## Are there degrees of truth ?

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I do not assume bivalence - that every proposition or declarative statement - is either true or false but not both. I do not raise the issue of 3-valued logics or offer or invite any theory of truth. But I do assume that at least some propositions or declarative statements are true. Question is, are there degrees of truth in the sense that some propositions are 'more true' than others ? My inclination is to say that any two true propositions are equally true - it is equally the case that they are true and in this sense (call it 'a') that there are not degrees of truth - but that they can differ in the information they contain - can differ in accuracy. For instance, it is true that the the population of Russia is larger than that of France. It is true that the population of Russia is over 100,000,000. It is true that the population of Russia is more than 140,000,000 and less than 160,000,000. I venture to suggest that of these three propositions the third is more true than the first or the second in the sense (call it 'b') that it contains more of the truth, is more informative, than the others.

It may be that I am posing an unintelligible question or suggesting muddle-headed answers to it. But I put it in good faith and would ask you to treat it in that light, whatever you think of it. This is not a request for soft treatment.

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See Fuzzy logic and SEP's entry on Fuzzy Logic.

– Mauro ALLEGRANZA – 2017-12-17T13:22:57.887

@Mauro ALLEGRANZA. Very grateful for comment, thanks. Could you settle one point, though ? Is fuzzy logic concerned with closeness to truth or with degrees of truth. If something is close to truth then it isn't true. I was just wondering whether propositions (statements, &c.) could be genuinely true but capable of truth, not just of closeness to it, in degrees. Excuse this inquiry. I much appreciate your help already. – Geoffrey Thomas – 2017-12-17T15:35:56.637

We can see Merrie Bergmann, An Introduction to Many-Valued and Fuzzy Logic, Oxford UP (2008), page 176: "A logical system in which sentences may have any of an infinite number of degrees of truth (e.g., values between 0 and 1) is an infinite-valued logical system. When the bases for assigning the degrees of truth are fuzzy sets, we call the system a fuzzy logic."

– Mauro ALLEGRANZA – 2017-12-17T15:45:45.203

I would think given the same data, a result with a probability of 0.8 is more true than one with a probability of 0.5. Or did I not understand your question correctly? – jjack – 2017-12-17T15:46:34.767

See Popper's theory of "closeness to the truth, i.e. degree of truthlikeness, or verisimilitude".

– Mauro ALLEGRANZA – 2017-12-17T16:28:01.620

I believe you don't want to tag this question to 'Truth'. Is this question purely on logic? I give great importance to your last statement (about soft treatment). – SonOfThought – 2017-12-19T04:30:23.237

@SonOfThought. Thanks for comment, thought-provoking as always. I certainly didn't intend the question as epistemological. I was not asking about the possibility of degrees of belief. JJ's comment (above) accepts the idea of degrees of truth as does TB's answer (below) on mathematical accuracy as distinct from precision. It would certainly help me if you would clarify the grounds for re-taggging the question. Again : comment appreciated, – Geoffrey Thomas – 2017-12-19T10:23:36.203

@Geoffrey Thomas: I believe the Truth can cover logic. But logic can't cover the Truth. That was why I said so. From you explanation I understand you are asking about truth; not the Truth...especially for analyzing some problems. So I don't wish to insist you to tag your question to Truth. If you tagged to it you can't channelize your question to your specific route. – SonOfThought – 2017-12-19T14:05:44.627

I'm not sure if it's enough for an answer, but I found this question and I think I can suggest some distinction in the logical statements themselves that might help. In your example, the past statement is "more true" because, I suppose, we can put it as two different statements, compared to the first two which only contains one statement. Now you can say "well yeah but I can suppose the second statement as population over 100k and under 170k which will give me 2 statements", but then I can argue that I can provide more true statements in the third than in the second. – Yechiam Weiss – 2018-03-11T23:16:13.597

