On logical equivalence

TheMadFool

My logic is rusty so kindly bear with me. I have a question on logical equality or equivalence.

The law of identity comes to mind first. Basically the law of identity states that for any given proposition A, A = A. I'm fine with that as without it we wouldn't be able to do any thinking at all.

However, my question is about logical equivalence i.e. the equality of two two different propositions. Say I have the proposition A and a different proposition B. A = B would be an indication of A <-> B being a tautology i.e. true under all circumstances in all possible worlds. In such a case one could say A = B because their truth values synchronize perfectly (T,T) or (F,F) in all possible worlds. I guess this has to do with truth functional interpretation of propositions.

However...

I'd like to discuss the word ''is'' which logically translates to ''=''. When I say ''Trump is the POTUS'' I mean Trump=POTUS.

In logic equivalence is commutative i.e. is to say, in my example, there's no difference between Trump=POTUS and POTUS=Trump.

But there seems to be a difference between T: Trump=POTUS and P: POTUS=Trump.

If I say T then I mean what everyone usually means in that Trump is the POTUS now. However, when I say P:POTUS=Trump there's an added meaning to what T says. P seems to have the additional meaning of ''Trump is the POTUS'' implying that there is no POTUS better than Trump.

So, in effect I'm saying that T is NOT equivalent to P although logically they are.

What are your views? Thanks.

Jeremiah

I don't think this has anything to do with logic, and this is posted in the wrong section.

tim wood

↪TheMadFool

I'll take a shot at it. The "=" is a strict operator. What's on the left equals what is on the right - whatever that means in the context.

But Trump is POTUS and (the) POTUS is Trump are not equivalent propositions. Two quick examples: 1) Dogs are four-legged. (The) four-legged are dogs. One is subject, the other is predicated of the subject. All s are p: all p are s. Do you see where some problems lie? Turning the propositions around this way is called conversion. Not all propositions are convertible.

There is also a substance/accident issue. Trump is a substance (Socrates is a substance); substances cannot be an accident (hahahahahahahahha - lol); i.e., substances cannot be predicates*. Logic can be ridiculous! Whether POTUS is a substance is an argument for someone else.

So you're reading the proposition as two kinds of propositions read in two different ways. Your ear and your sense tell you something went wrong. What went wrong is that the second, the "is" proposition, is (in this case) not convertible, whereas the "=" proposition is always necessarily convertible.

*The man is Socrates. Someone else can make this clearer. The idea is that Socrates, here, is not an accident of the man, whereas "is tall," or "has a hat" would be accidents (quality, possession).

Srap Tasmaner

↪TheMadFool

The short answer is that logical equivalence is just a matter of truth value, which in turn is just a matter of extension. All the other nuances of language are deliberately left out.

jkg20

↪Srap Tasmaner

T: Trump=POTUS and P: POTUS=Trump.
You talk about meaning being added by carrying out the change of position from T to P, so let us assume that meaning is indeed added. What a proposition expresses is its meaning and different meanings can be expressed only by different propositions. If T expresses one meaning and P expresses that meaning + some added meaning, then P expresses a different meaning to T and so is a different proposition. Different propositions, under propositional calculus, can take truth values independently of each other, so T <-> P is not a tautology in this case and so they are not logically equivalent.
Logical equivalence does not always lead to identity, or at least not straightforwardly - the connection is complex. A coin's head exists if and only if a coin's tail exists, but a coin's head is not a coin's tail, so that a coin's head exists must express a different proposition than that a coin's tail exists. Of course, I'm making the fatal error of treating exists as a predicate here, may Kant forgive me.
@TheMadFool I seem to remember that old fraud Quine suggesting that one could turn proper names into predicates (maybe in "On What There Is"?): "Socrates" becomes "the unique Socratizer" or some such nonsense.

