4 very easy arguments. Are they valid?

[fast]

... There is an underlying theme at play and it has to do with collective exhaustivity. ...

I think together we've succeeded in that effort. I concede ...

P2' and P2" are contradictory contrary to each other. Or else I just do not get it.

ETA- Either that or an elephant is a type of giraffe in which case C' is false, as well as C"'.

ETA - I hope I'm using the word concede correctly.

Lol.

I want to say something that might help illuminate matters. What may appear as a tangent on my part with arguing that the sentences aren’t contradictory isn’t as much of a tangent as one may realize.

There are a couple overarching rules to be mindful of when categorizing things, and it pertains to mutual exclusitivity and collective exhaustivity. Let’s say I wanted to track the hair color of everyone entering a building. I might make a form that reads: a) blonde, b) brunette, and c) redhead. As each person enters, I would observe the hair color and simply record the observation, but in the end, it’s very important that there’s a match between the number of observations made and the number of checks made in the boxes.

The categories should be mutually exclusive, meaning the categories I write down should accommodate the observations such that each person belongs in one and only one box. That would of worked in the 1950’s when whoever came in the building would be observed as either being one and never more than one. Today in San Francisco, a person might be part blonde and part brunette. That might require being more specific with the form: a) originally blonde, b) originally brunette, etc. the point is still that the categories should be such that for each person, there should be only one box that is applicable to them. At any rate, that has to do with mutual exclusivity.

However, it’s the other rule of categorizing that I would like to focus on: collective exhaustivity. If a person came in and was bald and was born with a defect such that she has no hair, there would be no category written on my paper to cover the instance. What people ordinarily do (and we see it all the time) is that they will tag on another catch-all category, often dubbed as “other.” For instance, if we were to write down the following categories:

A) completely blonde
B) completely brunette
C) completely red head

That would not be collectively exhaustive, but if we added

D) other

Then, no matter who came in, there would be a box for the person entering the building no matter what.

Second example: whoever enters the building is either

A) shorter than four feet tall
Or
B) taller than four feet tall

That would work out great except for the rare times someone enters and happens to be exactly four feet tall.

The problem is that the list isn’t collectively exhaustive, and what needs to be done is that the list needs to be remedied such that no matter who enters, there’s a category for them to belong to.

A versus not A is collectively exhaustive
A, B, and neither A nor B is collectively exhaustive.
A, B, C and everyone else is collectively exhaustive

Valid and not valid is exhaustive, but valid and invalid is not. This speaks to form, so it’s mighty important. If the argument is valid, then it’s a deductive argument such that the conclusion can be trusted IF the premises are true.

If it’s false that an argument is valid, and if that’s all I know, I cannot discern whether the argument is deductive or not; as far as I know, it could be an extremely strong inductive argument that would be foolish to immediately dismiss. An argument that is not valid might be deductive, or it might be nondeductive; if an argument is invalid, then I know immediately that we’re dealing with a deductive argument whose conclusion cannot be guarenteed. Validity allows for trust. Sprinkle a deductive argument that is valid with true premises, and the conclusion is guarenteed. Remember, never does nondeductive arguments give rise to that trust because they never have true guarentee-producing validity.

A contradiction such as I am in SC and I am not in SC guarentees the truth of one because there is collective exhaustivity.

An argument with true premises that has a valid form provides the same guarentee because of the precise definition of validity that allows for there to be collective exhaustivity between valid and invalid when only pertaining to deductive arguments. It’s getting late; I hope I worded that halfway right.

 

[Speakpigeon]

I don't want to know your answers.
EB

In light of that, I let you know that in order to address some of the points made by other posters, I will disclose the part that is hidden by quoting it in the next post.

I'm stil waiting... What is the justification given by professional specialists, mathematicians, logicians, philosophers etc. that would support your claim that the definition of logical validity you use is the correct one.
EB

 

[Treedbear]

P2' and P2" are contradictory. Or else I just do not get it.

ETA- Either that or an elephant is a type of giraffe in which case C' is false, as well as C"'.

No, P2' and P2'' are not contradictory, at least not by their form, though you might argue that it is a contradiction to claim that something is both a giraffe and an elephant by the meaning of the words, but that would not allow you to derive a contradiction without further premises that are not (by their form) tautologies, since you would need the semantic claim.