(I can say, "under 169k", "under 168k"... "under 160k" which will give me 10k more true statements than the claim for 170k. Keep in mind that's something I have nothing to base on, I just thought of it, if you know of someone who's made something out of this bizarre theory I'd love to hear.) - so, in a sense, yes, it absolutely contains "more of the truth", because it contains more true statements. – Yechiam Weiss – 2018-03-11T23:17:20.740

By the way, note that it can be greatly demonstrated in the way programming languages work, especially the "if" statements - the more accurate "if" statement will include more conditions - same as here the "truer" statement will include more "true statements". – Yechiam Weiss – 2018-03-11T23:24:02.737

1I am an old school proponent if the concept of propositions that are only true or false by definition. With proper & specific details in place all declarative sentences express propositions. When you undercut specific details you have the modal logic and other logics to resort to because one may purposely leave out details that would change the outcome. So if you know the actual population of Russia then why use statements that seem to indicate you might not know It? This is called sandbagging in the card game of spades. You have 4 aces and you say you can win 2 rounds of cards called books. – Logikal – 2018-07-10T03:21:11.150

@Logikal. I'm not unsympathetic. I grew up in the same 'old school'. I think what I had in mind didn't contradict T/F exclusiveness but centred on ideas we have about 'the truth, the whole truth, and nothing but the truth' in the legal formula. If there is 'the whole truth' (contextualised, of course) then, I thought, there are degrees of truth short of that. Also 'Are there degrees of truth ?' was an open question. It didn't assume that there are degrees of truth even if I had some vague ideas about the possibility. Btw : I have followed your always valuable comments elsewhere. Best - GT – Geoffrey Thomas – 2018-07-10T08:18:45.497

@user 170039. Thank you. This was an early question and I would phrase it differently now. I appreciate the reference : GT – Geoffrey Thomas – 2018-10-08T07:21:54.310

degrees of belief / confidence / certainty surely exist as per @YechiamWeiss's comment

– None – 2019-04-07T08:01:50.347

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In mathematics, and by extension science and computing, we have concepts called Accuracy and Precision.

Accuracy refers to whether or not the answer can pass for the correct answer. Often, this is talking about degrees of magnitude.

3 * 10^0 (3) is in the right order of magnitude for Pi
3 * 10^2 (300) is not, and neither is
4 * 10^0 (4) because 3 is closer to PI than 4.

On the other hand, precision refers to how close to the 'truth' you can get with a specific number. This refers to the Mantissa, or the numbers before the *10^x in scientific notation.

3 = pi if the precision is 1
3.14 = pi if the precision is 3

Most importantly, if the precision is 3, then the top number fails the accuracy test because 3.00 does not equal 3.14.

There is also the Platonic planes of truth to consider here; put simply, this concept states that nature can only ever approximate perfection, never replicate it. In this concept, there's no such thing as an observable perfect circle, because there will always be minute flaws in the production of the circle in nature. On the other hand, X^2 + y^2 = r^2 (simplified equation for centre at origin) defines a perfect circle in Plato's 'plane of truth'; meaning we can conceptualise perfection, but never observe it.

In such a world, degrees of truth are all we have because no observable phenomena can ever perfectly match the conceptualisation. To that end, I don't know what the non-mathematical philosophical concepts that translate to accuracy and precision are, but that is the place I would start to define degrees of truth.

An example of this would be;

All killing of other people is bad. Is it less 'bad' to kill to protect my family? Myself? My Nation?

Ultimately, degrees of truth and context start to look rather similar to my (admittedly) untrained eyes.

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In philosophy, these are concepts of "modality" and still hotly disputed. You've provided a very nice example of how epistemology and metaphysics can lead to greater degrees of questioning... then (maybe) have applications in real life mathematics or scientific deduction.

To summarize / bastardize:

"what sort of logical proposition exists, such that we know "the sun will rise tomorrow"?"

Well. We don't. We never know that. And we can't ever know that. But, does that matter?

The question becomes whether that's a function of our language, understanding of the world around us, or something completely different. Russell provides a number of avenues to this end that sort of bridge the gap between math, language, knowledge, etc. And, he writes well.

