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Is this Argument Tautologically Valid?

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I have to identify the sentence that each capital letter will abbreviate. Then I need to construct a truth-table for the conditional corresponding to the argument before determining whether this argument is tautologically valid?

"In an hour, a man's heart throws out more blood than his own weight. If this is so and if the blood flows only outward from the heart, then the heart creates more blood in an hour than the weight of a man. But this is impossible. If a man's heart throws out more blood than his weight in an hour's time, the blood must circulate through the body and reenter the heart. It follows that the received doctrine (blood flowing only out of the heart) is false, and that the new doctrine (circulation and reentry) is true.

So let's say
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.

@Horace phil,

Horace;67240 wrote:

I have to identify the sentence that each capital letter will abbreviate. Then I need to construct a truth-table for the conditional corresponding to the argument before determining whether this argument is tautologically valid?

a man's heart throws out more blood than his own weight(A).
A
If this is so(A) and if the blood flows only outward from the heart (B), then the heart creates more blood in an hour than the weight of a man(C).
(A & B) --> C
But this is impossible.
~ [(A & B) --> C]
If a man's heart throws out more blood than his weight in an hour's time(A), the blood must circulate through the body and reenter the heart(D).
A --> D
It follows that the received doctrine (blood flowing only out of the heart(B) is false, and that the new doctrine (circulation and reentry)(D) is true.
|- B & D

This is the truth table;

The green box shows all the main connective values and shows that none of the rows contain all true premises and a false conclusion, so the overall truth value of the truth table is valid. In terms of a tautology, logic dictates that a tautological truth value have matching components to be to true. So, T=T and F=F are tautologies whereas T=F and F=T is not valid as a tautology. In this sense, the whole paragraph is too diverse to be considered a tautology unless it were constructed in a specific way, which would be difficult to construct in the first place.

@Horace phil,

Horace;67240 wrote:

... Then I need to construct a truth-table for the conditional corresponding to the argument before determining whether this argument is tautologically valid?

I'm not sure exactly what you mean when you say 'tautologically valid'. All valid arguments can be restated as tautologies - that is, necessarily true conditional statements in which the antecedent is the conjunction of the premises and the consequent is the conclusion. Such conditional is the corresponding material conditional to which you refer. But if there is any distinction between 'tautologically valid' and just plain 'valid', I have no idea what it is. It seems to me that they are the same.

Horace;67240 wrote:

So let's say
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

I agree with your interpretation that the semantics of the sentence 'But this is impossible' is such that 'this' refers only to the consequent of the proceeding conditional. Namely, the intended meaning is to convey a claim that 'the heart creates more blood in an hour than the weight of a man' is what is impossible, rather than the proceeding conditional in its entirety. It is a bit of a sticky wicket, and could be interpreted to mean the negation of the proceeding conditional in its entirety. Note that the choice of interpretation changes the truth values.

Given the above, I interpret the argument as follows:

P1: A
P2: If (A & B) then C
P3: ~C
P4: If A then D
C: ~B & D

Fully symbolized:

P1: A
P2: (A & B) -> C
P3: ~C
P4 A -> D
C: ~B & D

The argument is valid. The corresponding conditional is:

[(A & ((A & B) -> C)) & (~C & (A -> D))] -> (~B & D)

So, that is the statement to use in order to create a truth table for the corresponding conditional.

@goapy,

so you're gonna have a different truth table than the previous post because you have interrpretted things differently. i think your interpretation corresponds to mine and so im working on a table

@Horace phil,

Horace;67240 wrote:

Is this Argument Tautologically Valid?

A 'tautology' is a logical/cognitive fallacy.

@nameless,

Goapy I did a truth-table based on your interpretation and in two places i have premises which are true, and a false conclusion. As I do not have a straight line of "t", and assuming I did the table correctly, that would not make this valid, right?

@Horace phil,

Horace;69573 wrote:

Goapy I did a truth-table based on your interpretation and in two places i have premises which are true, and a false conclusion. As I do not have a straight line of "t", and assuming I did the table correctly, that would not make this valid, right?

There are a couple of ways to use a truth table to test an argument for validity. If you keep the statements in argument form, with the premises and conclusion as separate statements, then the argument is valid if no line has all true premises and a false conclusion.

But if the argument is restated as the corresponding material conditional, then the argument is now just one statement. It is valid (or, more precisely, the statement is a tautology) if the truth values for the main connective are all true.

It was my understanding that you were to test the corresponding material conditional.

Code:





[(A & ((A & B) -> C)) & (~C & (A -> D))] -> (~B & D)

  T T   T T T  T  T   F  FT F  T T  T    T   FT F T   

  T T   T T T  T  T   F  FT F  T F  F    T   FT F F

  T F   T T T  F  F   F  TF T  T T  T    T   FT F T

  T F   T T T  F  F   F  TF F  T F  F    T   FT F F

  T T   T F F  T  T   F  FT F  T T  T    T   TF T T

  T T   T F F  T  T   F  FT F  T F  F    T   TF F F

  T T   T F F  T  F   T  TF T  T T  T    T   TF T T

  T T   T F F  T  F   F  TF F  T F  F    T   TF F F

  F F   F F T  T  T   F  FT F  F T  T    T   FT F T

  F F   F F T  T  T   F  FT F  F T  F    T   FT F F

  F F   F F T  T  F   F  TF T  F T  T    T   FT F T

  F F   F F T  T  F   F  TF T  F T  F    T   FT F F

  F F   F F F  T  T   F  FT F  F T  T    T   TF T T

  F F   F F F  T  T   F  FT F  F T  F    T   TF F F

  F F   F F F  T  F   F  TF T  F T  T    T   TF T T

  F F   F F F  T  F   F  TF T  F T  F    T   TF F F

                                         ^ main connective values are all true

If you were to test it the other way, then the premises would not be joined and the main conditional connective would not be present.

@nameless,

nameless;69553 wrote:

A 'tautology' is a logical/cognitive fallacy.

Why ever would you say such a thing. A tautology is a statement, and only arguments can be fallacious. So tautologies cannot be fallacious. In any case, tautologies are necessarily true statements like, "All dogs are dogs".

@Horace phil,

Finally figured it out (I think).
I was reading [(A & ((A & B) -> C))... as [((A&(A+B))->C...
Kinda makes a difference...

@Horace phil,

Horace;67240 wrote:

I have to identify the sentence that each capital letter will abbreviate. Then I need to construct a truth-table for the conditional corresponding to the argument before determining whether this argument is tautologically valid?

"In an hour, a man's heart throws out more blood than his own weight. If this is so and if the blood flows only outward from the heart, then the heart creates more blood in an hour than the weight of a man. But this is impossible. If a man's heart throws out more blood than his weight in an hour's time, the blood must circulate through the body and reenter the heart. It follows that the received doctrine (blood flowing only out of the heart) is false, and that the new doctrine (circulation and reentry) is true.

So let's say
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.

I don't know what the truth values are of all the variables. Once you provide those, I think I can help you.

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