Defining Synonyms and Tautologies: Exploring Logical Equivalence in Language

[Pavel]

If I have two words with different meaning, but which stand to each other in the relation of logical equivalence, do they form a tautology when one predicates the other? For example, if “the fittest” is both sufficient and necessary condition for “survives”, is the “the fittest survives” statement a tautology then?

Also, is there a formal definition for synonyms? Do they have to be logically equivalent but different in meaning? Do they have to imply one another, be the same in meaning, but different logical relation?

Thanks,

Pavel.

 

[AKG]

When you say "the fittest" is both a necessary and sufficient condition for "survives," are you basing this fact strictly on the definitions of "the fittest" and "survives," or is it conceivable that the fittest does not survive, but it has always been seen experimentally that the fittest does survive, and those that survive are the fittest? If you define "the fittest" as "those that survive" then "the fittest survives" is, of course, tautology. If you define an animal to be "the fittest" if, say, it is strongest, most intelligent, and fastest, and you determine that in all experimental cases, the fittest, and only the fittest survives, then you have not in fact shown that "the fittest" is both a sufficient and necessary condition for "survives," so, of course, it won't be a tautology.

Look into the various problems with induction, and also the analytic/synthetic distinction.

 

[Pavel]

When you say "the fittest" is both a necessary and sufficient condition for "survives," are you basing this fact strictly on the definitions of "the fittest" and "survives," or is it conceivable that the fittest does not survive, but it has always been seen experimentally that the fittest does survive, and those that survive are the fittest? If you define "the fittest" as "those that survive" then "the fittest survives" is, of course, tautology. If you define an animal to be "the fittest" if, say, it is strongest, most intelligent, and fastest, and you determine that in all experimental cases, the fittest, and only the fittest survives, then you have not in fact shown that "the fittest" is both a sufficient and necessary condition for "survives," so, of course, it won't be a tautology.

Hmm, well, I see your point, and I agree with your analysis. What's troubling me in this example is that it's kind of in the middle between being a definition and an empirical observation. I don't define the "fittest" as the "one who survives". Obviously, there woudn't be any problem, definitions are stipulations, they can't be right or wrong. However, here's the chain that makes me believe it's a sort of "necessary contigent" statement. The fittest is defined as the one who adapts the most to the environment. What is "adapts". Well, it's the one who manages to breed and continue its kind. What's "continue"? In other words, how can you tell that that specific species was able to continue and that one was not? Well, that one survived and that one didn't! That's the only way you can tell that that species managed to procriate or not. See my point? So, I don't really define the "fittest" as the "one who survies", but when you start the eximanation of what it implies, it appears that something is the fittest if, and only if, it survives. I'm not sure if it's the result of empirical observation or how we define the fittest... that's why I asked the question.

 

[AKG]

If I define A = B, and B = C, and C = D, is A iff D any less tautological than A iff B? In other words, whereas you say that you, "don't really define the "fittest" as the "one who survies," that is precisely what you're doing, just in a more roundabout way, giving the (false) impression that it is not what you're doing.

 

[Canute]

If you define "the fittest" as "those that survive" then "the fittest survives" is, of course, tautology. If you define an animal to be "the fittest" if, say, it is strongest, most intelligent, and fastest, and you determine that in all experimental cases, the fittest, and only the fittest survives, then you have not in fact shown that "the fittest" is both a sufficient and necessary condition for "survives," so, of course, it won't be a tautology.

I'm afraid it's still a tautology. When you say an animal is 'strongest, most intelligent and fastest' all you are doing is giving it traits that you assume will make it fitter. You might as well just say it's the fittest. After all, what if being strong, intelligent and fast does not make an animal fitter? (Perhaps in some cases such animals are more attractive to predators or viruses). Then you would choose different characteristics to stand in for 'fitness'.

To put it another way. We cannot escape the tautological nature of our concept of fitness by breaking an animals fitness down into the traits that we think bestow fitness on it. It doesn't change anything, the tautology is just less obvious.

