Validity vs. soundness

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In [[logic]] there's an important distinction between validity and soundness. A logical [[argument]] or [[syllogism]] is valid if true premises always lead to a true conclusion. Validity refers to the structure or form of the argument and not to its contents, while soundness considers the structure and content. An argument is sound if, '''and only if''', the argument is valid '''and''' all of the premises are true.
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In [[logic]] there is an important distinction between '''validity''' and '''soundness'''. A logical [[argument]] or [[syllogism]] is '''''valid''''' if true [[premise]]s always lead to a true conclusion. An argument is '''''sound''''' [[if and only if]] the argument is valid '''and''' all of the premises are true. Thus validity refers to the structure or form of the argument and not to its contents, while soundness considers the structure and content.
  
 
==Examples==
 
==Examples==
 
Consider this logical syllogism:
 
Consider this logical syllogism:
  
:P1: All '''G''' are '''S'''
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: P1: All '''G''' are '''S'''
:P2: All '''S''' are '''D'''
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: P2: All '''S''' are '''D'''
:C1: Therefore, all '''G''' are '''D'''
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: C1: Therefore, all '''G''' are '''D'''
  
 
The form is valid and this particular syllogistic form is named "[[syllogism|Barbara]]". If P1 and P2 are both true, C1 must be true. If we insert some "common knowledge" content into the argument, we can demonstrate an argument which is both valid and sound:
 
The form is valid and this particular syllogistic form is named "[[syllogism|Barbara]]". If P1 and P2 are both true, C1 must be true. If we insert some "common knowledge" content into the argument, we can demonstrate an argument which is both valid and sound:
  
:P1: '''''I''''' (G) am a '''''man''''' (S)
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: P1: '''''I''''' (G) am a '''''man''''' (S)
:P2: All '''''men''''' (S) are '''''mortal''''' (D)
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: P2: All '''''men''''' (S) are '''''mortal''''' (D)
:C1: Therefore, '''''I''''' (G) am '''''mortal''''' (D)
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: C1: Therefore, '''''I''''' (G) am '''''mortal''''' (D)
  
 
What happens when the premises are untrue? Consider the following example:
 
What happens when the premises are untrue? Consider the following example:
  
:P1: All '''''toothpicks''''' (G) are '''''made of metal''''' (S)
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: P1: All '''''toothpicks''''' (G) are '''''made of metal''''' (S)
:P2: All '''''metal objects''''' (S) are '''''toasters''''' (D)
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: P2: All '''''metal objects''''' (S) are '''''toasters''''' (D)
:C1: Therefore, all '''''toothpicks''''' (G) are '''''toasters''''' (D)
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: C1: Therefore, all '''''toothpicks''''' (G) are '''''toasters''''' (D)
  
 
We can prove that P1 and P2 are false by finding either a toothpick which isn't made of metal, or a metal object that isn't a toaster. In this particular case, P1 and P2 are not only false, they directly contradict each other (if all metal objects are toasters, clearly toothpicks can't be made of metal) and no external verification is required - the argument is valid, but the conclusion is unsound.  
 
We can prove that P1 and P2 are false by finding either a toothpick which isn't made of metal, or a metal object that isn't a toaster. In this particular case, P1 and P2 are not only false, they directly contradict each other (if all metal objects are toasters, clearly toothpicks can't be made of metal) and no external verification is required - the argument is valid, but the conclusion is unsound.  
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Let's look at an example where only one of the premises is untrue:
 
Let's look at an example where only one of the premises is untrue:
  
:P1: All '''''mammals''''' (G) have '''''backbones''''' (S)
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: P1: All '''''mammals''''' (G) have '''''backbones''''' (S)
:P2: All '''''creatures with backbones'''''(S) have '''''scales''''' (D)
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: P2: All '''''creatures with backbones'''''(S) have '''''scales''''' (D)
:C1: Therefore, all '''''mammals'''''(G) have '''''scales''''' (D)
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: C1: Therefore, all '''''mammals'''''(G) have '''''scales''''' (D)
  
 
In this example, P1 is true, but P2 is not. This one false premise renders the argument unsound. Let's modify this latest argument just a bit to demonstrate an important point:
 
