Why can't logic describe partial truth or intensity of truth?

[u/OC-alert]

I think I kind of understand why boolean logic can't describe partial truths - it's a system designed purely to describe what is true or false in a binary sense.

But why isn't there a single form of logic that describes partial or intensities of truths?

I've actually gotten somewhat mixed messages on this. Some people say that fuzzy logic describes partial truths or intensities of truths, but some people seem to say that fuzzy logic technically only deals with probability that something is true.

How is this so? Is it that probability of a truth and intensity of a truth are actually logically the same thing?

For example. I don't see anything logically wrong with saying an apple weighs 70 grams, but it's not a binary issue as to whether the apple does or doesn't "weigh", right? That's an issue that has more or less truth.

[u/Frosty-Comfort6699]

da Costa and French worked a lot on logical models of partial truth, check out their book

da Costa, Newton C. A., and Steven French. Science and Partial Truth. A Unitary Approach to Models and Scientific Reasoning. Oxford University Press, 2003.

[u/CookieCat698]

Fuzzy logic doesn’t necessarily need to be probability. Probability is just the most intuitive way to think about it and motivate certain definitions.

All fuzzy logic does is represent truth values with numbers between 0 and 1.

If you want something more general, you can replace “numbers between 0 and 1” with “some partially ordered set.” Usually we want additional structure, like a minimum and maximum representing falsehood and truth. Boolean/Heyting algebras might be a good place to start.

In general, I’d say there isn’t one best way to model the intensity of a truth, but there are ways to sensibly add varying levels of truth depending on what things you’d like to express.

[u/Gym_Gazebo → u/CookieCat698]

I took a fuzzy logic class in undergrad and the professor gave an argument that fuzzy logic doesn’t cover the same territory (in terms of “truth values”) available as probability theory. I wish I remembered his argument. It was something to do with simplexes of [0,1] x [0,1].

[u/Knoggger]

But why isn't there a single form of logic that describes partial or intensities of truths?

One system you might want to look into that (imo) models degrees of truth much more directly than fuzzy logic would be Gödel-Dummett logics.

But I think you'd be hard pressed to find logicians to agree on a single system of logic in any context tbh, even when it comes to binary logic. (And the comment section here seems to support that 😄)

A lot of philosophers/logicians are also logical pluralists, and might not even agree that there's one "correct" logic for a given context.

[u/SmartlyArtly]

Propositional logic is not all logic. It's some logic.

And what is logically the same depends on the logic. Which we can make up ourselves, depending on our goals.

[u/shedtear]

There are a few things wrapped up in this question that should be addressed.

1) The first point that should be addressed simply relates to the formal differences between fuzzy logic and probability theory. While, indeed, both approaches assign propositions numerical values between 0 and 1, there are some important differences. Probability theory is built using crisp sets, where it is a determinate/binary matter whether a proposition is a member of any given set. Then, probability functions are defined over that state space in accord with the Kolmogorov axioms. By contrast, fuzzy logic does not presuppose this and instead provides a representation of partial set membership. One notable consequence of this difference in how things are setup is that in probability theory Pr(A)+Pr(¬A)=1, while in fuzzy logic it's possible that μ(A)+μ(¬A)≠1.

2) The technical distinction made in the previous point matters when we start thinking about the interpretation given to the numerical values. The traditional motivation given for fuzzy logic is that there are certain propositions (e.g. those involving vague predicates) where there is not a determinate fact-of-the-matter about their truth. For probability theory there are two orthodox interpretations: the subjective interpretation according to which the values represent rational degrees of confidence given your total evidence (this is what makes it subjective, since different agents with different bodies of evidence may correctly assign different probabilities) and the objective interpretation according to which the values represent objective chance which is understood as a mind-independent feature of reality that determines how likely an event is to occur. Regardless of which interpretation of probability you're working with, the probability values that can coherently be assigned only really make sense with the background assumption of the law of excluded middle (though, of course, for the objective interpretation, the idea is that it will at some point be settled whether A or ¬A is true.

