Validity of a simple logical argument

[parmalee]

Clearly, Aristotle himself would have baulked at any argument with necessarily false premises. But remember he was the one who started the ball rolling so you can't expect him to have formalised or even articulated all there is to say on the subject.
So, all we can do is interpret his thinking.
As per the conventional interpretation of Aristotle, the validity of a syllogism is not decided on whether the premises are or not actually true. Instead, you just assume them true and decide for yourself if that somehow compels the conclusion to be true as well. In the case of the argument here, that's not the case since the last premise assumed true contradicts the conclusion, as Write4U correctly observed and probably why Quantum Quack and RainbowSingularity voted "not valid".
That's simple and it exactly fits with our intuition. The aficionados of modern "classical" mathematical logic here, like Sarkus, Yazata and Baldeee, all suppress the expression of their intuition so as to comply with the dogma they submit to even though it doesn't make sense. That is unfortunately something human beings can choose to do.
EB

That's why you need a fifth option on your poll:

Is the argument valid? That depends...

But now I understand why you are going with the early 20th century, rather than, say, mid/late 19th century, so thank you for the clarification.

 

[Speakpigeon]

But now I understand why you are going with the early 20th century, rather than, say, mid/late 19th century, so thank you for the clarification.

Please clarify and/or explain.
EB

 

[James R]

Also worth re-writing this in terms of a boolean expression. We have:

1. squid implies not giraffe.
2. giraffe implies not elephant.
3. elephant implies not squid.
4. (Joe is) squid or giraffe.
5. (Joe is) elephant.
(Conclusion. Joe is) squid.

Using the symbols "&" (conjunction), "or" (disjunction), "!" (negation), and "=>" (implication), the argument above, symbolically, is:

[(S => !G) & (G => !E) & (E => !S) & (S or G) & E] => S

Now we replace the implications, noting that (A => B) = (!A or B). The argument is equivalent to:

![(!S or !G) & (!G or !E) & (!E or !S) & (S or G) & E] or S

Continue, using various rules of Boolean algebra:

![(!S or !G) & (!G or !E) & (!E or !S) & (S or G) & E] or S
= !(!S or !G) or !(!G or !E) or !(!E or !S) or !(S or G) or !E or S
= (S & G) or (G & E) or (E & S) or (!S & !G) or !E or S
= (S & G) or (G & E) or [(E & S) or !E] or [(!S & !G) or S]
= (S & G) or (G & E) or [(E or !E) & (S or !E)] or [(!S or S) & (!G or S)]
= (S & G) or (G & E) or (S or !E) or (!G or S)
= (S & G) or (G & E) or S or !E or !G
= S or !E or (G & E) or !G
= S or !E or (G or !G) & (E or !G)
= S or !E or E or !G
= S or !G or TRUE
= TRUE

Thus we see that the given premises logically imply the given conclusion, thereby making the argument valid.

Well, that was fun. I haven't done a formal proof like this in years.

 

[James R]

As per the conventional interpretation of Aristotle, the validity of a syllogism is not decided on whether the premises are or not actually true. Instead, you just assume them true and decide for yourself if that somehow compels the conclusion to be true as well. In the case of the argument here, that's not the case since the last premise assumed true contradicts the conclusion,

The problem here is that you cannot assume that all of the premises are true simultaneously.

1. A squid is not a giraffe
2. A giraffe is not an elephant
3. An elephant is not a squid
4. Joe is either a squid or a giraffe
5. Joe is an elephant
Conclusion: Therefore, Joe is a squid

Note firstly that premises (4) and (5) mandate that Joe must be a squid or a giraffe, and also that Joe must be an elephant.

If Joe is a squid, then this contradicts premises (3) and (5), taken in combination.
If Joe is a giraffe, then this contradicts premises (2) and (5), taken in combination.
If Joe is an elephant, then this contradicts premises (2) and (3) and (4), taken in combination.

Since you can't do as Aristotle directs - i.e. assume all the premises are true - then the Aristotlean validity of the argument is undecidable under Aristotle's criterion. We need a better definition of what makes a valid argument if we want every argument to be either valid or invalid, rather than having a third "undecidable" category.

 

[Speakpigeon]

Thus we see that the given premises logically imply the given conclusion, thereby making the argument valid.