 

[AKG]

Canute, you missed the point. I suggest you re-read what I've posted, because what you've said has no relevance. I defined "fittest" to be "fastest, strongest, most intelligent," so of course it is tautological that the fittest has those characteristics. No one's disputing that, that is not even the topic at hand (which is why I say you missed the point). What isn't a tautology, given this defintion, is "the fittest survives."

 

[Pavel]

If I define A = B, and B = C, and C = D, is A iff D any less tautological than A iff B? In other words, whereas you say that you, "don't really define the "fittest" as the "one who survies," that is precisely what you're doing, just in a more roundabout way, giving the (false) impression that it is not what you're doing.

Fair enough. I'm of the same opinion as you are, but if this was that easy, no scientist would open their mouth and preach "the survival of the fittest". That ain't the case, and you have to give them some benefit of the doubt. There's something bothering me about this, and I don't think it's just an impression of a roundabout way, as you put it. I think there's some legitimacy behind my concern, but I can't pinpoint it and I was wondering if anybody could do it.

Let me throw you another example. I don't believe "The first prime number" and the "the first natural even number" refer to exactly the same thing. While it appears to be 2, it's not the same 2 as an ontological entity that has a property of being the first prime number and the first even number. It's two separate conceptual entities, yet which, in the same number theory, stand to each other in a relation of logical equivalence (it's first prime, if and only if it's first natural even number) . To me, I thought, "adapt" and "survive" are completely different ontological entities, yet which are logically equvalent...

Pavel.

 

[AKG]

Fair enough. I'm of the same opinion as you are, but if this was that easy, no scientist would open their mouth and preach "the survival of the fittest". That ain't the case, and you have to give them some benefit of the doubt. There's something bothering me about this, and I don't think it's just an impression of a roundabout way, as you put it. I think there's some legitimacy behind my concern, but I can't pinpoint it and I was wondering if anybody could do it.

Let me throw you another example. I don't believe "The first prime number" and the "the first natural even number" refer to exactly the same thing. While it appears to be 2, it's not the same 2 as an ontological entity that has a property of being the first prime number and the first even number. It's two separate conceptual entities, yet which, in the same number theory, stand to each other in a relation of logical equivalence (it's first prime, if and only if it's first natural even number) . To me, I thought, "adapt" and "survive" are completely different ontological entities, yet which are logically equvalent...

Pavel.

If you define numbers in a given way, and you define "even" "prime" and all those other properties in a given way, then it will necessarily be the case that "the first prime number" is "the first natural even number." Those properties will necessarily belong to the same number. Again, it follows from definition (from the definitions of a great number of things, like "natural", "number", "even", "prime", "first", etc.), but it follows from definition nonetheless. Prima facie we can sort of imagine "the first prime" and "the first even natural" as having different referrants, but this would not have any basis in reason.

In an intuitive sense, we see the two predicates as being different, and this helps us. Sometimes, it is helpful to focus on the primeness of 2, sometimes, its evenness. But "x is the first prime" implies "x is the first prime" and "x is the first even natural." The implications of the statement are the same regardless of if you say, "x is the first prime" or "x is the first even natural." In a strictly logical sense, they mean the same thing (as the statements yield the same implications), but intuitively, we distinguish the two.

If I look at a bottle which is black on one side, and white on the other, it might appear to be black one day, and white the next. Prima facie, it appears that I am observing two different things. When viewed in full context, they are the same. At first glance, it appears that the two properties for "2" are different, and sometimes, it might be useful to just focus on one aspect, or it might be useful to just talk about the black half of the bottle, and ignore the white half entirely, but that white half will necessarily be there, and it is no less part of the bottle than the black part, even though we like to focus on one part at a time. Similarly, the two properties, in truth, mean the same thing, we simply like to focus on one aspect at a time.

As for scientists who claim "survival of the fittest," yes, we do give them the benefit of the doubt. Induction is inherently imperfect, but in every day life, in a practical sense, we treat inductive reasoning as justified reasoning. In an absolute sense, it is not, but we're normally not concerned with such things. So, scientists can claim "survival of the fittest," and if they show that this always holds true in experiment, we'll accept "survival" and "fittest" to have some sort of "practical equivalence," and in our everyday lives, we'd treat it no different than logical equivalence, even though we know that in truth (and in a less pragmatic sense) they are not equivalent, or at least, inductive reasoning has not, and can not, show them to be.

