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If (A+B) then C

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If (A+B) then C
Horace
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Posted 06/16/09 - 09:37 AM:
Subject: If (A+B) then C 		quote post
#1
I am confused a few answers in this truth table:



http://www.philosophyforum.com/for...-tautologically-valid.html



We have the (A+B) then C



A/B



F/T then F

T/F then F

F/F then F



Supposedly the first of these is False, and the next two are true and I can't figure out why. I would assume the person is mistaken.



My reasoning is that the fact that A+B are bracketed would make (A+B) False in each example, which would leave us with



F then F

F then F

F then F



Which in my mind, in the construction of a truth table would make each True, and not simply the latter two. Some help??? I would be grateful.
Cadrache
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Posted 06/16/09 - 09:52 AM:
quote post
#2
For easier reference:


A=In an hour, a man's heart throws out more blood than his own weight
B=Blood flows only outward from the heart
C=The heart creates more blood in an hour than the weight of a man
D=Blood circulates through the body and reenters the heart

So I see
If A and B then C

AND

Since we know that C is false ("But that is impossible)

Then

If A then D

And D is the Opposite of B

So if A is true than D is true...

This does seem tautological, but not valid, even though it's true. How am I doing.
Thanks for any help. I appreciate it.

[/quote]
Words the wiser become the name you do not know. - Some guy with a surname Blanchard.
_____________________________________________

Truth is want. - The internal state of matters.

Truth is Need. - The external state of affairs.
xzJoel
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Posted 06/16/09 - 10:01 AM:
quote post
#3
Can you give more details as to what you mean?

Identify the specific statement that is at issue and the relevant truth values.

Generally speaking, the statement (A + B) reads "A and B". The truth tree is as you suggested.

P | Q | P & Q

T | T | T
T | F | F
F | T | F
F | F | F

If a conjunction is used as the antecedent of a conditional, the only time that the conditional could possibly be false is in a line where the statement (P & Q) is TRUE. Therefore, anytime P is false or Q is false, the conditional is TRUE.


Looking at a proper truth tree

P | Q | R | (P & Q) | (P&Q)>R

T | T | T | T | T 1
T | T | F | T | F 2
T | F | T | F | T 3
T | F | F | F | T 4
F | T | T | F | T 5
F | T | F | F | T 6
F | F | T | F | T 7
F | F | F | F | T 8

You can see that lines 3-8 are true regardless of the value of R.

Line 2 is the only FALSE line because the antecedent (P&Q) is true and the consequent R is false.

Where a 4th variable is added, you would expect to see a doubling of Line 2 - that is you'd expect to see a truth tree with two occasions of the conditional being false.

He has the conditional being false 4 times in 16 lines, so he quite clearly has something wrong. (Lines 5, 6, 13 and 14) He should have the conditional as false in lines 5 and 13. His lines 6 and 14 are wrong. He has a false for the antecedent, so he cannot have a false conditional.


Edited by xzJoel on 06/16/09 - 10:47 AM. Reason: What is a valuable?
"For there is such a distance between how one lives and how one ought to live, that anyone who abandons what is done for what ought to be done achieves his downfall rather than his preservation." Niccolo Machiavelli, "The Prince", Chapter XV.
Cadrache
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Posted 06/16/09 - 10:42 AM:
quote post
#4
In the case from the link: C is a combination value, and therefore does not quite follow the standard system.

Each statement contains multiple variables.

A = a -> b -> c
B = f(b)
C = f(d) -> e
D = d

There is no A ^ B -> C.

The form:

A = a -> x
B = b -> x
C = b -> x

ie.

All dogs bark.
"Cristy" barked.
Therefore Cristy is a dog.

Then you can argue the validity of the statements using the form (A+B) then C.
Words the wiser become the name you do not know. - Some guy with a surname Blanchard.
_____________________________________________

Truth is want. - The internal state of matters.

Truth is Need. - The external state of affairs.
xzJoel
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Posted 06/16/09 - 11:09 AM:
quote post
#5
By the by, the argument is not factually good even if logically adequate.

The heart pumps blood. Where does it get the blood from? a) from within or b) from without.

If the heart pumps volume x of blood, x must have come from a) or b).

Even taking the statement "It is impossible for the heart to create more blood in an hour than the human body weighs", how do you get to circulating blood? All you have done is ruled out creation of blood.

