Validity vs. soundness

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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.


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: ( ) The form is not valid. All Arguments must pass a validity test. The Ts stand for True and Fs stand for False. An argument with 2 premises and a conclusion have 8 possible worlds in which none may make an invalid inference or the entire argument is invalid. The number of worlds in an argument is calculated by 2 to the power of N= variables. Every letter is a variable. So in this case 2 to the power of 3 is 8.

           G is 1. All G are  S , S is 2. All S are D, and D is 3.  All G are D
                         G              S             D
                          T              T             T
                          F               T            T
                          T               F            T
                          F                F           T
                        (  T               T            F  ) Invalid  All true premises lead to a false conclusion making this an invalid argument. 
                          F               T            F
                          T               F            F
                           F              F            F
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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