Re: Vacuously TRUE vs Vacuously FALSE? [ properties of elements of the empty set ]

[olcott]

On 10/24/2021 9:55 AM, André G. Isaak wrote:
> On 2021-10-23 23:09, olcott wrote:
>> On 10/23/2021 10:14 PM, André G. Isaak wrote:
>>> On 2021-10-23 21:01, olcott wrote:
>>>> On 10/23/2021 9:32 PM, André G. Isaak wrote:
>>>>> On 2021-10-23 19:03, olcott wrote:
>>>>>> On 10/23/2021 6:10 PM, Dan Christensen wrote:
>>>>>>> On Saturday, October 23, 2021 at 6:15:15 PM UTC-4, olcott wrote:
>>>>>>>> On 10/23/2021 4:58 PM, Dan Christensen wrote:
>>>>>>>>> On Saturday, October 23, 2021 at 3:31:36 PM UTC-4, olcott wrote:
>>>>>>>>>> On 10/23/2021 1:44 PM, Dan Christensen wrote:
>>>>>>>>>>> On Saturday, October 23, 2021 at 12:42:41 PM UTC-4, olcott
>>>>>>>>>>> wrote:
>>>>>>>>>>>
>>>>>>>>>>>> If we want to have actual correct reasoning then we get rid
>>>>>>>>>>>> of the
>>>>>>>>>>>>
>>>>>>>>>>>> Material conditional
>>>>>>>>>>>> p q p → q
>>>>>>>>>>>> T T T
>>>>>>>>>>>> T F F
>>>>>>>>>>>> F T T
>>>>>>>>>>>> F F T
>>>>>>>>>>>>
>>>>>>>>>>>> and replace it with if-then
>>>>>>>>>>>> if P then q
>>>>>>>>>>>> p q if p then q
>>>>>>>>>>>> T T T
>>>>>>>>>>>> T F F
>>>>>>>>>>>> F T undefined
>>>>>>>>>>>> F F undefined
>>>>>>>>>>>
>>>>>>>>>>> Here is a formal proof of ~A => [A =>B], the basis for the
>>>>>>>>>>> last two lines of the truth table for A => B. To prevent this
>>>>>>>>>>> derivation, somehow you will also have to ban or restrict the
>>>>>>>>>>> application of one more of the rules of inference used here.
>>>>>>>>>>> Which will it be?
>>>>>>>>>
>>>>>>>>>> I am saying for symbolic logic is defined incorrectly when
>>>>>>>>>> symbolic
>>>>>>>>>> logic is required to be the basis for correct reasoning.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> You haven't answered the question. Which line(s) in the above
>>>>>>>>> proof would be invalid in your proposed alternative system of
>>>>>>>>> logic? Somehow, you want to make it impossible to derive
>>>>>>>>> ~A=>[A=>B].
>>>>>>>>>
>>>>>>>> The => implication operator is tossed out on its ass, thus
>>>>>>>> unavailable
>>>>>>>> for any proof.
>>>>>>>
>>>>>>> Let's start with something REAL easy. How would you prove A & B
>>>>>>> => B & A?
>>>>>>
>>>>>> I reject material implication and the principle of explosion.
>>>>>
>>>>> You do realize that even if you "eliminate" material implication
>>>>> and replace it with your version (whatever that might be), you'd
>>>>> still be able to prove anything from (A & ¬A). The principle of
>>>>> explosion is usually illustrated using implication but it isn't
>>>>> actually tied to implication.
>>>>>
>>>>> André
>>>>>
>>>>>
>>>>
>>>> I reject material implication and the principle of explosion
>>>> separately.
>>>
>>> Unless you plan on rejecting ∧, ∨ and ¬, you're not going to be able
>>> to get rid of the principle of explosion since it is a direct
>>> consequence of the logical definitions of these operators.
>>>
>>
>> We simply forbid any syntactic entailment that is contradicted by
>> semantic entailment. We put the semantic relevance back into logic
>> that was removed from Aristotle's syllogism.
>
> How exactly do you 'forbid' something which follows directly from the
> rules of the system without ending up with an inconsistent system?
>

We adapt symbolic logic so that the semantic meaning of propositional
variables is specified. Aristotle's syllogism does this with Categorical
propositions:

