Philosophy Now Forum

I'm no logician, but as I understand it, 2+2=4 has the same logical structure as 2-2=0 hence they are logically equivalent. Clearly though, they mean different things.

Or my goodness, someone (5 red apples) understands the statements in the same way as me !!

I don´t have to despair any more ...

uwot wrote:I'm no logician, but as I understand it, 2+2=4 has the same logical structure as 2-2=0 hence they are logically equivalent. Clearly though, they mean different things.

Why are they logically equivalent ?
Because the result is "true" in both cases ?

uwot wrote:I'm no logician, but as I understand it, 2+2=4 has the same logical structure as 2-2=0 hence they are logically equivalent. Clearly though, they mean different things.

These are not logic statements but self-defining statements and thus completely circular. Symbolic logic is inherently tautologous.

x + 2 = 4
x - 2 = O

What is x ?

2 !

uwot wrote:I'm no logician, but as I understand it, 2+2=4 has the same logical structure as 2-2=0 hence they are logically equivalent. Clearly though, they mean different things.

Interesting distinction. Let me see if I can unpack what you're getting at.

As syntax, they are equivalent because you can start from either one and derive the other via formally legal manipulations.

In terms of semantics, you might say they're different because 2 + 2 = 4 expresses 4 as the result of a particular computation, while 2 - 2 = 0 expresses -2 as the additive inverse of 2. But that's a stretch, I don't think that's really any kind of accepted reasoning.

Perhaps you can explain why you think their logical structures are equivalent but their meanings different. You said "clearly," but your comment is far from clear to me.

wtf wrote:Perhaps you can explain why you think their logical structures are equivalent but their meanings different.

It's probably because I'm not a logician.

Are they mathematically equivalent ?

I suppose that the sign "=" made you think of "equivalent".

Two red apples and two green apples are equivalent to four red or green apples.

Two yellow apples on a plate and two apples from this plate being eaten is equivalent to no apples on the plate.

duszek wrote:Are they mathematically equivalent ?

I suppose that the sign "=" made you think of "equivalent".

Two red apples and two green apples are equivalent to four red or green apples.

Two yellow apples on a plate and two apples from this plate being eaten is equivalent to no apples on the plate.

You make a good point. Mathematical logic can only be applied to mathematical objects and not to physical objects. In a physical sense not only are there no two apples the same but even a single apple is not physically the same apple from one Planck interval to the next. This principle is as old as the Ship of Theseus and serves to illustrate the illusory nature of what we think of as an object. An object is actually a PROCESS in continuous transition so equivalence is not a valid physical construct. Reality is only definable in the language of its changes.

Yes, Leo.
But we do use mathematical logic and logic as in language when we talk about real objects in the physical world.

I sell a hog and weigh it. The hog is 105 pounds and I get 80 Australian dollars for it.

If the hog weighed 156 pounds what would be a fair price for it ?

We can make meta-language statements (definitions for example) and statements referring to the real world (what is the case now).

When I say: I see one hog now.
Then the hog has not enough time to change (procreate, develop into a new species) and to make my statement false.

I wasn't objecting to the general principle of equivalence in the sense of our common language usage but rather in applied metaphysical terms. As a process philosopher I regard our commonsense notion of the "object" as physically illusory although it obviously retains some utility in our everyday human discourse. Where it has no utility at all is in sub-atomic physics, where the subatomic particles are routinely described by physicists as being "fundamental", despite the fact that Einstein disproved this assumption with his famous equation E=mcc. Subatomic particles with mass can only be understood as discrete quanta of energy which have been configured in a particular way and we already know that quanta of energy can only move at the speed of light. Therefore not only does the apple become a different apple in each successive Planck interval so too does every subatomic particle within the apple become a different subatomic particle at the speed of light. Therefore atoms are not specified by what these particles ARE but rather atoms are specified by what these particles ARE DOING.

wtf wrote:

uwot wrote:I'm no logician, but as I understand it, 2+2=4 has the same logical structure as 2-2=0 hence they are logically equivalent. Clearly though, they mean different things.

Interesting distinction. Let me see if I can unpack what you're getting at.

As syntax, they are equivalent because you can start from either one and derive the other via formally legal manipulations.

In terms of semantics, you might say they're different because 2 + 2 = 4 expresses 4 as the result of a particular computation, while 2 - 2 = 0 expresses -2 as the additive inverse of 2. But that's a stretch, I don't think that's really any kind of accepted reasoning.

Perhaps you can explain why you think their logical structures are equivalent but their meanings different. You said "clearly," but your comment is far from clear to me.

Whilst it is true that 2+1=3, and 3+4=7 has the same logical structure, I think it not fair to suggest that 3-1=2 has the same logical structure. I think the operator must indicate a different logical structure in this case a logical subtraction, rather than addition.
It is the number values that are arbitrary not the logical operators.

Mortician wrote:
Enjoy music less
Enjoy less music

The statements are mutually compatible if
You enjoy less music and listen to less of it

The statements are mutually incompatible if
You enjoy less music and listen to more of it
You enjoy more music and listen to less of it

These sentences are not logically equivalent.