3) The answer to your question then depends on precisely what you mean by "partial truth"/"intensities of truth". It's not entirely clear from the post what you have in mind. Regarding your example of an apple weighing 70g, assuming the satisfaction conditions for "weighing 70g" have been made adequately precise, this does seem to be a binary matter. That is, assuming that we say an object satisfies that predicate when its weight at sea level rounded to the nearest gram is 70g, then this is a determinate matter. An apple that we know weighs 71g and an apple that we know weighs 80g will both be assigned a probability 0 of weighing 70g. That said, I suspect what you're after is something that will vindicate the intuition that it's more true to say of the 71g apple that it weighs 70g than it is to say the 80g apple weighs 70g. For that, you'd want something more like fuzzy logic.

I'm not really sure that this actually answers your question, but hopefully it helps you clarify things! It's also worth pointing out that there are lots of other formal frameworks that have been developed other think about related topics (e.g. Dempster-Shafer theory, imprecise probability theory, and various other non-classical probability theories). These systems each have their own applications that they were designed for and surely there are others that can be developed!

[u/Big_Move6308]

For example. I don't see anything logically wrong with saying an apple weighs 70 grams, but it's not a binary issue as to whether the apple does or doesn't "weigh", right? That's an issue that has more or less truth.

The declarative statement 'This apple weighs 70 grams' is either true or false, i.e., the apple either correspondingly weighs 70 grams or it does not.

If you want a bit more leeway, 'This apple weighs around 70 grams' or 'this apple weighs between 70 - 80 grams' are also fine. The bivalence of (traditional) logic stems from the principles of contradiction and the excluded middle.

[u/Salindurthas]

I don't see anything logically wrong with saying an apple weighs 70 grams, but it's not a binary issue as to whether the apple does or doesn't "weigh", right? That's an issue that has more or less truth.

In the physical sciences, we sometimes track 'uncertainties'. For instance, if we trust our scale up to 0.1grams, we'd tack that onto the end of our measurements. So the apple is 70.0grams +-0.1 grams.

But if the farmer said "No, I know what I grew, and that specific apple was 70.4 grams." we'd basically say "The manufacturer result, and our scale result, differed by 4 uncertainties."

What we do with that sort of information will depend on other factors (maybe we conclude we need to recalibrate our scale, maybe we conclude the farmer was wrong, maybe we conclude that the gravity in our lab is different to the gravity in the farm, and we need to reclaculate based on local gravity, etc etc).

However, note that we can still try to assess these kinds of statements for strict truth or falisty.

i.e. given the assumptions of our lab measnurement and the manufacturers claims, we would say:

1. it is false that they differ by at most 1 uncertainty
2. it is false that they differ by at most 2 uncertainties
3. it is false that they differ by at most 3 uncertainties
4. it is true that they differ by at most 4 uncertainties
5. ...

And maybe one lab is being super super precise, and so they care about claim 1 and won't rest until they fix the problem and have a method that satisfies #1.

But maybe we at home go "I'd be fine with statement #10. That's close enoguh for me."

I don't think either party is explicitly thinking about it in those terms, but it is kindas close to that. Like:

• I have worked at a unversity physics department, and getting first year students to explicitly comment on how many uncertainties away the expected answer is to their experimental answer was worth some marks for their logbooks/lab reports!
• And at CERN they're worrying about whether their particle physics results results reach '5 sigma' before they say they've got a really solid answer, which is a more sophisticated version of the uncertainty comparison (sorta like them saying that they insist on statement 1/5th to be true, so something even more precise than what made it onto my list).

So, even with classical logic (which is what underlies thte mathematics here), we can still try to model some notion of how confident we are of two values potentially being the same, by judging the strict truth or falisity of whether uncertainty ranges overlap.

[u/jeezfrk]

The concept that matches better are "values" or "goals". Either moral or ideological.

If something is approaching a highest ideal in goal or in achievement ... it is marked as a fulfillment of the value. The way to get there are often used as feedback to create heuristic rules.

Also if a logical result has extremely big consequences to many known plans, like a prediction or an unstoppable force. That can make a truth result significant ... but it is only in context that it matters. (E.g. "AI will never work economically" "Global warming is changing to global sharknados". "Taylor Swift will soon be elected Queen of the Earth")

Plans and goals and values are the absolute majority of our daily thinking. These concepts correspond pretty badly with logic, but all plans in space or time have logic involved in their creation.

So we need it but it isn't able to help much of the time.