 

[Pavel]

In an intuitive sense, we see the two predicates as being different, and this helps us. Sometimes, it is helpful to focus on the primeness of 2, sometimes, its evenness. But "x is the first prime" implies "x is the first prime" and "x is the first even natural." The implications of the statement are the same regardless of if you say, "x is the first prime" or "x is the first even natural." In a strictly logical sense, they mean the same thing (as the statements yield the same implications), but intuitively, we distinguish the two.

We'll just have to disagree on this. I don't consider "x is the first prime" and "x is the first even" to mean the same thing, even in a logical sense. You'd have to show me how they're logically the same a priori. You know that they are the same how? Which a priori method, a necessary relationship, did you use to derive one from the other? Specifically please. Or did you find out that they're the same by examining the prime number line and the even number line? Do you hope perhaps there's a necessary relation between the two? Or do you have some conjecture about the realtionship between the two number lines?

If I look at a bottle which is black on one side, and white on the other, it might appear to be black one day, and white the next. Prima facie, it appears that I am observing two different things. When viewed in full context, they are the same. At first glance, it appears that the two properties for "2" are different, and sometimes, it might be useful to just focus on one aspect, or it might be useful to just talk about the black half of the bottle, and ignore the white half entirely, but that white half will necessarily be there, and it is no less part of the bottle than the black part, even though we like to focus on one part at a time. Similarly, the two properties, in truth, mean the same thing, we simply like to focus on one aspect at a time.

I can't remember the formal term for this fallacy, just to be pedantic ,but you're making a circlular argument - you presume the conclusion and then use it as a premise. Namely, you presume it's the same bottle and then use it as your premise to derive the conclusion that it's the same bottle. My whole argument that it is NOT the same bottle, that we're looking at two different bottles that stand in a relation of logical equivalence to each other.

As for scientists who claim "survival of the fittest," yes, we do give them the benefit of the doubt. Induction is inherently imperfect, but in every day life, in a practical sense, we treat inductive reasoning as justified reasoning. In an absolute sense, it is not, but we're normally not concerned with such things. So, scientists can claim "survival of the fittest," and if they show that this always holds true in experiment, we'll accept "survival" and "fittest" to have some sort of "practical equivalence," and in our everyday lives, we'd treat it no different than logical equivalence, even though we know that in truth (and in a less pragmatic sense) they are not equivalent, or at least, inductive reasoning has not, and can not, show them to be.

Heh, that was the whole point of my original post. Somehow, I don't believe they are making an inferential jump from the fittest to survival. I think they're commiting a tautology using terms that don't mean the same thing, but yet logically equivalent. It's because of this equivalence, I think it's a tautology. You're trying to make it either purely analytic (one simply defines the other), or purely synthetic (scientists concluded through experiments). I'm saying it's the middle, a necessary contigent, if you will. The terms do not define one another, yet they imply one another.

 

[AKG]

We'll just have to disagree on this. I don't consider "x is the first prime" and "x is the first even" to mean the same thing, even in a logical sense. You'd have to show me how they're logically the same a priori. You know that they are the same how? Which a priori method, a necessary relationship, did you use to derive one from the other? Specifically please. Or did you find out that they're the same by examining the prime number line and the even number line? Do you hope perhaps there's a necessary relation between the two? Or do you have some conjecture about the realtionship between the two number lines?

The easiest way is to show that the first prime is 2, and the first even natural is 2, and then use transitivity to show that the first even natural is the first prime. Now, when we learn math, and when we approach it as humans, it may feel like it has an experimental nature to it. We just so happen to discover that the first prime and the first even natural have the same referrent. We feel this to be true a posteriori. However, there is no justification to treat this as a posteriori knowledge and the knowledge that (x > 2) <--> (-x < -2) as a priori, it only feels that way. Why is "2 is the first prime and 2 is the first natural" any more coincedental than "x > 2 iff -x < -2?" Since the first is not obvious to us, it seems like more of a coincidence and almost an "empirical" discover, but in fact, it is no more of a coincidence than the second truth. It seems you're equivocating that which isn't immediately obvious with that which isn't necessarily true, a priori.