In this example, P1 is true, but P2 is not. This one false premise renders the argument unsound. Let's modify this latest argument just a bit to demonstrate an important point:
  
:P1: All '''''mammals''''' (G) have '''''backbones''''' (S)
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: P1: All '''''mammals''''' (G) have '''''backbones''''' (S)
:P2: All '''''creatures with backbones'''''(S) have '''''three bones in each ear''''' (D)
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: P2: All '''''creatures with backbones'''''(S) have '''''three bones in each ear''''' (D)
:C1: Therefore, all '''''mammals'''''(G) have '''''three bones in each ear''''' (D)
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: C1: Therefore, all '''''mammals'''''(G) have '''''three bones in each ear''''' (D)
  
 
P1 is still true and P2 is still false (there are vertebrates with only one bone, the stapes, in each ear) however, the conclusion (C1) in this example happens to be true. If an argument is unsound, the conclusion may be either true or false - there's simply no way to tell from the argument alone. This issue is seen in many common [[logical fallacies]] and can be confusing to those who aren't skilled in assessing logical arguments.
 
P1 is still true and P2 is still false (there are vertebrates with only one bone, the stapes, in each ear) however, the conclusion (C1) in this example happens to be true. If an argument is unsound, the conclusion may be either true or false - there's simply no way to tell from the argument alone. This issue is seen in many common [[logical fallacies]] and can be confusing to those who aren't skilled in assessing logical arguments.
  
 
It's possible to reach the correct conclusion by accident, but in order to actually demonstrate that the conclusion is true, the argument '''must''' be both valid and sound.
 
It's possible to reach the correct conclusion by accident, but in order to actually demonstrate that the conclusion is true, the argument '''must''' be both valid and sound.
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[[Category:Logic]]

Revision as of 10:22, 1 July 2010

In logic there is an important distinction between validity and soundness. A logical argument or syllogism is valid if true premises always lead to a true conclusion. An argument is sound if and only if the argument is valid and all of the premises are true. Thus validity refers to the structure or form of the argument and not to its contents, while soundness considers the structure and content.

Examples

Consider this logical syllogism:

P1: All G are S
P2: All S are D
C1: Therefore, all G are D

The form is valid and this particular syllogistic form is named "Barbara". If P1 and P2 are both true, C1 must be true. If we insert some "common knowledge" content into the argument, we can demonstrate an argument which is both valid and sound:

P1: I (G) am a man (S)
P2: All men (S) are mortal (D)
C1: Therefore, I (G) am mortal (D)

What happens when the premises are untrue? Consider the following example:

P1: All toothpicks (G) are made of metal (S)
P2: All metal objects (S) are toasters (D)
C1: Therefore, all toothpicks (G) are toasters (D)

We can prove that P1 and P2 are false by finding either a toothpick which isn't made of metal, or a metal object that isn't a toaster. In this particular case, P1 and P2 are not only false, they directly contradict each other (if all metal objects are toasters, clearly toothpicks can't be made of metal) and no external verification is required - the argument is valid, but the conclusion is unsound.

Let's look at an example where only one of the premises is untrue:

P1: All mammals (G) have backbones (S)
P2: All creatures with backbones(S) have scales (D)
C1: Therefore, all mammals(G) have scales (D)

In this example, P1 is true, but P2 is not. This one false premise renders the argument unsound. Let's modify this latest argument just a bit to demonstrate an important point:

P1: All mammals (G) have backbones (S)
P2: All creatures with backbones(S) have three bones in each ear (D)
C1: Therefore, all mammals(G) have three bones in each ear (D)

P1 is still true and P2 is still false (there are vertebrates with only one bone, the stapes, in each ear) however, the conclusion (C1) in this example happens to be true. If an argument is unsound, the conclusion may be either true or false - there's simply no way to tell from the argument alone. This issue is seen in many common logical fallacies and can be confusing to those who aren't skilled in assessing logical arguments.

It's possible to reach the correct conclusion by accident, but in order to actually demonstrate that the conclusion is true, the argument must be both valid and sound.

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