TheMadFool

So you're reading the proposition as two kinds of propositions read in two different ways. Your ear and your sense tell you something went wrong. What went wrong is that the second, the "is" proposition, is (in this case) not convertible, whereas the "=" proposition is always necessarily convertible. — tim wood

Does that mean logic can't handle this particular nuance of "is" in language? Thank you. Your answer is the most sensible.

he short answer is that logical equivalence is just a matter of truth value, which in turn is just a matter of extension. All the other nuances of language are deliberately left out. — Srap Tasmaner

:up:

I seem to remember that old fraud Quine suggesting that one could turn proper names into predicates (maybe in "On What There Is"?): "Socrates" becomes "the unique Socratizer" or some such nonsense. — jkg20

Can you explain what that means? I understand that predicates usually have to be properties and proper names are more like arbitrary labels given to some objects. Using a proper name as a predicate would be confusing unless the particular proper name is an archetype.

andrewk

I'd like to discuss the word ''is'' which logically translates to ''=''. When I say ''Trump is the POTUS'' I mean Trump=POTUS. — TheMadFool

You think that's bad. What about 'She is hungry'?

The problem is with the verb 'to be', which is a jumble of vagueness and equivocations. That's why somebody invented E*Prime to avoid its use.

The French have the right idea. They say 'I have hunger', which avoids the whole problem. The E*Prime way of saying that Macron 'is' the president of the Republic of France would be something like 'Monsieur Emanuel Macron holds the office of president of the Republic of France.'

jkg20

↪andrewk

Can you explain what that means?

You'd need to read up on Quine's writings on ontological commitments and how to avoid them to get the details. Basically, Quine's idea was that the "ideal" language of metaphysics should have no singular terms such as names or constants, and consist just of variables, quantifiers, predicates and rules of logical inference.

Jeremiah

The stress is subjectivitaly beinging put on the word the, and that stress is ambiguous. This is a grammar issue, not one of logic.

Jeremiah

Holy crap! Trump is the president. We are all doomed!

tim wood

Does that mean logic can't handle this particular nuance of "is" in language? Thank you. Your answer is the most sensible. — TheMadFool

Yours is actually not a simple question. Short answer is both yes and no. It depends on what you want. Logic is a tool that aims for precision in the the development of arguments to certain conclusions. Like any tool, it can be misused and abused. And it requires materials of a certain quality. The "is," as andrewk puts best above, is problematic and used carelessly can introduce error and absurdity.

Parallel to, and similar but different, is rhetoric, aka persuasion. While the proper material for logic is matters that are always and universally so, and understood to be so. The proper material for rhetoric is matters that are contingent and could be this way or that way.

Nuance, or ambiguity, or imprecision, can blow up logic. But the same are among the basic materials of rhetorical propositions.

andrewk

↪jkg20

I think it must be somebody else that said that. I can't see it in what I wrote.

Banno

The law of identity comes to mind first. Basically the law of identity states that for any given proposition A, A = A. I'm fine with that as without it we wouldn't be able to do any thinking at all. — TheMadFool

The law of identity is better phrased as every thing is the same as itself. that is,

U(x)(x=x)

were x is an individual, not a proposition.

But "Trump is President" is represented as P(t), a predicate relation.

In English both relationships are parsed using "is". The logical parsing shows that the English parsing is ambiguous.

So Trump is Trump, and unfortunately Trump is also the President. The first "is" is the "is" of identity; the second, the "is" of predication.

When I say ''Trump is the POTUS'' I mean Trump=POTUS. — TheMadFool

No, you don't.

Banno

↪Jeremiah

Not all of us. China is doing fine.

jkg20

↪andrewk

Mea culpa - the reply was for @TheMadFool's question concerning Quine's idea that names can be converted to predicates.

TheMadFool

↪andrewk

:grin: :up:

TheMadFool

↪Banno

Thanks

TheMadFool

Holy crap! Trump is the president. We are all doomed! — Jeremiah

:joke:

EnPassant

I think the confusion is purely semantic. POTUS is a position in a hierarchy that can be occupied by many people. Trump is only one person. T = P, is meaningless if P = 'the top position in the hierarchy'.
T = P has meaning if P = 'the present occupier of the top position in the hierarchy'

On logical equivalence

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