Testing Syllogisms for Validity

Using Truth Trees*

 
We have seen that a categorical sentence can be understood to be making a claim about a Venn Diagram.  It's easier to see how this is so if we label the discrete areas of the circles which comprise the Venn Diagram:

 

Consider the categorical sentence form "All S are P."  The sentence says that the area of S outside of P is empty. On our numbered Venn Diagram, the empty areas are 1 and 4.  So "All S are P" says the same thing as "Area 1 is empty and Area 4 is empty."  "No S are P" commits us to areas 2 and 5 being empty.  What about particular categorical sentences? Consider "Some S are P."  When "Some S are P" is true, there is something in the overlapping area of S and P, namely areas 2 and 5. But we don't know whether something is in area 2 or area 5, so we express that by saying "Area 2 is non-empty or Area 5 is non-empty." To sum up, we can notice the following equivalences:

All S are P Area 1 is empty and Area 4 is empty.
No S are P Area 2 is empty and Area 5 is empty.
Some S are P Area 2 is non-empty or Area 5 is non-empty.
Some S are not P Area 1 is non-empty or Area 4 is non-empty.

Now, with the above point in mind we can turn syllogisms into arguments in propositional logic, and then test them for validity using the truth tree method (or any other semantic method).  To do that we introduce the following convention. We will use numerals to abbreviate sentences about Venn Diagram areas. For example,  we'll let "1" abbreviate "Area 1 is empty." Then to say that area 1 is non-empty will just require that we negate it. Here are the abbreviations:

1 Area 1 is empty.
2 Area 2 is empty.
3 Area 3 is empty.
4 Area 4 is empty.

Now we can give the full translations of the four categorical sentence forms:

All S are P Area 1 is empty and Area 4 is empty. (1 & 4)
No S are P Area 2 is empty and Area 5 is empty. (2 & 5)
Some S are P Area 2 is non-empty or Area 5 is non-empty. (~2 v ~5)
Some S are not P Area 1 is non-empty or Area 4 is non-empty. (~1 v ~4)

 

Let's translate the following syllogism into our modified propositional logic:

All fish are swimmers.
All bass are fish.
All bass are swimmers.

We use "S" for "bass", "P" for "swimmers", and "M" for fish, and the numbered Venn Diagram above, we have:

All M are P
All S are M
All S are P.

which is tranlated as:

(4 & 7)
(1 & 2)
(1 & 4)

 

Now we treat this as a normal argument in propositional logic, and evaluate it by the truth-tree method.  We do a truth tree for the premises and the negation of the conclusion. All the paths close on the tree, and so the argument is valid.

Consider our dog syllogism:

 

Some dogs are happy
Some dogs are not brown.

No brown things are happy.

 

We translate this syllogism as follows:

Some M are P
Some M are not S

No S are P

and finally:

(~5 v ~6)
(~6 v ~7)

(2 & 5)

When we do our truth tree, we discover that the syllogism is invalid.

Exercise: Test Syllogisms for validity using the Truth-Tree Method.

*This method is due to Nuel Belnap.