Why Are Vacuous Truths in Logic So Counterintuitive?

[disregardthat]

There is absolutely no correlation between logical implications and causality. This point is extremely important to stress. Even in logical models of physical phenomena the difference is still there.

Furthermore, pure logic can quite easily deal with actual things as longs as we treat our statements as logical propositions. This is the crux of the issue, we often perfectly well know what a proposition is; a statement, an assertion, but still confuse it with hypothetical situations in some "If A, then B" type of statements.

It can be interpreted as "If A were to happen", then "B would happen". But "A were to happen" isn't the same thing as "A is true". And "B would happen" is not the same as "B is true". We just transitioned from logical propositions to something completely else, but still confuse it with logical propositions. The structure of these types of statements is completely different with radically different means of verification. A logical implication can be verified by looking at the respective truth-values of propositions. A causal consequence is verified by other means, such as empirical investigation, or intuition.

 

[cmb]

I put my small observation on causality or co-incidence as an aside as I think it is off-topic. There is no 'cause and effect' in mathematics, only in the world we use mathematics to describe. Whereas this thread is about the mathematical nature of 'a true/false implication'.

 

[Deveno]

There is absolutely no correlation between logical implications and causality. This point is extremely important to stress. Even in logical models of physical phenomena the difference is still there.

Furthermore, pure logic can quite easily deal with actual things as longs as we treat our statements as logical propositions. This is the crux of the issue, we often perfectly well know what a proposition is; a statement, an assertion, but still confuse it with hypothetical situations in some "If A, then B" type of statements.

It can be interpreted as "If A were to happen", then "B would happen". But "A were to happen" isn't the same thing as "A is true". And "B would happen" is not the same as "B is true". We just transitioned from logical propositions to something completely else, but still confuse it with logical propositions. The structure of these types of statements is completely different with radically different means of verification. A logical implication can be verified by looking at the respective truth-values of propositions. A causal consequence is verified by other means, such as empirical investigation, or intuition.

one still sees (more frequently in older literature, to be sure) the terms "necessary and sufficient conditions". if B (state of being, action, chemical reaction, whatever) only happens after A, one says "A is a necessary condition for B", or: A "only if" B, or A→B. one might argue that our language is poorly suited to logical discourse, as individual words have rich connotative meanings not in accordance with the precision we wish to express mathematically. i will note, somewhat dryly, that this did not apparently stop the ancient greeks, or hume or kant from using this imprecise means of communication, nor did mr. boole suddenly "create" logic when he devised his algebra.

and apparently, scientists have used logical reasoning to guide them in deciding which emprical results were worth investigating. how absurd of them, don't the poor saps know there's no connection?

if i know that turning on the gas on a stove under a pot of water will cause it to boil, and i know that when it boils away the pan itself will start to burn, and i know further that burning that pot will ruin it for further use, i don't need to observe the intermediate steps empirically to reason that neglecting the pot of water for several hours will result in a ruined pot. do you?

i humbly submit that the connection between logical entailment and cause-and-effect is not null, neither is it 1-1, but something in-between. perhaps the statisticians should quit now, while their dubious application of logic to "real world" events has done little damage, so that mathematical logic can retain its somewhat pure and unsullied state.

 

[Jamma]

one still sees (more frequently in older literature, to be sure) the terms "necessary and sufficient conditions".

I would still say that this has nothing to do with causality though- I read necessary and sufficient as just being an if and only if statement.

