruth', and define truth *as* supertruth. Thus, some propositions are neither true nor false, although the law of excluded middle is preserved.) As I recall, Timothy Williamson's book Vagueness is very good and very clear on this point. But (and here I am less sure) the supervaluationist approach to vagueness is originally due to Kit Fine.

[BV]

John,

Thanks for the helpful response. Now let me see if I understand you.

Fine was the first to work out the details of a supervaluationist approach, but he was anticipated by Henryk Mehlberg.

Fine and Co. proceed on the assumption that vagueness is not a merely epistemic matter: it does not reflect our ignorance. It is not that we don't know where the boundary is between the non-tall and the tall; the problem is that the extension of the predicate 'tall' is not wholly determinate or determinate across the board: in some of its uses it has a clear application and in others it does not. And the same holds for 'not tall.' Suppose Tom is not clearly tall and not clearly non-tall. He is at the borderline. Max Black said that at the borderline we have a TV glut; Fine that we have a TV gap. On Black's view, the borderline predication is both true and false. On the Fine view, the borderline predication is neither true nor false.

But we can precisify 'tall' and when we do we get a sharp boundary between the tall and the non-tall. We can then say that 'either Tom is tall or Tom is not tall' is true on all precisifications of 'tall' and thus that the disjunction is true as a substitution instance of LEM. In this way LEM is upheld.

Bivalence, however, fails. (BV) says that every sentence is either true or false. 'Tom is tall or Tom is not tall' is a counterexample thereto.

In this way (LEM) and (BV) come apart.

[BV]

Ostrich,

>>But what of my proposal that we can define negation in such a way that LEM is true?<<

You are missing the point of the discussion. The question is not whether LEM is true, but how it differs from BV, if it does differ.

[The Bad Ostrich]

>>You are missing the point of the discussion. The question is not whether LEM is true, but how it differs from BV, if it does differ.

Then define 'false' as 'not true', where 'not' has the meaning that I have defined above. Then it follows from my definition of 'not' that every proposition is either true or not true, and from my definition of 'false' that every proposition is either true or false. Thus LEM = BV.

A subtlety: is 'p is not true' equivalent to 'not-p is true'?

>>We can then say that 'either Tom is tall or Tom is not tall' is true on all precisifications of 'tall' and thus that the disjunction is true as a substitution instance of LEM. In this way LEM is upheld.

Again, you need to distinguish 'Tom is not tall' from 'it is not the case that Tom is tall'. These are equivalent in predicate calculus, but not in English I think.

[The Bad Ostrich]

The Wikipedia article says

The statement "Pegasus likes liquorice or Pegasus doesn't like liquorice", however, is an instance of the valid schema p or not-p

That is the same error we discussed earlier. ‘Pegasus likes liquorice’ which implies someone likes liquorice, and and ‘Pegasus does not like (i.e. dislikes) liquorice’, which implies someone dislikes liquorice, are contraries, not contradictories. Thus the statement "Pegasus likes liquorice or Pegasus doesn't like liquorice" is not an instance of the valid schema p or not-p, at least where ‘not’ is interpreted as the true negation operator.

[David Gudeman]

Mr. Ostrich, the question is whether

1. for all p . p=T v p=F

means the same as

2. for all p . p v ~p

I claim that 1 is the traditional definition of LEM and it was on this ground that I questioned whether there is a distinction between LEM and bivalence. (I should note that some authors say that bivalence is

3. for all p . p | ~p

where "|" is exclusive or--this is clearly not the same as LEM).

So the issue of negation that you bring up is pivotal. If one can assume that ~F=T and ~T=F, and that asserting that p is the same as asserting that p=T, then 1 and 2 are equivalent. These conditions, however, do not hold for all logics.

I was assuming that this was understood, otherwise there is no issue to discuss, so my question to Mr. Vallicela was really whether he believes that 2 is the traditional way to define LEM (I think it is not), or if not, what his grounds are for preferring the new-fangled definition. It seems to me, that some logicians have change the definition of LEM from 1 to 2 in order to claim that their multi-value logics still satisfy LEM although they do not satisfy bivalence. That seems to me like a, let us say "questionable", motivation to change the definition of LEM.