The argument A4 is valid, though, and the premises are contradictory together, even if they are pairwise not contradictory.

In that case I just don't understand the value of syllogisms when they allow you to create a paradox by simply inserting a contradictory premise into an argument for no other purpose. That seems to be the sole purpose of P2". I can see the value of creating sorites ("a form of argument in which a series of incomplete syllogisms is so arranged that the predicate of each premise forms the subject of the next until the subject of the first is joined with the predicate of the last in the conclusion.") But I don't see that happening here. They're simply unconnected arguments.

 

[Angra Mainyu]

I don't want to know your answers.
EB

In light of that, I let you know that in order to address some of the points made by other posters, I will disclose the part that is hidden by quoting it in the next post.

I'm stil waiting... What is the justification given by professional specialists, mathematicians, logicians, philosophers etc. that would support your claim that the definition of logical validity you use is the correct one.
EB

First, you have failed to answer the question: are these arguments valid?

Second, if you are waiting for that you are being irrational. I did not claim that they do that, and I have no interest in trying to say that that is their justification. In fact, I claimed that most of them do not give any justification for the concepts they use, including this one: they just use them.

Third, the arguments are valid even under the definition you yourself provided in the other thread.

 

[Angra Mainyu]

P2' and P2" are contradictory. Or else I just do not get it.

ETA- Either that or an elephant is a type of giraffe in which case C' is false, as well as C"'.

No, P2' and P2'' are not contradictory, at least not by their form, though you might argue that it is a contradiction to claim that something is both a giraffe and an elephant by the meaning of the words, but that would not allow you to derive a contradiction without further premises that are not (by their form) tautologies, since you would need the semantic claim.

The argument A4 is valid, though, and the premises are contradictory together, even if they are pairwise not contradictory.

In that case I just don't understand the value of syllogisms when they allow you to create a paradox by simply inserting a contradictory premise into an argument for no other purpose. That seems to be the sole purpose of P2". I can see the value of creating sorites ("a form of argument in which a series of incomplete syllogisms is so arranged that the predicate of each premise forms the subject of the next until the subject of the first is joined with the predicate of the last in the conclusion.") But I don't see that happening here. They're simply unconnected arguments.

They do not allow you to "create paradoxes". It is not a paradox that from some premises, a contradiction follows. The reason I'm doing this is to try to help reduce some of the confusion caused by Speakpigeon in the other thread. It seems to me there might be room for some progress here, but I would like to know where you are, so I would like to ask you two questions:

1. Do you realize why A1, A2, and A3 are valid?
2. Do you realize why A4 is valid?

 

[steve_bank]

[

P1: Joe is either a squid or a giraffe.
P2: Joe is not a squid.
C: Joe is a giraffe.

s OR g, NOT s
c = [ s OR g ] AND [ NOT s ]
c = [ s XOR g ] AND [ g ] = g
XOR exclusive OR, one or the other but not both
s g c
0 1 0
1 0 1

P1’: A giraffe is not an elephant.
P2’: Joe is a giraffe.
C’: Joe is not an elephant.

g = NOT e, j = g
c = [(NOT e) AND g]
c = g AND g = g = NOT e

 

[steve_bank]

P1: Joe is either a squid or a giraffe.
P1’: A giraffe is not an elephant.
P1’’: An elephant is not a squid.
P2’’: Joe is an elephant.
C’’’: Joe is not an elephant, and Joe is an elephant.

Regardless of preceding logic, a valid argument can not be both true and not true. Can't be ambiguous if rules of logic are properly applied.

Given P2 all of P1 is irrelevant. I'd say it is non sequeter, C does not follow from P1,P2.

 

[DBT]

A4 appears to be another version of the Liars Paradox.....

 

[Speakpigeon]

It is not a paradox that from some premises, a contradiction follows. The reason I'm doing this is to try to help reduce some of the confusion caused by Speakpigeon in the other thread.

Initially, since Aristotle, the Stoics, through the Scholastics and people like Leibnitz, l'école de Port-Royal in France etc. and broadly up until Frege, formal logic was conceived as the formalisation of logical reasoning, where logical reasoning was thought of as a capability of the human mind. All during this very long period, formal logic was understood as a practical discipline, a tool. The Scholastic in particular developed a debating procedure which was essentially a procedure to prove claims during theological debates. The method of proof mathematicians use today is a mathematical formalisation of this debating procedure, though with some major differences. Even Frege developed his method of formal logic to do exactly the same thing as what human logical reasoning does, only formally, which would have allowed mathematicians to formalise the proofs of their theorems, something Frege thought would help improve rigour.

In that context, the notions of argument and validity of argument were at the heart of the discipline. The term "argument" referred to all reasoned arguments, and "validity" referred to all deductive arguments. These were actual argument people had. Not the kind of abstract formulas we find in maths textbooks which are disconnected from human reasoning.

Read Boole and Frege. That's the way they still expressed themselves. Boole titled his own book, published in 1854, "An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities". The idea, clearly was to model what all human beings do when they reason logically, including by using logically valid arguments: Socrates is a man,; All men are mortal; Therefore, Socrates is mortal. That was 2,400 years ago! Contrary to Plato, Aristotle was interested essentially in the idea of developing empirical sciences. His formal logic was meant to be an operational model of human logical reasoning, not some abstraction cut off from the reality of life.

However, once mathematicians, starting with Boole, Frege and Russell, started the formalisation of logic using a mathematical formalisation, they also started to cut corners. Unable to reproduce the logic of human reasoning, they opted for the best approximation they could find, which was essentially the material implication. The irony is, this makes 1st order logic, in effect, a first order approximation of the logic of human reasoning. Since then, no progress has been made in that direction. The whole field of mathematical logic is based on this approximation. Approximation here means that the more complex the formula, the more likely the result will be different from human logical reasoning, and therefore what we would all qualify as just plain wrong.

There is now, and since several generations, everywhere in the world, a class of self-appointed ignoramuses who assert against all empirical evidence that logic, argument and validity are nothing but what mathematicians have decided to formalise as such. Whatever human logical reasoning might be, it's just not logic and not valid if it doesn't comply with this first order approximation that is the mathematical model. What a load of bullshit.
EB

 

[steve_bank]

P1: Joe is either a squid or a giraffe.
P1’: A giraffe is not an elephant.
P1’’: An elephant is not a squid.
P2’’: Joe is an elephant.
C’’’: Joe is not an elephant, and Joe is an elephant.

Regardless of preceding logic, a valid argument can not be both true and not true. Can't be ambiguous if rules of logic are properly applied.

Given P2 all of P1 is irrelevant. I'd say it is non sequeter, C does not follow from P1,P2.

I understand the paradox. In digital logic liars paradox would be called an actable multivibrator. Cross coupled logic. It continually switches between two states.

Joe says all Greeks are liars. Joe says he is a Greek. We go back and forth between two states. A paradox.

The way it is stated I see it as is mutual exclusion that I think precludes reducing to formal logic. If mutual exclusion fits Liar's Paradox then so it is.

 

[Speakpigeon]

P1: Joe is either a squid or a giraffe.
P1’: A giraffe is not an elephant.
P1’’: An elephant is not a squid.
P2’’: Joe is an elephant.
C’’’: Joe is not an elephant, and Joe is an elephant.

Regardless of preceding logic, a valid argument can not be both true and not true. Can't be ambiguous if rules of logic are properly applied.

Given P2 all of P1 is irrelevant. I'd say it is non sequeter, C does not follow from P1,P2.

I understand the paradox. In digital logic liars paradox would be called an actable multivibrator. Cross coupled logic. It continually switches between two states.

Joe says all Greeks are liars. Joe says he is a Greek. We go back and forth between two states. A paradox.

The way it is stated I see it as is mutual exclusion that I think precludes reducing to formal logic. If mutual exclusion fits Liar's Paradox then so it is.

Not quite the same thing. No sentence is a paradox in or by itself. For a sentence to become a paradox, you need a human being to see it as a paradox. The sentence itself doesn't actually "switch" between true and false. The "switching" is all yours. Switching is in the eyes of the beholder. I say stop switching. You don't have to.
EB

 

[Angra Mainyu]

Regardless of preceding logic, a valid argument can not be both true and not true. Can't be ambiguous if rules of logic are properly applied.

A deductive argument is not the sort of thing that can be true or false. I guess you could say it is not true in the sense it is not the case that it is true, but that would be misleading, like saying that, say, a banana is not true.

Given P2 all of P1 is irrelevant. I'd say it is non sequeter, C does not follow from P1,P2.

That is false. C follows from P1 and P2. Why do you think otherwise?
Let us try to formalize it a little bit:

A1:

P1: Joe is either a squid or a giraffe.
P2: Joe is not a squid.
C: Joe is a giraffe.

The argument has the following general form:

1: A or B.
2: ¬A.
3: B.

Now, 3. follows from 1. and 2. regardless of what A or B are. In the particular case of A1, 1 is P1, 2 is P2, A is "Joe is a squid", and B is "Joe is a giraffe".
Why does 3. follow from 1. and 2.?

One way to see this is that whenever 1. and 2. are true, B is true - no matter what you pick as A or B.
Do you object to that inference rule?

 

[Angra Mainyu]

It is not a paradox that from some premises, a contradiction follows. The reason I'm doing this is to try to help reduce some of the confusion caused by Speakpigeon in the other thread.

Initially, since Aristotle, the Stoics, through the Scholastics and people like Leibnitz, l'école de Port-Royal in France etc. and broadly up until Frege, formal logic was conceived as the formalisation of logical reasoning, where logical reasoning was thought of as a capability of the human mind. All during this very long period, formal logic was understood as a practical discipline, a tool. The Scholastic in particular developed a debating procedure which was essentially a procedure to prove claims during theological debates. The method of proof mathematicians use today is a mathematical formalisation of this debating procedure, though with some major differences. Even Frege developed his method of formal logic to do exactly the same thing as what human logical reasoning does, only formally, which would have allowed mathematicians to formalise the proofs of their theorems, something Frege thought would help improve rigour.

In that context, the notions of argument and validity of argument were at the heart of the discipline. The term "argument" referred to all reasoned arguments, and "validity" referred to all deductive arguments. These were actual argument people had. Not the kind of abstract formulas we find in maths textbooks which are disconnected from human reasoning.

Read Boole and Frege. That's the way they still expressed themselves. Boole titled his own book, published in 1854, "An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities". The idea, clearly was to model what all human beings do when they reason logically, including by using logically valid arguments: Socrates is a man,; All men are mortal; Therefore, Socrates is mortal. That was 2,400 years ago! Contrary to Plato, Aristotle was interested essentially in the idea of developing empirical sciences. His formal logic was meant to be an operational model of human logical reasoning, not some abstraction cut off from the reality of life.

However, once mathematicians, starting with Boole, Frege and Russell, started the formalisation of logic using a mathematical formalisation, they also started to cut corners. Unable to reproduce the logic of human reasoning, they opted for the best approximation they could find, which was essentially the material implication. The irony is, this makes 1st order logic, in effect, a first order approximation of the logic of human reasoning. Since then, no progress has been made in that direction. The whole field of mathematical logic is based on this approximation. Approximation here means that the more complex the formula, the more likely the result will be different from human logical reasoning, and therefore what we would all qualify as just plain wrong.

There is now, and since several generations, everywhere in the world, a class of self-appointed ignoramuses who assert against all empirical evidence that logic, argument and validity are nothing but what mathematicians have decided to formalise as such. Whatever human logical reasoning might be, it's just not logic and not valid if it doesn't comply with this first order approximation that is the mathematical model. What a load of bullshit.
EB

That is a mixture of some truth and falsehood, and part of what leads you astray. But in any case, that does not address the matter at hand: Are the arguments valid?

 

[steve_bank]

There is no special inductive or deductive logic.

The difference is the starting point. A car crash occurs and you are a forensic investigator. You work your way back from the observed event to the causes. Is that deductive or inductive?

You arrive on the scene, like the CSI TV show, and see blood, a knife on the ground and other evidence. From the observed evidence you conclude an event occurred. Is that deductive or inductive? Specific to the general and the general to the specific, some call it top down and bottom up reasoning.

Real world reasoning is never perfectly deductive or inductive, it is a mix of the two. You can not have one without the other. Logic is never proven true in a general sense. An argument can be tested for validity, meaning the conclusion proceeds from the hypothesis and premise. A logically valid argument does not have to be tied to reality.