Some alternatives:

a)1) The heart already has a supply of blood in storage and uses that to supplement its production. What blood is created in the heart is supplied by external resources.
b)1) The heart is attached to a source of new blood exterior to the body(such as a direct link to a giant blood tank).
b)2) The heart receives blood created by another internal organ/process, the resources for which come from a source external to the body.

These available alternatives to D indicate that D does not necessarily follow from the premise A. You may wish to state all of your assumptions about the human body upfront and remove them from your proof only after you see that they are never used in the steps of your proof.

For example:

A: The heart pumps in an hour a greater weight in blood than the entire human being weighs.
B: The human being does not consume or otherwise ingest more weight than his weight in any given hour.
C: The blood pumped from the heart remains inside the body.
D: Human beings occasionally have hangnails.
Therefore
E: The blood pumped from the heart must go back into the heart within an hour for it to be pumped out of the heart again.

Notice that this is not a formal proof in the sense that the conclusion necessarily follows from the assumptions, but you come closer to specifying the conditions required for us to intuitively agree that your proof is complete.

Without a closed circulatory system, there is no reason why your proof doesn't lend itself to the imagery of the heart "throwing" blood all over the floor.

Also, D may be true, but it has nothing to do with the proof. On revision, you could remove it.

Edited by xzJoel on 06/16/09 - 12:36 PM
"For there is such a distance between how one lives and how one ought to live, that anyone who abandons what is done for what ought to be done achieves his downfall rather than his preservation." Niccolo Machiavelli, "The Prince", Chapter XV.
Horace
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Posted 06/16/09 - 11:45 AM:
quote post
#6
The specifics come from the truth table found in my opening post. The 6th, 7th, 8th, 14th, 15th, and 16th lines under the first "therefore" sign confused me. To my mind, each of the six should be true...
xzJoel
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Posted 06/16/09 - 12:03 PM:
quote post
#7
I have previously dealt with 6 and 14.


7th line

A | B | C | (A&B) | (A&B)->C
T | F


Knowing nothing else, you know that (A & B) is false (at least one conjunct is false), therefore the conditional is TRUE. He is correct. (Antecedent is false, conditional MUST be true.)


8th line

A | B | C | (A&B) | (A&B)->C
F


Knowing nothing else, you know that (A & B) is false (at least one conjunct is false), therefore the conditional is TRUE. He is correct. (Antecedent is false, conditional MUST be true.)


15th line

A | B | C | (A&B) | (A&B)->C
T | F


Knowing nothing else, you know that (A & B) is false (at least one conjunct is false), therefore the conditional is TRUE. He is correct. (Antecedent is false, conditional MUST be true.)


16th line


A | B | C | (A&B) | (A&B)->C
F


Knowing nothing else, you know that (A & B) is false (at least one conjunct is false), therefore the conditional is TRUE. He is correct. (Antecedent is false, conditional MUST be true.)
"For there is such a distance between how one lives and how one ought to live, that anyone who abandons what is done for what ought to be done achieves his downfall rather than his preservation." Niccolo Machiavelli, "The Prince", Chapter XV.
worldlogicleague
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Posted 06/16/09 - 12:09 PM:
quote post
#8

Cadrache wrote:




All dogs bark.



"Cristy" barked.



Therefore Cristy is a dog.



Then you can argue the validity of the statements using the form (A+B) then C.






In this case this is an invalid syllogism, for nowhere does it say only dogs bark.
Cadrache
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Posted 06/16/09 - 12:20 PM:
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#9
Yup. grin

It's this word I don't particularly like. "creates". To be a correct 3 system function; this word should be "moves".


In other words:

If C is false; the falsity of C does not falsify A and B because C is referencing altogether a different thing.

You could however possibly argue A and B then D; of a sorts.
Words the wiser become the name you do not know. - Some guy with a surname Blanchard.
_____________________________________________

Truth is want. - The internal state of matters.

Truth is Need. - The external state of affairs.
Cadrache
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Posted 06/16/09 - 12:31 PM:
quote post
#10
Ie: There are 3 large errors in this.

1. The wording in C.
2. (A+B) then C is invalid to solve a 4 predicate system. (count them... A,B,C,D...)
3. A form similar to syllogism.
Words the wiser become the name you do not know. - Some guy with a surname Blanchard.
_____________________________________________

Truth is want. - The internal state of matters.

Truth is Need. - The external state of affairs.
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