David Lewis is sort of a hallmark (for me anyway) in these discussions. You could summarize "degrees of truth" through the notion of "possible worlds" in which a statement like "I believe the moon is made of green cheese" means that you're committing to the actual existence of moon made of green cheese, just in another time and space. Something like a "square'd circle," couldnt possibly exist.

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Here are four things I've heard used in the discipline of history that they throw around when trying to decide the best theory that explains the evidence at hand:

Plausible (it doesn't contain impossibilities, mistakes, etc.)

Explanatory scope (the theory fits many points of evidence all at once)

Explanatory power (the evidence well matches what the theory would cause you to expect)

"Less ad-hoc" (the theory doesn't have too many exceptions, assumptions, terms, etc.)

While not directly addressing your question, you can choose the best theory (according to this) by finding one that is the best in these four categories.

Here's a quote from a recent article about science and alarmism regarding nuclear radiation (http://journals.sagepub.com/doi/pdf/10.1177/1559325818779651). "Over 50 years ago, Sir Austin Bradford Hill established a set of objective criteria that help determine when causation can be legitimately concluded from an observed correlation. These criteria are (1) temporal relationship (eg, exposure must occur before the disease), (2) strength (eg, size of the association between exposure and disease), (3) dose–response relationship, (4) consistency, (5) plausibility..." continued

– elliot svensson – 2018-07-10T18:31:54.890

from previous "...(6) consideration of alternate explanation (eg, confounding effects), (7) experiment (eg, the condition can be altered by an appropriate experimental regimen), (8) specificity, and (9) coherence (eg, associated compatible with existing theory and knowledge?)." (Hill AB.The environment and disease: association or causation? Proc R Soc Med. 1965; 58:295-300. http://journals.sagepub.com/doi/full/10.1177/0141076814562718) – elliot svensson – 2018-07-10T18:32:01.393

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Mother ask "would you like an apple or a banana?", and the little logician says "Yes."

We are reminded of two things: Working with formal languages changes how we think about truth, and there are sometimes a supplier and/or consumer of information. Now let me proposes here: If natural language was meant to convey only 'Formal Truth' then it would have evolved on the basis of logical structures. Since formal languages are amply dealt with by better minds than mine I will restrict my answer to the more 'natural' meaning of 'truth', that of information transfer. Although I don't claim exclusivity in the idea, I think that when the 'truth' can be seen as a transactional item, is when we can most properly define "degrees of truth". Here are some possible metrics:

-The degree to which the supplier uses language appropriate to the information.

-The degree to which the supplier uses language understandable to the consumer.

-The degree to which the supplier gives full account of their knowledge of the 'truth'.

-The degree to which the supplier is cognizant of the full measure of 'truth'.

-Intent and measure of deception by the supplier.

-The consumer's degree of diligence and commitment to receiving the 'truth'. This implies a feedback loop so the supplier can adjust their language.

-The consumer's degree of reticence. That is willingness to accept the 'truth', willingness to accept information from the supplier, etc.

-The degree of information channel noise.

-The degree of information channel interference. That is "putting a spin on the story", "controlling the narrative", and other third party attempts to modify the truth.

This may infringe heavily on communication science but illustrates that at least some incarnations of 'truth' can be evaluated to 'degree'.

*If anyone can add references to this, I'll be indebted.**

+1. This is an interesting angle on the question, not one I had expected but all the better for that. Thanks : GT – Geoffrey Thomas – 2018-07-09T08:29:33.310

1@GeoffreyThomas I may add that, as I imagine any student of logic would be, I was shaken upon discovering Godel. It occurred to me that natural language must have evolved to sacrifice precision for utility, which is why it is so intractable from a formal perspective. – christo183 – 2018-07-09T08:41:57.897

2I'm not sure if what you're presenting could be said to represent "degree of truth", rather than perhaps "degree of certainty". – Yechiam Weiss – 2018-07-10T03:56:06.287