 

[Deveno]

I would still say that this has nothing to do with causality though- I read necessary and sufficient as just being an if and only if statement.

the association of causality and implication isn't just relegated to "misguided" applications of logic to "real life".

for example, it is common practice in computer programming to have satements of the form:

if (condition) then (instruction).

i daresay the condition check happens temporally prior to the instruction being carried out, and furthermore that this sequential chain of events (which is directly caused by the configuration of the logical gates) is based entirely on a system of circuitry which attempts to as faithfully as possible, realize abstract logic in physical form.

to go a step further, the entire scientific method, is an application of logic to causality. one posits a hypothesis, P. one reasons if P, then Q. one then checks for Q or not Q. if not Q, one discards the hypothesis as falsified.

now, I'm not saying that using our natural language as a guide can't lead you into logical hot water...it CAN, especially if one isn't careful about the distinction. but the two aren't divorced from each other (perhaps they're just going through a trial separation, i dunno).

in fact, certain causal chains, have the structure of a lattice, and form posets isomorphic (as categories) to posets of wff's using implication as arrows. for SOME processes, A comes before B is the same "ordering" as P implies Q. we just have to be careful about the domains we choose, not every causal chain has a "realization" as a predicate implication, even though in english, they may sound superficially similar.

in short, the origination of our language, and our usage of it, isn't so misguided as it seems. sure, a "logically true" statement may not be the same as a "linguistically true" statement, the statement: "if mice are dinosaurs, then 2 is an odd number" is logically true, but linguistically, it's another story. in (classical) logic, things are true or false, truth is bivalent. in ordinary (natural) language, statements which are neither abound.

we don't use similar language for causality and logical implication by accident, we do it on purpose.

 

[AlephZero]

to go a step further, the entire scientific method, is an application of logic to causality.

I disagree with that. The notion of causality is irrelevant to science. It's a leftover from Greek philosophy, where Aristotle constructed an elaborate theory of different types of "causes" including the idea that everything had an "original cause", which was used as a "proof" of the existence of god.

Causality was certainly at the heart of Greek science, and arguably was the reason most of it never progressed beyond the level of "astronomical bodies move in circular orbits because the circle is the most perfect geometrical figure".

Modern science is based on correlation, not causality.

 

[cmb]

Modern science is based on correlation, not causality.

I agree that some is, and that is bad.

If your statement is true for science generally, then I consider that to be a terrible indictment of modern science.

Unless we have, create, and execute the opportunities and experiments to investigate the mechanisms between cause and effect, then we are doing nothing more than exercising a sophisticated form of superstition.

 

[Deveno]

I disagree with that. The notion of causality is irrelevant to science. It's a leftover from Greek philosophy, where Aristotle constructed an elaborate theory of different types of "causes" including the idea that everything had an "original cause", which was used as a "proof" of the existence of god.

Causality was certainly at the heart of Greek science, and arguably was the reason most of it never progressed beyond the level of "astronomical bodies move in circular orbits because the circle is the most perfect geometrical figure".

Modern science is based on correlation, not causality.

pfft! modern. i hate that word. it reeks of hubris, as if we're somehow smarter now than we used to be.

anything modeled by a function is cause and effect. we don't use functions in science anymore do we? oh, wait...

hmm, let me guess: we take a statistical sampling of data points for analysis, and then use some sort of curve-fitting to determine, oh, say the trajectory of a communications sattelite. because we don't actually believe that gravity has an effect on its orbit, we've just noticed a high degree of correlation, so our approximations are justified. dang...you know, I'm sure glad we've got high-speed computers now, it's a wonder those archaic old fogeys ever managed to do any proper astronomy at all, know what i mean?

and we deduced that our actual planetary orbit was elliptical, because...well, that's what the curve-fitting came up with, right? that darn kepler, if he had just freed his mind, he might have seen there's only a correlation...

i call bs. seriously.

yes, at sub-atomic levels, notions of causality, perhaps even of time itself, break down. but if you even suggest to me that the water in my boiling pot doesn't change phases because i applied heat...i'll...i'll...oh gee, i shouldn't say.

oh sure, we've gotten more sophisticated, instead of cause and effect, we use fancy words like supersymmetry, and genetic encoding. i guess our entire theory of disease is just wrong, viruses and bacteria don't cause infections, they just correlate with them.

and what's with this "big bang" stuff, anyway? why do we feel a need to try to explain the why of anything, after all, nothing causes anything, it just happens.

but sure, the people of antiquity just "thought wrong", now that we're new and improved, we've got better ideas.