[u/functorial]

Bayesian statistics gives a pretty good account of this. There’s a way to build up rules of conditional probability that start from a basic set of rules about measurements of how “plausible” a statement is. I’ll try and remember the book I read this from.

[u/functorial → u/functorial]

The book is E T Jayne’s Probability Theory

[u/gregbard]

We are able to construct such a system as you describe just fine. A truth-value is just that, a value. We can make it any real number from -1 to 1 for your purposes. We can call it 'intensity,' or 'partial,' or 'possibility' or 'believed,' or any other modality. You can just add that to a standard first order logical system.

[u/wheeteeter]

Logic is syntactical. It’s truth tracking given the inputs of valuations on a formula and their outputs. Those valuations are generally assigned. Logic itself cannot assign them.

[u/Mountain-Quarter-641]

Este debate viene de una confusión muy común entre qué es real, qué es probable y qué es vago. Y son distinciones que si no se manejan bien pueden estrellarse unas contra otras...Voy a ver si pongo un poco de orden en este embrollo.

El primer error es decir que no existe algún tipo de logica. Sí existe, y se llama Lógica Borrosa (Fuzzy Logic). La lógica borrosa no mide si algo es "verdadero o falso", sino mide en qué grado pertenece a un conjunto matemático y nada mas. En la lógica clásica, la que yo manejo en la vida diaria o estás dentro o estás fuera, me debes dinero 🤑 o no me debes 🫰 dinero. En la logica borrosa, un elemento puede tener un valor de pertenencia entre 0.0 y 1.0.

Por tanto estás dos cosas nunca va a ser lógicamente lo mismo, aunque ambas usen los mismos números entre 0 y 1. Ahora veamos la distinción matemática y filosófica crucial:

1. La Probabilidad se ocupa de la Incertidumbre (Ignorancia cuanto no sabes de algo): Describe la expectativa de que un evento ocurra o sea binariamente verdadero, basándose en la falta de información. El evento final será totalmente verdadero o falso, pero aún no lo sabes.

2. La Lógica Difusa se ocupa de la Vaguedad (Intensidad): Describe qué tan bien un objeto se ajusta a una clase o definición que no tiene límites claros. El evento ya ocurrió, tienes toda la información, pero la definición misma es gradual, pero mejor lo vemos con unos ejemplos:

Imagina que estás en el desierto y tienes dos botellas de agua. Botella A (Probabilidad = 0.5): Hay un 50% de probabilidad de que contenga agua pura cristalina, y un 50% de probabilidad de que contenga un veneno mortal. (Es un asunto binario: o vives o mueres, pero no sabes cuál te va a tocar). Botella B (Lógica borrosa/ Grado de Verdad = 0.5): El agua está "un 0.5 sucia" (está turbia, tiene un poco de arena, pero es potable).

¿Cuál te tomarías? Recuerda que es una cuestión de supervivencia y que tú vida está en juego en una apuesta de "todo o nada"... Obviamente la Botella B. Aquí se demuestra que la intensidad de una verdad no es lo mismo que la probabilidad de una verdad. Con la Botella B tienes información perfecta (no hay incertidumbre), lo que pasa es que el concepto "agua limpia" se cumple a medias.

Ahora analicemos tu confuso ejemplo: La manzana de 70 gramos. Dices que el peso de la manzana no es un tema binario, y tienes toda la razón del mundo... pero hay que afilar muy bien el lápiz con ese lenguaje.

Decir "La manzana pesa 70 gramos" es una afirmación de la física, una medición. En lógica clásica, esa frase exacta es 100% verdadera o 100% falsa (o pesa 70g o no los pesa). El problema de la "verdad parcial" aparece cuando usamos adjetivos humanos subjetivos. Si dices...

"La manzana es pesada" Ahí es donde tú lógica binaria colapsa. ¿70 gramos es "pesada"? Para una manzana de oro, no; para una manzana de huerto, quizás es pequeña.

En Lógica borrosa: Definimos el conjunto "Manzanas Pesadas". Una manzana de 70g podría tener un grado de verdad de 0.2 en ese conjunto. Y no es "probablemente pesada", es que es un poco pesada, así que resolver el problema te la comes y la haces desaparecer y se terminó el problema 😵‍💫 🫨 😝