 

[Speakpigeon]

There is no special inductive or deductive logic.

???

Are you saying all those people who have thought about and contributed to formal logic from Aristotle to today were not terribly smart after all?! Or that you have all understood what they haven't been able to, in 2,400 years of history?!

Whoa. Ego or sheer stupidity?
EB

 

[Speakpigeon]

Regardless of preceding logic, a valid argument can not be both true and not true. Can't be ambiguous if rules of logic are properly applied.

A deductive argument is not the sort of thing that can be true or false. I guess you could say it is not true in the sense it is not the case that it is true, but that would be misleading, like saying that, say, a banana is not true.

???

And argument is valid or invalid. But an argument is a particular kind of implication and as such it may be true and false, more precisely it will be either true or false in each specific logical case.

Given P2 all of P1 is irrelevant. I'd say it is non sequeter, C does not follow from P1,P2.

That is false. C follows from P1 and P2. Why do you think otherwise?

Whoa! Go easy here! Who are you to tell people they are wrong?! On what basis?! You're just a dogmatic fool!

And I'm still waiting for you to provide the justification by professional specialists for the definition of validity you use.
EB

 

[fast]

P1: Joe is either a squid or a giraffe.
P1’: A giraffe is not an elephant.
P1’’: An elephant is not a squid.
P2’’: Joe is an elephant.
C’’’: Joe is not an elephant, and Joe is an elephant.

Regardless of preceding logic, a valid argument can not be both true and not true. Can't be ambiguous if rules of logic are properly applied.

Given P2 all of P1 is irrelevant. I'd say it is non sequeter, C does not follow from P1,P2.

I understand the paradox. In digital logic liars paradox would be called an actable multivibrator. Cross coupled logic. It continually switches between two states.

Joe says all Greeks are liars. Joe says he is a Greek. We go back and forth between two states. A paradox.

The way it is stated I see it as is mutual exclusion that I think precludes reducing to formal logic. If mutual exclusion fits Liar's Paradox then so it is.

Not quite the same thing. No sentence is a paradox in or by itself. For a sentence to become a paradox, you need a human being to see it as a paradox. The sentence itself doesn't actually "switch" between true and false. The "switching" is all yours. Switching is in the eyes of the beholder. I say stop switching. You don't have to.
EB

Say what?

Sentences, statements, and propositions.

Sentences that fail to express a proposition are not statements, so any sentence used that does not express a proposition is neither true nor false.

Otherwise, sentences are true or false, statements are true or false, and propositions are true or false.

Case 1: the cat is on the mat.

The physical state of affairs is such that the cat is on the mat.

The sentence, “the cat is on the mat,” the statement, “the cat is on the mat,” and the proposition, “the cat is on the mat” are all true.

Case 2: on the is can

Some might say that’s incoherent, but I can read the words just fine. Some might say it’s meaningless, but regardless, whether meaningless, incoherent, or sensibly short for an unbeknownst reason, it is a sentence, and it’s comprised of words where each individual word has a meaning, but taken together, it doesn’t make much sense, so some might say it’s not sensible. Oh, the things we can say!

It’s a sentence that fails to express a proposition and is not a statement. So, although its a sentence that’s not true, it’s not a sentence that is false.

Case 3: Joe is not an elephant, and Joe is an elephant.

That’s two sentences, and each is an independent statement with its own proposition.

 

[Speakpigeon]

Not quite the same thing. No sentence is a paradox in or by itself. For a sentence to become a paradox, you need a human being to see it as a paradox. The sentence itself doesn't actually "switch" between true and false. The "switching" is all yours. Switching is in the eyes of the beholder. I say stop switching. You don't have to.
EB

Say what?

Sentences, statements, and propositions.

Sentences that fail to express a proposition are not statements, so any sentence used that does not express a proposition is neither true nor false.

Otherwise, sentences are true or false, statements are true or false, and propositions are true or false.

Case 1: the cat is on the mat.

The physical state of affairs is such that the cat is on the mat.

The sentence, “the cat is on the mat,” the statement, “the cat is on the mat,” and the proposition, “the cat is on the mat” are all true.