 

[AlephZero]

hmm, let me guess: we take a statistical sampling of data points for analysis, and then use some sort of curve-fitting to determine, oh, say the trajectory of a communications sattelite. because we don't actually believe that gravity has an effect on its orbit, we've just noticed a high degree of correlation, so our approximations are justified. dang...you know, I'm sure glad we've got high-speed computers now, it's a wonder those archaic old fogeys ever managed to do any proper astronomy at all, know what i mean?

You are making another basic philosophical mistake here: you are assuming that because something (e.g. "gravity") has a name, therefore it is a thing.

If you compare Newtonian and relativistic mechanics, you will see the flaw in that position.

 

[disregardthat]

A model of a physical phenomena doesn't allow us to logically infer an effect from an event. There is a lot of logical inference when reasoning within a given mathematical model, but when transitioning from the model to propositions of physics, there is never logical inference. I.e. a causal implication is not a logical logical implication.

Neither Hume nor Kant said that. From my understanding: Kant's transcendental logic referred to the necessary forms of experience, he said we could use logic to infer some things about nature, but then in the context of these things being necessary for human experience and understanding. Examples such as a massive object pulls other objects due to gravitational force does not fit into this context (no matter how sophisticated our mathematical models are).

Hume on the other hand denied any necessary relationship between events. A causal relationship could not in any way, shape or form be a priori (which is required at least for any logical relationship). He said that a causal relationship is "inferred" purely due to habit, a psychological thing. In other words causality is merely a term used for something we get used to.

I do sympathise with Kant's view, but I doubt he would ever characterize any physical proposition of a causal relationship as synthetic a priori. However, he said that causality is a necessary form of experience, meaning that to experience (experience: understanding sensory input) something, you must be able to put in a causal context. But this does not mean that any particular causal relationship can be inferred logically.
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Also, I think it's wrong to distinguish logic with language; logic is the structure of language. Our logic defines our criteria for truth in sentences, so you can't say that a proposition is logically false, but linguistically true. Then you aren't talking about a proposition at all.

 

[AlephZero]

and we deduced that our actual planetary orbit was elliptical, because...well, that's what the curve-fitting came up with, right? that darn kepler, if he had just freed his mind, he might have seen there's only a correlation...

Check out some history before you accuse Kepler of doing anything more than correlation.

The only "cause" he proposed for the spacing of planetary orbits was that the 5 platonic solids fitted nicely in between the orbits of the 6 planets he knew about. Oh, and there is a section of the "music of the spheres" (including musical notation of what it sounds like) in another of his published works...

yes, at sub-atomic levels, notions of causality, perhaps even of time itself, break down. but if you even suggest to me that the water in my boiling pot doesn't change phases because i applied heat...i'll...i'll...oh gee, i shouldn't say.

So you accept that observations blow holes in your logical position, but you are going to stick with it anyway? Sorry, but I don't need to "call" bs over that attitude to science.

 

[Deveno]

you guys are funny.

i don't "believe" in some synthetic a priori anything. nor do i see causation as some sort of primitive force changing anything. i am a structuralist, i don't really care what, or why, just how. I'm interested in the shapes things take, nothing more.

there is a similarity of shape in causal chains, and logical structures. there is. this is not necessarily explanative of anything, but it can be manipulated, purely on a formal level. we call it deductive reasoning. it really doesn't matter whether energy potential fields or subatomic spins and momentum or microbes are actually "doing" something, or if our understanding of underlying relationships is even correct, we can abstract the structure as if our models were correct, and use logic on the model, and transfer back. we do this all the time, with varying degrees of success.

sometimes this isn't an optimal approach, or even appropriate. i am not arguing that everything is "caused". but in some frames of reference, such a description makes sense. does it do so, because our synaptical patterns are conducive to that way of thinking? perhaps, I'm no neurobiologist, but it seems plausible that is one explanation.

i am not a determinist, but neither am i a phenomenalist, either. I'm really not much of an -ist kind of guy. i believe in local maps, partial explanations, within context.