In logic, a categorical proposition, or categorical statement, is a
proposition that asserts or denies that all or some of the members of
one category (the subject term) are included in another (the predicate
term). https://en.wikipedia.org/wiki/Categorical_proposition

> And unless you can provide some actual *rules* which allows us to decide
> whether or not something is contradicted by 'semantic entailment', the
> above is worthless. Note that giving examples of things which you think
> do or do not involve 'semantic entailment' is not the same thing as
> providing actual explicit rules. So far, any time I've asked you about
> your notion of 'semantic relatedness' or other things you've responded
> by giving one or two examples of things you consider related or
> unrelated, but no actual rule which would allow us to decide whether two
> arbitary things count as related.
>
>>>> I am not sure how to best express the set of changes that are required.
>>>>
>>>> A good heuristic might be that when semantic values are assigned to
>>>> propositional variables and then when rules of logic are applied to
>>>> these variables derive semantic nonsense then this is a rule that
>>>> must be discarded.
>>>
>>> There are only two semantic values that can be assigned to
>>> propositional variables: true and false. I have no idea what you can
>>> derive from these two values that could possibly objectively count as
>>> 'semantic nonsense'.
>>>
>>> André
>>>
>>
>> That is not exactly true. Truth conditional semantics is anchored in
>> true and false yet has a whole additional supporting infrastructure.
>
> That 'supporting architecture', if I understand what you are claiming is
> *not* part of logic.
>
>> It is true that an X is a Y is the propositional level.
>> When we plug semantics in the we get truth conditional semantics.
>> It is true that a dog is an animal.
>
> What your referring to here isn't 'semantics'. The only semantic values
> available to logic are 'true' and 'false'. What you are referring to is
> 'content'.
>
> The entire point of formal logic is that it looks exclusively at the
> form which an argument takes while ignoring the content altogether.
>
> Formal logic has no knowledge whatsoever about dogs or animals, nor
> should it.
>
> André
>


--
Copyright 2021 Pete Olcott

"Great spirits have always encountered violent opposition from mediocre
minds." Einstein

[olcott]

On 10/24/2021 9:59 AM, Dan Christensen wrote:
> On Sunday, October 24, 2021 at 1:17:12 AM UTC-4, olcott wrote:
>> On 10/24/2021 12:06 AM, Dan Christensen wrote:
>>> On Saturday, October 23, 2021 at 11:21:12 PM UTC-4, olcott wrote:
>>>
>>>>
>>>> If natural language conditionals were understood in the same way, that
>>>> would mean that the sentence "If the Nazis won World War Two, everybody
>>>> would be happy" is true.
>>>> https://en.wikipedia.org/wiki/Paradoxes_of_material_implication
>>>>
>>>
>>> An educated person might point out that such a statement is pure speculation. And that it is an opinion, and not a verifiable fact. As such it is not a logical analysis of the current state of the world.
>>>
>>> Dan
>>>
>>> Download my DC Proof 2.0 freeware at http://www.dcproof.com
>>> Visit my Math Blog at http://www.dcproof.wordpress.com
>>>
>> Logical implication

>> p q p ⇒ q
>> T T T
>> T F F
>> F T T
>> F F T

>> When we apply semantics to the variables material implication asserts
>> that the is a relationship between unrelated things. This is an error.
>> The last two rows of the truth table are mistaken.
>
> You really need to be able to formally prove that A & B => B & A. Your system is DOA otherwise.

A = dogs are animals
B = dogs are not animals
A & B => not one damn thing

A = it is raining outside
B = you go outside unprotected from the rain
C = you get wet
A & B ⊨ C

>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

[Peter]

Not only do you know no formal logic, you know informal logic either.
One cannot prove a generality (here that A & B => B & A) by appealing to
individual cases (also, your use of the individual case above is silly).
If Dan asks you to prove that A & B => B & A, just say that you can't.
(The reference to "Your system" is so much noise. You have no system.)

>
>>
>> Dan
>>
>> Download my DC Proof 2.0 freeware at http://www.dcproof.com
>> Visit my Math Blog at http://www.dcproof.wordpress.com
>>
>
>


--

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