I guess you understand that the notion of validity you use here is the one defined in material logic (modern mathematical "classical logic").
And so, how do you feel about the argument? Do you really feel that the conclusion follows from the premises?
EB

 

[Speakpigeon]

The problem here is that you cannot assume that all of the premises are true simultaneously.
(...)
Note firstly that premises (4) and (5) mandate that Joe must be a squid or a giraffe, and also that Joe must be an elephant.
If Joe is a squid, then this contradicts premises (3) and (5), taken in combination.
If Joe is a giraffe, then this contradicts premises (2) and (5), taken in combination.
If Joe is an elephant, then this contradicts premises (2) and (3) and (4), taken in combination.
Since you can't do as Aristotle directs - i.e. assume all the premises are true - then the Aristotlean validity of the argument is undecidable under Aristotle's criterion. We need a better definition of what makes a valid argument if we want every argument to be either valid or invalid, rather than having a third "undecidable" category.

Sure, the premises here could not all be true.
Yet, you're wrong when you claim here that we cannot assume that they are all true. In fact, there's absolutely no difficulty in that.
For Aristotle, premises are just stated, i.e. asserted as true, i.e. assumed true. If the conclusion then follows necessarily from the premises, the argument is valid.
Clearly, assuming all premises are true, that Joe is a Squid doesn't follow necessarily and therefore the argument is invalid.
You have to remember that the definition of validity given by Sarkus is that used for the material implication: "if, and only if, it is impossible for (all) the premises to be true and the conclusion at the same time to be false".
As such, you can't use it for validity as Aristotle described it.
In effect, the definition of validity in modern mathematical "classical logic" has been tweaked to comply with Russell's definition of the material implication. As such, it doesn't work like in Aristotle's notion of syllogism, understood as a valid argument.
Given premises 3 and 5, the conclusion not only doesn't follow but it is necessarily false.
Indeed, no conclusion based on the premises as stated could possibly be true. So, no conclusion follows at all. And I dare say, this seems exactly what sanity requires.
So, in effect, Aristotle's notion of syllogism seems to capture perfectly what we feel intuitively is a valid argument.
EB

 

[parmalee]

Please clarify and/or explain.
EB

My initial query arose from a misreading of your post #66, so nevermind that. Anyhows:

It's the notion of logical validity as it is used since Aristotle.
Modern "classical" mathematical logic tweaked the definition of validity to make it compatible with the truth-table definition of the material implication, which, as it's name clearly suggests, is not the logical implication.
EB

Your "it," in the first sentence, referring to Yazata's initial formulation in post #40 (paraphrasing, "an argument is valid iff, all premises being true, the conclusion necessarily follows").

Yeah, I can appreciate your criticisms of what is now commonly referred to as "classical logic"--well, namely that it is referred to as such. An old prof, way back when--a Russian guy who was a dead ringer for Kyle Machlachlan--pretty much shared your perspective. It kinda complicated what he was teaching, his repeated (and, at a certain stage, unnecessary ) insistence on how such and such runs counter to intuition and, well, logic; nonetheless, it was salient point.

 

[Speakpigeon]

My initial query arose from a misreading of your post #66, so nevermind that. Anyhows:

Aw sucks. It seemed to be a promising kerfuffle...

Your "it," in the first sentence, referring to Yazata's initial formulation in post #40 (paraphrasing, "an argument is valid iff, all premises being true, the conclusion necessarily follows").

Not quite...
Yazata's initial version: "An argument (premises and conclusion) is valid iff, when all the premises are interpreted as being true, the conclusion must also be true."
Are interpreted as being true... Which is always possible, and very different from "all premises being true", which is impossible in the case of this thread's Squid argument.
And that's precisely what most people fail to notice. It's not even something you will read in any logic textbook.

Yeah, I can appreciate your criticisms of what is now commonly referred to as "classical logic"--well, namely that it is referred to as such. An old prof, way back when--a Russian guy who was a dead ringer for Kyle Machlachlan--pretty much shared your perspective. It kinda complicated what he was teaching, his repeated (and, at a certain stage, unnecessary ) insistence on how such and such runs counter to intuition and, well, logic; nonetheless, it was salient point.

Thanks for sharing the anecdote. I would guess many people have some similar story. I realised there was a problem I was 19 and a maths and physics student at university. My very first time exposure to formal logic. Everything goes smoothly, conjunction, disjunction, negation etc. Arrive the implication. I stare at the truth table the guy had written on the blackboard and my brain just pukes. I can still remember this moment. So, I realised there and then there was something seriously wrong with mathematical logic. It's only later that I realised many people had a similar story to tell and indeed many philosophers have kept criticising the material implication since its inception. Without any perceptible effect. And the reason is first that the material implication is not really used to do anything (and I would certainly hope so). The second reason is that there is the Gentzen method of proof, which in effect very similar to and a generalisation of Aristotle. Seems to work well enough. The main reason is that mathematicians themselves don't even use any formal logic to prove their theorems. Meanwhile, we have this ridiculous situation where cohortes of students are taught logic as based on the material implication. This is seriously idiotic and most likely damaging in some respect.
EB

 

[arfa brane]

The main reason is that mathematicians themselves don't even use any formal logic to prove their theorems.

OMG, it's all just made-up as they go?

 

[Speakpigeon]

OMG, it's all just made-up as they go?

Nearly all mathematicians use their logical intuitions. One or two use formal logic through theorem provers. I know of only one proof done by a mathematician using formal logic. And it's not just me who say it. One guy working on theorem provers says exactly that. And one philosopher recently wrote a paper saying the same thing. It's not exactly a secret.
EB

 

[Write4U]

Nearly all mathematicians use their logical intuitions. One or two use formal logic through theorem provers. I know of only one proof done by a mathematician using formal logic. And it's not just me who say it. One guy working on theorem provers says exactly that. And one philosopher recently wrote a paper saying the same thing. It's not exactly a secret.
EB

Do computers use "formal" logic?

 

[arfa brane]

Nearly all mathematicians use their logical intuitions. One or two use formal logic through theorem provers.

A question: is mathematics a formal system? If it is, does it have a formal logic?

Are there any sets of rule-based "mathematical logic"? can I just write down an intuitive equation that says 2 + 3 = 4?

 

[Speakpigeon]

Do computers use "formal" logic?

Good question. I would encourage you to start a thread on this. It's a very interesting point. I'm sure you and many people will have things to say.
EB

 

[Speakpigeon]

A question: is mathematics a formal system?

Yes.

If it is, does it have a formal logic?

It's complicated.
There is something called "mathematical logic", but this is a field of study, not a method of logic.
There are different methods which are considered and studied in this field and are considered by mathematicians as "logics", like paraconsistent logic, relevance logic, first order logic, etc. but most of them are not what we originally mean by logic. They are best understood not as logic proper but as mathematical theories that have some similarities with logic. 1st order logic, in particular, since it is the de facto standard as far as mathematical logic goes, is at best an approximation of logic. In fact, there is an irony in the name itself, "1st order logic", because it is exactly what it is, a 1st order of approximation of logic as we think of it. As such, it has been shown to give the correct results for a number logical formulas that are basic, which is why it got selected in the first place, but you won't necessarily know if the result is still correct whenever you try with more complex formulas, and many formulas are known to give the wrong result, as exemplified in my Squid argument.
Then there is the entirely different Gentzen method of formal proof. It is used to prove mathematical theorems and produce formal proofs. However, it is still based on logical formulas, namely "rules of inference", that are themselves not proven true and therefore just admitted as true on the face of them, i.e. we all agree intuitively they must be true (e.g. "A and B implies B"). Still, as such, it is a method which is both formal and probably logical, i.e. consistent with logic as we think of it, although I couldn't possibly guaranty that.
So, Gentzem is indeed a formal method of logic for proving mathematical theorems, but it works like Aristotle's syllogism in that it is based on rules of inference admitted as true, not proven true. As far as I can tell, it is a mathematical formalisation and generalisation of Aristotle's method of formal logic.
However, very few mathematicians actually use it. It is used mainly in the context of theorem provers.

Are there any sets of rule-based "mathematical logic"?

Yes, that's Gentzen.

can I just write down an intuitive equation that says 2 + 3 = 4?

Mathematicians start with any arbitrary axioms they want and then they try to work out all the logical consequences of those, and nearly always that will be on the basis of their own logical intuitions, not any method of formal logic.
A mathematician could start an axiomatic system with an axiom saying 2 + 3 = 4. As long as it is not logically inconsistent with another of his axioms, he will be happy. No problem with that.
However, me, I'm talking only of logical intuitions. And there's nothing logically intuitive in 2 + 3 = 4 (and there is nothing intuitive in 2+2 = 4 until you get trained in the addition). Logical intuitions are not just arbitrary ideas. A logical intuition is the subjectively certain impression that a logical relation, like for example "A and B implies A", is true. I have very good reasons to think that logical intuitions are not dependent on training in formal logic. Rather the reverse. We understand formal logic because we have logical intuitions prior to being exposed to it.
Although, I can't guarantee this applies to everybody, or even to most of us.
EB

 

[Write4U]

Are there any sets of rule-based "mathematical logic"? can I just write down an intuitive equation that says 2 + 3 = 4?

Is that not using a logical function based on a false premise? The premise that 5 = 4?
After all the answer is known.
Logic would demand that 2 + 3 = 5 as codified in the decimal number system (and the Fibonacci sequence), no?

 

[James R]

Arrive the implication. I stare at the truth table the guy had written on the blackboard and my brain just pukes. I can still remember this moment. So, I realised there and then there was something seriously wrong with mathematical logic.

Speaking of truth tables, here's the relevant table for A => B:

A B  A=>B
0 0  1
0 1  1
1 0  0
1 1  1

If you wanted to implement implication in a computer program, say, what truth table would you prefer to use?

 

[Write4U]

Speaking of truth tables, here's the relevant table for A => B:

A B  A=>B
0 0  1
0 1  1
1 0  0
1 1  1

If you wanted to implement implication in a computer program, say, what truth table would you prefer to use?

Roger Antonsen shows several truth tables dependent on the base used.

https://www.ted.com/talks/roger_antonsen_math_is_the_hidden_secret_to_understanding_the_world

 

[Write4U]

You should learn to discipline yourself. You need to be able to read an argument without adding anything to the premises.

Rather than stew over the logical validity of a proposition, we should perhaps concentrate on the real values which are presented and which will be mathematically (logically) processed;
"Joe is either a squid or a giraffe,
Joe is an elephant
Therefore, "Joe is a squid" = "Joe is an elephant" = NOT logically possible = Error.
This equation is forbidden by the physical mathematics.

Two opposing answers to an equation is mathematically impossible and therefore logically false even though the argument is logically valid until the statement of fact, a specific value, which introduces a relative constant, which now requires allowable other mathematical values of squids and giraffes, which do not exist, unless you want to go back to origins....

Try the experiment with Platonic solids if you doubt me.
Can you logically turn a sphere into a cube or an octahedron and maintain it's values and characteristics as a sphere?.............

 

[Speakpigeon]

Speaking of truth tables, here's the relevant table for A => B:
(...)
If you wanted to implement implication in a computer program, say, what truth table would you prefer to use?

I would want something that works.
1. So, why use this truth table?
2. Why use any truth table?
EB

 

[Speakpigeon]

"Joe is either a squid or a giraffe,
Joe is an elephant
Therefore, "Joe is a squid" = "Joe is an elephant" = NOT logically possible = Error.

I saw one guy produce a proof that Joe is a squid. His proof is 10 lines long but I can do it 8. He infers from his proof that the argument is valid, even though you can prove using his same method that Joe is not a squid in 3 lines. Yet, he persists. He exhibited what he thinks is a proof and that's it. His method is flawed but he doesn't know that and he certainly doesn't understand why it is flawed. Using apparently the same method, I proved correctly the argument not valid:

Proof
An elephant is not a squid..............P3
Joe is an elephant...........................P5
Therefore, Joe is a not squid..........P3, P5, R1: ((x ≠ y) ∧ (z = x) ) → (z ≠ y)​

Which is basically what you said.
Formally, for the argument to be valid, since equality is not a logical symbol, you need to add a premise with the R1 rule ((x ≠ y) ∧ (z = x) ) → (z ≠ y). But it's implicit anyway.
EB