i have to say as well, i disagree that (classical at least) logic covers the extent of what language is capable of. i have yet to see a syllogism that reads like poetry, but you're welcome to give it a go. the very rigidity of logic works against its expressive power. its main advantage is its precision and umabiguity, which is also its biggest weakness.

i've read a little lately about the notion that time is merely an artifact of our sensory apparatus. could be, but as I'm apparently stuck with my sensory apparatus, and my unrealistic fictionalizing brain, why not make the most of it? cause and effect may not be a sweeping universal explanation of how the world works, but it does come in handy, even if it isn't "true". i daresay you could say the same thing about logic.

i never said that causal relationships are logical implications. but some, under some circumstances, have the same kind of structure. yes, computers and robots do exist. we have made machines which transfer logical structure into mechanical interdependency. electricty flows, the bits flip, and lo! there is reaction to stimulus. it could be just a coincidence, but...really?

 

[disregardthat]

we can abstract the structure as if our models were correct, and use logic on the model, and transfer back. we do this all the time, with varying degrees of success.

The point is as I repeat here: what you describe is not a logical inference.

 

[Deveno]

i am not saying, nor have i been trying to say, that logical implication and causal dependence are "the same thing". i am pointing out that (in our language) and (symbolically) you can use similar terminology for both, and that (in some sense) this similarity of terminology is justified.

a woman might very well say to her husband: "if you cheat on me, i will divorce you."

it would be bad logic to conclude, adultery implies divorce.

one might also say: "if n is divisible by 4, n is even", but surely "divisible by 4"-ness doesn't "cause" evenness, a (natural number) either has this property, or it does not.

i get the distinct impression that you want to somehow extend this to some epistemological principle i believe in, which has somehow been "voided" by "modern science". i'll say it again, for clarity:

causal chains have a structure which is partially ordered.

logical implications have a structure which is partial ordered.

the hypothetical/consequential "if...then" is one way of expressing these two facts (rather economically, i might add). there is no reason whatsoever, to take results from one structure to another, BUT: there are certain rules we can apply equally well to BOTH.

if A causes B, which in turn causes C, we can (logically!) deduce that A will cause C. now there might also exist a similar wff like A, and one like B, and it may be the case as well that A implies B, but B might not imply C. the mapping between a set of causes/effects and a logical theory may only be order-preserving in one direction, and it need not be faithful.

natural language doesn't have the same "meaning" as "logical language" this is well-understood.

for example the statement A or B, in logic is quite common but the statement: "you are alive, or dead, or both", has a wtf? factor missing in logic.

"if...then" doesn't have the same meaning in english that it does it logic (the temporal element is usually absent), but the confusion between two similar types of consequential statements is perfectly understandable, our consciousness is rooted in time.

the recent "fast neutrino" episode is a case in point: the assumptions are: our data is error-free, and the standard model is correct. since that predicts a certain neutrino speed, when a higher speed is recorded, we reason (using...what? stochiastics? i think not), the data is not error-free, or the standard model is not correct (both could conceivably be true). what basis are we using for the furor, if not logic (a statistical anomaly would be no cause for concern, it just weakens the correlation, which presumably might be corrected by acquiring a larger data set)?

the very utility of mathematics in science pre-supposes a logically consistent system underlying our observations (or, why are we even bothering to observe? if there's no actual underlying structure, just this one random data-set, the whole ediface of science appears to be self-defeating). I'm not talking quantum versus Newtonian here, I'm talking deeper than that. no sense talking about eigenstates if you don't have a hilbert space. why calculate anything, if the meaning is an illusion, fostered by our own delusions? if you want to adopt that position, fine by me, but i somehow doubt that you do.

but, you know, i don't need to take "a theory of everything" into account to justify my observations. cause-and-effect is useful for analyzing SOME things, and i would be very surprised to learn that anyone who visits this forum believes that is never true. sure, you can misapply logic by "confusing domains", just like any kind of encapsulation of any information system (applying mathematics to romance could have, erm, humorous results).