Systematic Philosophy

to know what can be known

What Is Entailment?

In order to understand what an argument is, you need to understand entailment. What does it mean to say that one or more propositions entails some proposition?

What entailment is

The concept of entailment depends on a more fundamental concept, the concept of immediate entailment. Once you grasp the concept <immediate entailment>, the concept <entailment> is easy to understand.

In particular, to say that one or more propositions “entail” some proposition Q is to say that those propositions are related to proposition Q by a chain of immediate entailments. This means that like immediate entailment, entailment is a relation between propositions and relates one or more propositions to a given proposition.

Some examples

Consider the following list of propositions:

A.  <Socrates is a person>
B.  <all people are mortal>
C.  <Socrates is mortal>
D.  <all mortal things have parts>
E.  <Socrates has parts>
F.  <all things that have parts are made of particles>
G.  <Socrates is made of particles>

Here, propositions A and B immediately entail proposition C. This is a chain of immediate entailment one link long, so propositions A and B entail proposition C. Similarly, propositions C and D immediately entail proposition E and propositions E and F immediately entail proposition G. It follows that propositions C and D entail proposition E and that propositions E and F entail proposition G. It also follows that propositions A, B, D and F are linked to proposition G by a chain of immediate entailments. So it follows that propositions A, B, D and F together entail proposition G.

Next, consider these propositions:

H.  <all cats are animals>
I.  <all dogs are animals>
J.  <all cats are dogs>

Here, none of the propositions entail any of the others. How do we know? It is not because we cannot see how any of them entail any of the others. The axioms of geometry and arithmetic entail a very large number of propositions, including many propositions we cannot tell that the axioms entail. So our merely not seeing the entailments is not enough to show that the entailments are not there.

Non-obviously entailed by the axioms of geometry.

There are various ways to figure out which propositions entail one another and which do not. For instance, as we will note in a moment, entailment is truth-preserving. This means that true propositions never entail false propositions. From this it follows that propositions H and I do not entail proposition J, as <all cats are animals> and <all dogs are animals> are true while <all cats are dogs> is false.

Properties of entailment

The relation of entailment has a number of important properties. As we just said, entailment is truth-preserving. False propositions can entail true propositions or false propositions, but true propositions can only ever entail other true propositions. Entailment has other important properties as well.

Next up: […]

Written by Geoff Anders

May 27, 2011 at 2:25 am

Posted in Uncategorized

%d bloggers like this: