No, I'm not going to be able to get away with claiming that truth is beauty, and beauty, truth.
The first issue in understanding truth is recognizing that truth is a property of a proposition. (What's a proposition? A proposition is a claim.) A proposition that is true has a certain kind of correspondence with the world about which it is making a claim. A proposition that is false does not have this correspondence.
At the most basic level, what we want from this correspondence seems pretty obvious: what the propositions says about the world matches up with how the world actually is.
So, if the world we're talking about (which some philosophers or logicians will describe as our "universe of discourse") contains three green spheres and a red tricycle and nothing else, then the claim:
There are three green spheres and one red tricycle.
will be true, as will the claim
There are more spheres than tricycles.
and the claim
There are fewer red things than green things.
as well as the claim
There are no cubes.
All of these propositions are accurate in their descriptions of our universe of discourse (a universe of discourse which admittedly doesn't have much in it). The agreement between what the propositions are claiming and the world about which the propositions make their claims means that the claims are true.
On the other hand, for that same little world, the claims
There are three red spheres and one green tricycle.
There are only red things.
There are no green things.
There are more tambourines than spheres.
are all false. The descriptions these propositions offer of the universe of discourse do not correspond to how things really are in this universe.
When you have a complete inventory of the stuff in your world (and, if you like, of the things like spatial relations between objects, or parts-and-wholes relationships, and good stuff like that), it's not too hard to evaluate whether a given proposition is true or false. But in a world like ours, where you're not issued a full parts-list at the outset, telling whether a particular claim is true or false can get a lot harder.
There are some claims whose truth we can evaluate just on the basis of logical form (at least if we're not working with a non-standard logic). For example,
There is an elephant in the room or there is not an elephant in the room.
is a true claim, something we can know without even checking the room for elephants. The form of this claim can be expressed in more general terms, if we let P stand in for the proposition "There is an elephant in the room," as:
P or not-P
The underlying assumption is that propositions (like P) are either true or false (with no squishy in-between state of truthiness), which means either P is true, or it's false -- in which case not-P is true.
For similar reasons, we can judge a claim of the form
P and not-P
as false -- since a proposition and its negation can't both be true at the same time.
However, every now and then we have to face down a claim like
SchrÃ¶dinger's cat is dead or SchrÃ¶dinger's cat is not dead.
While the cat is in the box with the vial of cyanide rigged to open if the isotope decays -- that is to say, before we've opened the box and made an observation of the cat's state -- the official stand on this claim from quantum mechanics is that the cat is neither alive nor dead, but rather in a superposition of the two states. Indeed, prior to the measurement, what the QM folks hold as true is
SchrÃ¶dinger's cat is dead and SchrÃ¶dinger's cat is not dead.
In other words, non-standard logics end up playing a role in some realms of human endeavor (like quantum mechanics) that are concerned with getting a handle on what we can say about the world that's true. Quantum mechanics and other such fields of endeavor aside, though, standard logic usually gets the job done.
There are some other cases where we can assert that a claim is true (or false) on account of the definitions of the words in the claim. For example
All bachelors are unmarried.
is a true claim, at least in contexts where "bachelor" means "unmarried man". Similarly, the claim
At 1 atm pressure, water freezes at 0 oC and boils at 100 oC.
is true because these properties of water were built in to the definition of the Celsius temperature scale. There are plenty of other claims, though, where we need more than definitions in order to establish whether the claims are true, for example:
There are 6 liters of water in this kettle.
A given sample of water contains twice as many atoms of hydrogen as atoms of oxygen.
A bucket of water was used to melt the Wicked Witch of the West.
I love the sprogs.
To assess the truth of any of these claims, we need to be able to get some accurate information about the state of things in the world -- whether to measure the volume of water in the kettle (and to ensure that the liquid whose volume we're measuring is in fact water), or to determine the ratio of hydrogen to oxygen atoms in a water sample (which might involve drawing inferences from the volumes of gases collected in the electrolysis of water). Assessing the truth of the claim about the Wicked Witch of the West may require that we struggle with philosophical issues about fictional universes.
That claim about my feelings for the sprogs is an interesting one from the point of view of determining its truth. Assuming we can pin down a reasonable standard version of love, I am the only one with any direct access to the "data" of my feelings for the sprogs. Others may be able to detect clues in my behavior that are usually concomitants of love ... but that's not the same thing as being able to direct the feeling itself. Still, I assure you, the claim is true.
Many of us will also be inclined to say that
Murder is wrong.
is a true claim. However, it's harder to pin down precisely what kind of larger claim about the world this commitment of ours embodies. Is there a moral fact in the world about the wrongness of murder (and if so, can we expect the development at some future date of measuring devices that can detect moral facts like smoke-detectors detect smoke)? Are we making a claim about our emotional attitude toward murder (either individually or collectively)? Are we making a claim about the status of murder in our societal norms (which might be different from our emotional attitudes toward it)? Or about the advisability of murder given what we judge as our best interests?
Thinking hard about the metaphysical commitments that may be lurking beneath our moral assertions is the kind of thing that may send you back to thinking about something fluffier ... like SchrÃ¶dinger's cat
An important point here is that we can draw a distinction between how things are and what we know (or could know) about how things are. If you take empirical evidence to be important in helping us come to knowledge about how things are, then you will have good reason to believe that there are some true claims about the world that we don't have the data to recognize as true. Maybe someday we will get our hands on the evidence to establish some such claims as true. For other such claims, maybe the evidence will remain elusive.
There are at least a couple different ways people respond to the possibility that there may be quite a lot of truths out there that are hard for us to access. One response is to dispense with talk about truth or falsity in matters where there is not some reasonable way -- at least in theory -- to access the relevant facts about the world (or sensory data that would let us establish those facts). A different response is to entertain such talk while recognizing the possibility that we might never be in possession of data that would let us determine the truth values of some of our claims.
Some people are right at home dwelling in the possibilities. Other people will politely request that you give them just the established facts. A great deal may depend on whether we're talking about what we can establish about how things are, or if instead we're speculating about ways things could be.
This Basic Concepts post goes out, with my thanks, to Nick, who made a generous donation to my DonorsChoose challenge. The sprogs are still thinking about how to illustrate the concept of truth in a picture -- if they figure it out, we'll be getting you some artwork, too!
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Isn't this because for those people, the truth of this statement (like your examples of bachelors and the behavior of water) is built into the definition of "murder?" The wrongness of a particular killing is what makes it murder and not something else, such as self-defense or a tragic accident. This is why people can argue about whether abortion, capital punishment, killing of enemy soldiers in wartime, euthanasia, etc., are - or are not - murder.
Martin, that's a fair point. But if we go to an example like "Lying is wrong" or "Hitting your little brother is wrong" I think we end up in the same territory of uncertainty about where the truth-maker lies, with less grounds for arguing that the "wrongness" is definitional.
What about the (post-modern?) view that the real problem is with the idea of "correspondence with the world," especially when the "claim" can depend on language and on context such as who is saying it and who is hearing it? There could be many different ways people use the word "truth", related to each other by imperfect analogy, and only the logician's truth (or truths about the meanings of words) are reducible to propositions and predicates.
So, I realize that this is entirely tangential to the point of your post, but I have always been troubled by the SchrÃ¶dinger's cat example. It seems pretty clear to me that the superposition cannot hold until the scientists open the box. I mean, SchrÃ¶dinger's cat is an amusing example for physics students to imagine, but I'm pretty sure that nowhere in quantum theory is there an equation suggesting that an event has to be observed by consciousness to count as having been observed, and there is certainly no requirement for self-aware consciousness.
Sorry. This is just a thing I have. Carry on.
Ah, I've clearly had one glass of wine too many to comment extensively on *this* thread. At least tonight.
Thanks for the interesting post.
Indeed, prior to the measurement, what the QM folks hold as true is
I was more under the impression that any model of the cat must consist of a superposition of the states of alive and dead than that the cat is in fact both alive and dead.
One of your statements has brought up a pet obsession of mine, hopefully this isn't too long or off-topic:
| A given sample of water contains twice as many atoms of hydrogen as atoms of oxygen.
There's a sample of water on my desk right now (mmm, water), which I've obtained using my kitchen tap. And I haven't done the experiment, but I'd be pretty happy in saying that there are most likely less than twice as many hydrogen atoms as oxygen atoms, due to the presence of calcium bicarbonate (we're in a hard water area), dissolved oxygen and carbon dioxide, etcetera. I don't have access to a chemistry lab these days, but if I did, then I could get some ultrapure water from the Milli-Q (TM) machine, and de-gas it, and even then I don't think the ratios would be dead on, although I'd be hard pressed to say in which direction the deviation would be in, and I'd also be hard pressed to say which analytical techniques would be sensitive enough to tell you in whether there was more or less than twice as much hydrogen as oxygen.
You could dodge this problem in a variety of ways; you could admit to a vagueness in the meaning of "twice", allowing it to cover ratios which are not quite 2:1, or you could say the statement is about some hypothetical universe where the problem of impurities does not exist, or you could say that truth is testability (but that would mean that a previously true statement can suddenly become untrue when you get access to better analytical devices than you previously had). Alternatively, you could say that the "sample of water" in fact refers to the parts of that portion of liquid that are H2O molecules, in which case the statement is similar to the one about water freezing at 0 oC. Or something else entirely that I haven't thought of
So, coming up with a satisfying notion of truth turns out to be rather tricky; I'm not sure which of notions I've set out in the paragraph above I prefer. It's a tricky subject.
There's more in a blog post of mine.