1@YechiamWeiss Now you mention it "channel noise" would be much more appropriate as a 'certainty' metric. Some of the others can overlap between 'truth' and 'certainty'; anyone attempting to actually make a truth scale with these would have to be very careful indeed. But consider a person knowing 75% of the facts pertaining to some circumstance. They then tell you 100% of what they know with 100% certainty. Would it be more correct to ask how certain you are about the circumstance or how much of the truth you know? – christo183 – 2018-07-10T05:54:41.933

@christo183 well, if you're 100% certain of your knowledge then it may become truth, but the scale is still to the degrees of certainty, not of truth -- by your reasoning there's only one level of truth and you'd need to pass a certain "certainity" threshold to reach that level. Besides, the certainity of the knowledge doesn't necessarily represent the truth of the knowledge - I can be 100% be certain that a is true, while the reality is that a is false (Gettier problems are a classic example). – Yechiam Weiss – 2018-07-10T09:42:01.330

1@YechiamWeiss Not at all, certainty is about the quality of the information while 'degree of truth' is primarily about the quantity of information. More precisely about the fraction conveyed of the full set of information. The quality is secondary, yet necessary consideration. Remember these are proposed metrics to evaluate particular types of statements. It can't be universally applied, specifically not for formal truth. I didn't try to construct a scale, only show how it might be done. – christo183 – 2018-07-10T10:46:59.753

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I'm going to offer an answer that goes against the grain here, and say no, truth does not admit of degrees. Either a declarative statement is true, or it is not, in which case it is false. False statements may be a useful approximation, almost the truth, pragmatically useful for a given purpose, a conjunction containing more true statements than false, etc., but if it doesn't correspond to the way the world is, then it is false. If there's a room full of 99 people, and I say "there's a hundred people in that room", it's false. If I said "there's 4.65 people in that room", it's also false. The former is a falsehood that approximates truth better, or may be more useful for estimating how many kegs of beer I should purchase, but it's still false. The former is a falsehood that is closer to the truth than the latter, but it's still not the truth. (Maybe falsehood loosely admits of degrees depending on it's usefulness, approximation to truth, and so on?) If the particle is neither here nor there, but in some wavey superposed state, then "the particle is here" is false, and "the particle is there" is false (after all, we just said the particle is neither here nor there). If "here" and "there" (once properly defined) are classical notions, and the world is not classical (say quantum instead), any statement in physics that presupposes classical conceptions of the world are false. They might be falsehoods that are good enough to do engineering, or useful for most practical purpose, but they're still false.

I'm not 100% sure I personally believe what I just wrote, but I'm also not sure I've ever heard an example that forces me to abandon bivalence with respect to truth. Examples involving fuzzy or vague predicates, well, involve fuzzy and vague notions. Once the notions are analyzed and made precise, the vagueness can disappear. Not to say fuzzy logics aren't useful for analyzing fuzzy concepts, but fuzzy concepts don't correspond precisely to the world and so can't describe the world truthfully until the fuzziness is clarified. Examples opting to replace truth with provability are about provability, not truth. Quantum mechanics is a mathematical theory expressed in regular, bivalent mathematics, and can only be described accurately using the language of mathematics, not our everyday language.

Again, I'm not sure I believe all this, but I'm just offering a different perspective than the other answers!

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English. English is what brings us together today.

But seriously, English has huge possible context and connotation to the word "truth." For example, Scott Adams reports that he constantly gets letters claiming that an episode in "Dilbert" is "just like my office!" No. There is no alien with his head exploding in your office.

Logical truth is often bivalent. Then there's Goedel's theorem which tells us that some times you won't be able to demonstrate the truth or falsehood of some statements within a logic system. "Everything he says is a lie." "I'm lying."

Science and engineering truths are often expressed as probabilities.

Political truths are often based on tradition, culture, or expedience. And these are often the kind of issues that are most resistant to change. And as the OP points out, can be shaded and spun by such dodges as strategic amnesia.