I can't remember the formal term for this fallacy, just to be pedantic ,but you're making a circlular argument - you presume the conclusion and then use it as a premise. Namely, you presume it's the same bottle and then use it as your premise to derive the conclusion that it's the same bottle. My whole argument that it is NOT the same bottle, that we're looking at two different bottles that stand in a relation of logical equivalence to each other.

I'm not fallaciously presuming it's the same bottle, I'm assuming, for the sake of analogy, that it is the same bottle. If "the first prime" and "the first even natural" were really just two, different-coloured faces of the same bottle, then the two really are inseperable, we just might like to focus on one face and ignore the other face sometimes. If a bottle is painted completely white, and you look at it from one side, and I look at it from the other, when we describe it, we will have no disagreement that we're looking at the same thing. Similarly, if you say x > 2 and I say -x < -2, we will not disagree that we're talking about the same thing. When you talk about a number being the first prime, and I talk about the first even natural, we might thing we're talking about different things (like one person looking at the black half, the other looking at the white) but it's really the same thing that looks like two distinct things if we only concentrate on what's obvious to us, and not what is the true nature of the bottle.

Heh, that was the whole point of my original post. Somehow, I don't believe they are making an inferential jump from the fittest to survival. I think they're commiting a tautology using terms that don't mean the same thing, but yet logically equivalent. It's because of this equivalence, I think it's a tautology. You're trying to make it either purely analytic (one simply defines the other), or purely synthetic (scientists concluded through experiments). I'm saying it's the middle, a necessary contigent, if you will. The terms do not define one another, yet they imply one another.

What would a "necessary contingent" be, given that the terms are contradictory? I suppose you mean it in a Zen-ish, paradoxical, "I mean not what I say," sort of way. Unfortunately, I don't read minds, so I don't know what you mean. Anyways, you suggest that "the fittest survives" falls on some middle ground between analytic and synthetic. Similarly, I suppose you believe that "2 is the first prime" and "2 is the first even natural" falls on that same ground, and that "Venus is the evening star" and "Venus is the morning star" does as well. I believe you might be equating relationships which don't appear necessary to relationships that aren't necessary. I believe that to be the fundamental flaw in your argument. Perhaps you're right in that "[w]e'll just have to disagree on this."

 

[AKG]

Actually, I think heading in the direction of "a priori" and "a posteriori" is wrong. We're concerned with the inherent distinction between two properties, and the methods we employ to know whether or not there is a distinction is independent from whether the distinction is inherent or coincidental. You're confusing an epistemological problem with a metaphysical/ontological one. Whether we come to know of the sameness of the two properties in an a posteriori, "coincidental" fashion (making the sameness appear contingent) or in an a priori fashion is irrelevant to the whether the sameness is necessary or not. Again, what may appear coincidental need not be coincidental, and in mathematics, it makes little sense to speak of anything as coincidental. It's not as though things could have been different. Numbers don't have causal effects on each other. Perhaps "prime" could have been defined differently in a different possible world, but then we'd be dealing with an entirely different problem. Given the definitions we're working with, there is no contingency: the discovery of some theorem doesn't cause the corollaries to be true, they were all true in the first place, it just wasn't obvious to us. Some people, like I've mentioned, redefine truth to mean justification. For these people, a theorem is not true one minute, then true the next once someone proves it, and shortly after, its corolloraies become true. The previous sentence makes sense to me if "true" is replaced with "justified," but not left on its own. Are you replacing "true" with "justified", thereby causing our disagreement? Are you replacing "apparently contingent" with "actually contingent", thereby causing our disagreement? Or is it something else?

 

[Pavel]

...

Hey man, so you don't feel I ignored your reply, I think it'd be better if we just leave this topic. You and I think on somewhat different frequencies and I think it'd take a lot of time for us to find a common ground and agree on the basics. Perhaps we'll have better luck in a different topic.

Thank you for your comments though!

Pavel.