Case 2: on the is can

Some might say that’s incoherent, but I can read the words just fine. Some might say it’s meaningless, but regardless, whether meaningless, incoherent, or sensibly short for an unbeknownst reason, it is a sentence, and it’s comprised of words where each individual word has a meaning, but taken together, it doesn’t make much sense, so some might say it’s not sensible. Oh, the things we can say!

It’s a sentence that fails to express a proposition and is not a statement. So, although its a sentence that’s not true, it’s not a sentence that is false.

You think sentences are true or false even when nobody thinks what they mean?!

Well, sorry , but you'll to prove that.

Case 3: Joe is not an elephant, and Joe is an elephant.

That’s two sentences, and each is an independent statement with its own proposition.

Actually, there are three truth values here. That of each of the conjuncts and that of the conjunction.

So, we think we know the truth value of the conjunction, i.e. false.

Now, your turn, you tell me the truth value of each of the conjunct.

Hint

Show

It's either true or false.

EB

 

[Angra Mainyu]

There is no special inductive or deductive logic.

https://plato.stanford.edu/entries/logic-inductive/
https://www.iep.utm.edu/ded-ind/

There are deductive arguments and arguments that are not deductive. Inductive arguments are not deductive arguments.

The difference is the starting point. A car crash occurs and you are a forensic investigator. You work your way back from the observed event to the causes. Is that deductive or inductive?

Inductive. You are making probabilistic assessments about the causes (not necessarily with numbers) on the basis of observations.

You arrive on the scene, like the CSI TV show, and see blood, a knife on the ground and other evidence. From the observed evidence you conclude an event occurred. Is that deductive or inductive?

Inductive.

But if you prove that on the basis of ZF, the Axiom of Choice is equivalent to Zorn's Lemma and to the Well Ordering Principle, that is deductive. And the arguments A1, A2, A3 and A4 are all deductive ones too.

Real world reasoning is never perfectly deductive or inductive, it is a mix of the two.

That is false. It might be that inductive arguments involve deductive implicity supporting arguments in many cases, though I doubt this is universal. But deductive arguments certainly do not always involve inductive ones (even if one might intuitively make probabilistic assessments about your chances of having erred in the deduction).

An argument can be tested for validity, meaning the conclusion proceeds from the hypothesis and premise.

By that definition, A1, A2, A3 and A4 are all valid, because the conclusion is entailed by the premises.

A logically valid argument does not have to be tied to reality.

True.

 

[Angra Mainyu]

???

And argument is valid or invalid. But an argument is a particular kind of implication and as such it may be true and false, more precisely it will be either true or false in each specific logical case.

Again, a deductive argument is not the kind of thing that can be true or false, though the proposition, statement, assertion, etc., that the premises imply the conclusion is the sort of thing that can be true or false.

Given P2 all of P1 is irrelevant. I'd say it is non sequeter, C does not follow from P1,P2.

That is false. C follows from P1 and P2. Why do you think otherwise?

Whoa! Go easy here! Who are you to tell people they are wrong?! On what basis?! You're just a dogmatic fool!

First, I am a person, and as such, I have the right to free speech, which includes the right to tell people that they are wrong.
Second, I am of course correct. It is true that C follows from P1 and P2.
Third, it is epistemically irrational on your part to believe I am either dogmatic or a fool (well, in a sense I am a fool; it is not good for me to spend time in an interaction as hostile as this one, and I shouldn't be coming back to demolish your argumentation, so I guess it's bad for both of us. You are still colossally wrong, though).
Fourth, you ask "on what basis". Well, I already explained that in my reply to steve bank. So, here goes again:
Let us try to formalize it a little bit:

A1:

P1: Joe is either a squid or a giraffe.
P2: Joe is not a squid.
C: Joe is a giraffe.

The argument has the following general form:

1: A or B.
2: ¬A.
3: B.

Now, 3. follows from 1. and 2. regardless of what A or B are. In the particular case of A1, 1 is P1, 2 is P2, A is "Joe is a squid", and B is "Joe is a giraffe".
Why does 3. follow from 1. and 2.?

One way to see this is that whenever 1. and 2. are true, B is true - no matter what you pick as A or B.
Do you object to that inference rule?

And I'm still waiting for you to provide the justification by professional specialists for the definition of validity you use.

You are still being irrational, as I have told you many times I will not do that, and it is not proper of you to make such demand, for the reasons repeatedly stated.