[The Bad Ostrich]

Dave, note that Aristotle (Metaphysics IV.7) defines true and false as follows:

οὐδὲ μεταξὺ ἀντιφάσεως ἐνδέχεται εἶναι οὐθέν, ἀλλ᾽ ἀνάγκη ἢ φάναι ἢ ἀποφάναι ἓν καθ᾽ ἑνὸς ὁτιοῦν. δῆλον δὲ πρῶτον μὲν ὁρισαμένοις τί τὸ ἀληθὲς καὶ ψεῦδος. τὸ μὲν γὰρ λέγειν τὸ ὂν μὴ εἶναι ἢ τὸ μὴ ὂν εἶναι ψεῦδος, τὸ δὲ τὸ ὂν εἶναι καὶ τὸ μὴ ὂν μὴ εἶναι ἀληθές

… there cannot be an intermediate between contradictories, but of one subject we must either affirm or deny any one predicate. This is clear, in the first place, if we define what the true and the false are. To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true

From which LEM immediately follows. Either we say (1) of what is the case that it is the case, (2) of what is not the case that it is not the case, (3) of what is not the case that it is the case, (4) of what is the case that it is not the case.

But (1) and (2) are defined as being true, (3) and (4) are defined as being false. These are the only possibilities, ergo etc.

If Aristotle is the 'traditional interpretation', then it seems he begins with BV.

However in On Interpretation c.9, on the sea battle problem, he appears to go against this, but the passage has considerable difficulties of interpretation.

[The Bad Ostrich]

However I am not sure Aristotle’s proof works. Consider:

S says that it is boiling and it is boiling (TRUE) S says that it is freezing and it is freezing (TRUE) S says that it is freezing and it is boiling (FALSE) S says that it is boiling and it is freezing (FALSE)

Clearly these are cases of true and false. But do these cover all the cases? No, because there are many intermediate states between boiling and freezing. We want to add the assumption that there is nothing in between boiling and freezing. But that, by analogy, would be excluded middle.

That said, is it not built into the meaning of ‘false’ that the two Aristotelian cases of falsity as given cover all the cases? I.e. if S says that p then the only way for his statement to be false is if it is not the case that p, likewise if S says that not-p then the only way for his statement to be false is if it is the case that p. Isn’t that built into the meaning of ‘false’? If so, (a) we have established LEM and (b) we have neatly tied it to BV.

[David Gudeman]

Mr. Ostrich, Aristotle can only be construed as correct if one assumes that he was discussing a very limited subset of propositional sentences. It isn't just that there are lots of intermediate stages between boiling and freezing, it is that there are propositional sentences that have no definite truth value at all.

If S says that S is false, then is S true or false?

[The Bad Ostrich]

>>If S says that S is false, then is S true or false?

It depends what S means by 'false'. Aristotle says he is providing a definition of truth/falsity. Thus

S says that p and p (‘true’)
S says that not-p and not-p (‘true’)
S says that p and not-p (‘false’)
S says that not-p and p (‘false’)

These are the only 4 possibilities, on a syntactic basis. If this is the correct definition of true and false, there cannot be any propositional sentences ‘that have no definite truth value at all’, for ‘p’ is a placeholder for any sentence whatever.

You may question the definition of ‘true’ and ‘false’, but on what grounds? Clearly if S says that p and it is also the case that p, then what S says is guaranteed to be true. Likewise if S says that not-p and it is also the case that not-p, then S speaks truly.

The case is more difficult for falsity. If S says that p and not-p, does S necessarily speak falsely? Depends on the meaning of the negation operator. But as I have argued, this whole discussion hangs on the meaning of ‘not’. If I am right, ‘not’ rules out all the possible situations where p is true. Consider ‘the watch is on the table’. If that is true, the watch could be situated in any number of places on the table. Take all those situations P, then consider all the possible situations P* that are not one of them (watch on the floor, watch on my wrist etc). But P and P* are by definition all the possible situations, without remainder. So S either speaks truly, or falsely, tertium non datur.

You will object about situations, such as the watch being on my wrist, and my wrist on the table. Can we decide whether the watch is on the table, or not on the table? Well it may be difficult, but this does not touch Aristotle’s definition. If the watch is not on the table, and S says it is on the table, S does not speak the truth.

[David Gudeman]

Mr. Ostrich, sorry for being unclear. I was engaging in a sort of pun with "If S says that S is false". The letter S often refers to a sentence, so I was referring to a sentence like

1. This sentence is false.

The sentence 1 is a declarative sentence that is neither true nor false.

There are also non-paradoxical examples. If S says that Joe is tall and Joe is 5' 10", is S speaking truly? If S says that Andy Warhol's Soup-Can painting is an important work of art, is S speaking truly? If the label says that the bag of sugar is 5 pounds, but you measure it carefully and get 5.001 pounds, was the label in error?

These are just a few examples. There are various categories of declarative sentences that cannot be said to be definitively, precisely true or false.