Validity vs. soundness

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In logic there's an important difference between validity and soundness. An argument is valid if it's possible for the premises to be true but the conclusion to be false. An argument is sound if both the premises and the conclusion are true.
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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.
  
 
==Examples==
 
==Examples==
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Consider this logical syllogism:
  
# I either own a bicycle or a car.
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:P1: All '''G''' are '''S'''
# I don't own a bicycle.
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:P2: All '''S''' are '''D'''
# Therefore, I own a car.
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:C1: Therefore, all '''G''' are '''D'''
  
The premises can both be true, that is, it's possible for me to own a car but not a bicycle. However, it's not necessarily true that, just because I don't own a bicycle I must own a car. Thus, this argument is valid but not sound.
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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:
  
It's worth noting that the premises of an argument don't even need to be true in order for an argument to be valid.  For example:
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:P1: '''''I''''' (G) am a '''''man''''' (S)
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:P2: All '''''men''''' (S) are '''''mortal''''' (D)
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:C1: Therefore, '''''I''''' (G) am '''''mortal''''' (D)
  
# All toothpicks are made of metal.
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What happens when the premises are untrue? Consider the following example:
# All metal objects are toasters.
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# Therefore, all toothpicks are toasters.
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While the argument is obviously absurd, the premises to support the conclusion, thus the argument is valid but not sound.
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:P1: All '''''toothpicks''''' (G) are '''''made of metal''''' (S)
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:P2: All '''''metal objects''''' (S) are '''''toasters''''' (D)
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:C1: Therefore, all '''''toothpicks''''' (G) are '''''toasters''''' (D)
  
An example of an argument that is both valid and sound is this one:
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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.
  
# No moth is a spider.
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Let's look at an example where only one of the premises is untrue:
# All spiders are arachnids.
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# Therefore, no moth is an arachnid.
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In this argument the premises logically support the conclusion, plus both the premises and the conclusion are true.  Thus, this argument is both valid and sound.
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:P1: All '''''mammals''''' (G) have '''''backbones''''' (S)
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:P2: All '''''creatures with backbones'''''(S) have '''''scales''''' (D)
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:C1: Therefore, all '''''mammals'''''(G) have '''''scales''''' (D)
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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:
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:P1: All '''''mammals''''' (G) have '''''backbones''''' (S)
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:P2: All '''''creatures with backbones'''''(S) have '''''three bones in each ear''''' (D)
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:C1: Therefore, all '''''mammals'''''(G) have '''''three bones in each ear''''' (D)
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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.
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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.

Revision as of 21:29, 30 August 2006

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.

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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