The water case is even worse! The properties of water depend on impuities - pure h2o doesn't freeze at 0 or boil @ 100 & lacks other key properties of "water" ...
Not that this really messes w the main correspondance theory of truth stuff. But still...
(Yanking hyperlinks for faster posting; Wikipedia has decent coverage of the math.)
Janet D. Stemwedel: The underlying assumption is that propositions (like P) are either true or false (with no squishy in-between state of truthiness), which means either P is true, or it's false -- in which case not-P is true.
For the law of excluded middle, it's a hair more complicated, in that you can have a lattice which having more than just the two elements of TRUE and FALSE, but which nonetheless remains Boolean. (One can also have a trivial lattice, where TRUE and FALSE are two different names for the same thing. While this may be interesting, in its own context it is therefore also not interesting.) Despite the additional elements, the join of an element and its complement is the universal supremum element TRUE, and the meet is correspondingly the universal infimum FALSE.
Merely because you're dealing with fuzzy logics, does not mean you are dealing with non-Boolean fuzzy logics.
Of course, there are other logics; ones based on Heyting lattices, for example, where (P OR NOT P) does not necessarily hold. However, this requires considering what you mean by "OR" and "NOT". For Boolean logics, (P OR Q) is equivalent to ((P NAND P) NAND (Q NAND Q)), and (NOT P) to (P NAND P), where NAND is defined such that (((P NAND Q) NAND R) NAND (P NAND (P NAND R) NAND P))) is equivalent to R. (Whether this is the starting definition or a consequence depends on which of several equivalent axiom schemata you take as the starting point.)
Heyting lattices do not share this meaning of "OR" and "NOT".
Of course, lacking a prior agreement to such basic ideas as "AND", "OR", "NOT", and "IMPLIES", communication to define what is meant becomes problematic.
Janet D. Stemwedel: Thinking hard about the metaphysical commitments that may be lurking beneath our moral assertions is the kind of thing that may send you back to thinking about something fluffier ... like SchrÃ¶dinger's cat
Not really. Wolfram's axiom, self-consistency of ZF (or similar) axioms, and the presumption that reality produces experience in a pattern. Of course, most people aren't that comfortable with the math....
(a different) Nick: What about the (post-modern?) view that the real problem is with the idea of "correspondence with the world," especially when the "claim" can depend on language and on context such as who is saying it and who is hearing it?
I hid a nasty amount of mathematics in the word "pattern". While the claim may depend on language, a self-consistent language is expressible within the context of a universal language; that is: mathematics. You might look into the relation of Chomsky's hierarchy and formal automata theory -- in particular, how the universality of Church-Turing Automata ("Turing machines") allows expressing any arbitrary language in terms of one particular language with the property of being universal.
(a different) Nick: There could be many different ways people use the word "truth", related to each other by imperfect analogy, and only the logician's truth (or truths about the meanings of words) are reducible to propositions and predicates.
In such cases, yes, the "truth" there is not the same meaning of TRUE associated with (P NAND (P NAND P)). Such "truth" is more likely associated with something on the lines of "having correspondence to the universe of discourse with probability pâ1"... which is also reducible to propositions and predicates; there's just a metric buttload more math to the reduction.
Of course, there are philosophers who don't like the idea of reducing everything to mathematics, and whine a lot when you point out this option. I sometimes suspect this is because they are afraid they might be expected to have to do real work for a living.
What do you think of Sir Karl Popper's ideas about 'verisimilitude'? As I understand it, a statement can have varing amounts of verisimilitude. A completely false statement would have 0% verisimilitude and a completely true statement would have 100% verisimilitude. In the real world many statements fall somewhere in between.
I am troubled by the statement about the philosophers who don't like mathematics. I don't see why truth and logic must correspond, unless so-defined.
The one that always bugged me was:
I get this one a lot in any debate involving creationism, and often for theology in general. I view it as a classic case of "begging the question" since, at minimum, it assumes:
Getting someone who utilizes this argument to realize they are begging the question, however, is probably an NP-Complete problem. ;-)
"the official stand on this claim from quantum mechanics is that the cat is neither alive nor dead, but rather in a superposition of the two states."
I don't think that is quite right. AFAICT, the idea that the cat is in a superposition of states has to do with an interpretation of quantum mechanics, rather than quantum mechanics proper, and it is a controversial outcome of that particular interpretation.
see what you have to answer for Erwin Schroedinger? The cat was just an analogy to attempt to describe the behaviour of subatomic particles such as electrons. The behaviour of such particles at the subatomic level cannot be extrapolated to such objects as cats. Your statement is therefore an attempt to say "P and non-P is true". In fact the cat is either dead or not dead. It cannot be both at the same time. But you just don't know which it is until you open the box and find that the cat died at sometime during the test (BTW it's not an experiment because you know what the possible outcomes are) or did not die.