What is meant by the following: This law is well motivated in cases where we may be ignorant of the facts ...

Question

The folllowing text is taken from a part in a book (reference below) which are about mathematical philosophy.

"... the theorem of classical logic known as the law of excluded middle: for every proposition $P$, the disjunction of $P$ and its negation, $(P ∨ ¬P)$, is true. This law is well motivated in cases where we may be ignorant of the facts of the matter, but where there are facts of the matter. For example, the exact depth of the Mariana Trench in the Pacific Ocean at its deepest point at exactly 12.00 noon GMT on the 1st of January 2011 is, unknown, I take it. But there is a fact of the matter about the depth of this trench at this time. It was, for example, either greater than 11,000 metres or it was not. Contrast this with cases where there is plausibly no fact of the matter. Many philosophers think that future contingent events are good examples of such indeterminacies. Take, for example, the height of the tallest building in the world at 12.00 noon GMT on the 1st of January 2021. According to the line of thought we’re considering here, the height of this building is not merely unknown, the relevant facts about this building’s height are not yet settled. The facts in question will be settled in 2031, but right now there is no fact of the matter about the height of this building. Accordingly, excluded middle is thought to fail here. It is not, for example, true that either this building is taller than 850 metres or not."¹

I do not understand what the author means when he writes "This law is well motivated in cases where we may be ignorant of the facts of the matter, but where there are facts of the matter." (I do especially not understand what is meant by the dependent clause). Could you please help me?

¹ Colyvan, Mark. (2012). An introduction to the philosophy of mathematics. Cambridge University Press. Pages 7-8.

PS As the context is about mathematics and mathematical logic (and mathematical philosophy), I chose MSE. Please tell me if this question fits better on another site, such as https://ell.stackexchange.com/ or https://philosophy.stackexchange.com/.

Answer 1

It is my impression that the other answers may be starting at a level too advanced for the OP so I will try to clarify the picture. To follow Colivan here one needs at least a rudimentary knowledge of intuitionistic logic (IL). The main feature of an IL that distinguishes it from classical logic (CL) is the reliance on the law of excluded middle (LEM). This can be stated formally as $P\vee\neg P$ but to understand what this means think of your favorite proof by contradiction, for example, the equivalence of your pair of favorite definitions of compactness. Such proofs typically rely on the law of excluded middle as their key ingredient. More specifically, one "shows" that a certain mathematical entity "exists" by arguing that its non-existence would lead to a contradiction. This type of proof does not exhibit the entity whose existence is claimed. Mathematicians in the constructive mode object to this type of argument by saying that existence is construction, it is not the impossibility of non-existence.

More specifically, Colyvan's comment

This law is well motivated in cases where we may be ignorant of the facts of the matter, but where there are facts of the matter

seems to refer to a mentality where mathematical entities are assumed to exist in some ideal realm ("there are facts of the matter"), in which case we can make definite statements about them (a given property is either true or not true about these existing entities; there is no other possibility).

By contrast, if we don't assume that such "facts of the matter" are out there in any reliable sense, then it becomes plausible to challenge the reliance on LEM.

When Colyvan speaks about our "ignorance of these [existing] facts" perhaps he is alluding to the fact that a classical mathematician is satisfied with a proof of existence that's not constructive, i.e., that does not exhibit the entity claimed to exist in any convincing sense (i.e., convincing to a constructivist).

Answer 2

I think what Professor Colyvan is suggesting is, is that make sense when our-inability to determine the truth value is a result of ignorance, - merely epistemic, but, in cases o in-determina-cy, such as in accounts of vague-ness

-"how may hairs does a person need to lose to become bald"?,

or non existents, such as properties predicated of entities in fairy tales such as

" did 'tin tin like red meat? Where the book, does not specify the answer".

So he is speaking about cases like these where the fact is either not determinate, or not determin-ant.

He is also possibly considering cases of ill formed formulae "an incomplete statement, where x is a free variable,

• 'x is raining" where x is does uniquely specificy a maximally specific scenario.

However, he may also be worried about trouble-some case that might relate to our world in some sense

But I think he is more troubled by cases in reality, or in logic, such as the liars paradox, future contingents, or in-determinism/quantum probability where this is more controversial. Or Aristotles sea battle case, where the future under a presentist view point where only the present exists.

I think he is more troubled by the others case that might arise in reality and not. Such as counter-factuals, in-determinism, and in-determinate-ness about the future and past.

Its possibly in his view, that the law that might may only fail precisely when probability is a real thing , as he argues in a paper directed against coxs derivation of the probability calcululus. Such as in qm . That is controversial of course.

But if Cox theorem, for example, weree derivation of some other thing and not of prob-abilism, it may have not been an issue, or as much of one..

Even if it relied on it the law of excluded middle.

That is because arguably, the very probabilistic nature , in some sense, of qm that generates the possible counter-examples to the law of excluded middle.

Answer 3

For example consider schrodringer cat.; not considered as a transition probability,

"There is now a positive ontic probability in the actual world, for either outcome to occur now" That was in issue for instantaneous/simultaneous collapse theorist, outside the context of future contingents.

If positive probability, means actually possible not just possible, and this is construed to mean"at time,

"t, its now actually possible that A time t, and now actually possible that ~A at time t".

Where something is not actually possible, if its negation is the case. So if both events has positive probability and are actually possible, it cannot that its either true or its not, at that point in time.

In either case, it will not be true, that both outcomes have a positive probability, not equal to one and zero,. If A is true, one the other event will not be actually possible and have zero probability. if ~A, likewise, And thus it will not the case that both are 'actually possible simultaneous given the nature of that statement if it simultaneously described as non transition probability, under bi-valence.

Answer 4

See Colyvans article where he explicitly mentions it Colyvan, Mark, The philosophical significance of Cox's theorem, Int. J. Approx. Reasoning 37, No. 1, 71-85 (2004). ZBL1093.68113.

Of course so there is weird feedback loop because the in-determinism, no hidden variables often result from Gleason uniqueness theorem, KS, and so forth generally relies on the axioms of probability in some sense. Ie the axioms were never derived, only that the measure was unique.

Of course it would be in some sense odd, on the other hand, for the hard core HV non-contextual hidden variable proponent to deny classical probability, and logic, to deny probabilism and bi-valence, when that is often what they want to preserve. (at least the logic). Truth values are trivially additive. Although these proofs (Cox's) have been adapted, some argue that in some, sense there is a better justification for 'classical probability' are under non bivalent, non classical logic.

Adherence to classical probability is not necessarily the same thing as adherence to classical physics, the classical interpretation of probability or to deny in-determinism ot quantum mechanics. It just means kolmogorov's basic axioms, generally.