Systematic Philosophy

to know what can be known

What Is Immediate Entailment?

One of the most important and most fundamental concepts in philosophy is the concept of immediate entailment. Understanding immediate entailment is essential for understanding what an argument is, validity, transparent validity and how to assess arguments. What is immediate entailment?

What immediate entailment is

It is very difficult to define “immediate entailment”. It may be impossible. So we will not try to define it here. Instead, we will attempt to convey it by giving a bunch of examples and by stating some general truths about it.

Several examples

Here are some examples of immediate entailment. They are organized into various categories. Keep in mind that these are just some examples. It is possible to generate many, many more.

  • Logical Connectives: And, Or, Not, If/Then
    • <Steve is happy and Steve is friendly> immediately entails <Steve is happy>.
    • <Julia is clever> immediately entails <Julia is a cube or Julia is clever>.
    • <Xenophanes is at home or Xenophanes is abroad> and <it is not the case that Xenophanes is abroad> together immediately entail <Xenophanes is at home>.
    • <All cubes are red or green> and <no cubes are red> together immediately entail <all cubes are green>.
    • <It is not the case that it is not the case that everyone is benevolent> immediately entails <everyone is benevolent>.
    • <Some people are cruel> immediately entails <it is not the case that it is not the case that some people are cruel>.
    • <If all cats are dogs, then all dogs are rabbits> and <if all dogs are rabbits, then all rabbits are elephants> together immediately entail <if all cats are dogs, then all rabbits are elephants>.
  • Modus Ponens
    • <Plato is cryptic> and <if Plato is cryptic, then Plato does not tell us everything he knows> together immediately entail <Plato does not tell us everything he knows>.
    • <Some triangles are green> and <if some triangles are green, then some triangles are visible> together immediately entail <some triangles are visible>.
  • Modus Tollens
    • <if Aristotle wrote The Republic, then Aristotle is Plato> and <it is not the case that Aristotle is Plato> together immediately entail <it is not the case that Aristotle wrote the The Republic>.
    • <if Aquinas was an atheist, then he did not argue for the existence of God> and <it is not the case that Aquinas did not argue for the existence of God> together immediately entail <it is not the case that Aquinas was an atheist>.
  • Various
    • <the box is red> immediately entails <the box is some color>.
    • <A is identical to B> and <B is identical to C> together immediately entail <A is identical to C>.
    • <<Socrates is wise> is true> immediately entails <Socrates is wise>.
    • <Kant is a philosopher> immediately entails <Kant is a philosopher>.

Some general truths

Here are some general truths about immediate entailment. By themselves, these truths are not enough to convey the concept fully. But they can help you check to make sure you have picked out the right idea.

  • Immediate entailment is a relation between propositions. It relates one more more propositions to a given proposition.
  • While false propositions can immediately entail true propositions or false propositions, true propositions can only immediately entail true propositions. In other words, immediate entailment is truth-preserving.
  • Whether one or more propositions entail some propositions depends only on the intrinsic properties of those propositions.
  • If you possess the concept of immediate entailment, then if one or more propositions immediately entail some proposition, it is possible for you to recognize this merely by examining the propositions in question.
  • If you know that some proposition is true and you know that that proposition immediately entails some proposition, then you can know that the latter proposition is true as well.
  • The fact that proposition A immediately entails proposition B and proposition B immediately entails proposition C is not enough by itself to ensure that proposition A immediately entails proposition C.
  • Every proposition immediately entails itself.

How to test for immediate entailment

As stated above, if you possess the concept <immediate entailment>, then it is possible to recognize whether some propositions immediate entail a proposition merely by considering the propositions in question. This means that the best way to test for immediate entailment is simply to examine the propositions in question and see whether they fall under the concept <immediate entailment>.

Apart from this, there is another test you can use. This test will not show that any purported immediate entailments do hold, but it will show in some cases that the purported immediate entailments do not hold. The test is as follows. Consider the propositions that are supposed to immediately entail the proposition in question. Pretend that those propositions are true. Now try to conceive of those propositions as being true and the proposition supposedly entailed as being false. Can you do this without overtly contradicting yourself? If you succeed, then the supposed immediate entailment does not hold. If you do not, there may be various reasons for this. As a result, no definitive conclusion can be drawn.

For instance, consider the proposition <every spatial object has a shape>. Does this proposition immediately entail the proposition <every spatial object has a color>? Clearly not. We can conceive of objects existing in space while being perfectly colorless and hence invisible. This means we can conceive the supposedly immediately entailing proposition being true while the supposedly immediately entailed proposition is false. It follows that the immediate entailment in question does not hold.

Invisible pile of gold. Counterexample conceived.

Immediate entailment and logical form

We gave several cases of immediate entailment above. Now some people will only want to refer to some of those cases as being cases of “immediate entailment”. This is because some people want to use the term “immediate entailment” to refer only to cases of immediate entailment that fit a particular logical form. For instance, some will want to only use the term “immediate entailment” to refer to cases of modus ponens. Others will want to use the term “immediate entailment” to refer to all of the cases above except for the first one in the “Various” category.

Who is right? People are free to use terms however they would like. Thus people are free to use the term “immediate entailment” however they would like. There are various reasons to select one terminology over another. In this case, though, we believe that all of the options mentioned are perfectly fine. One can use the term “immediate entailment” as we initially presented it, or one can restrict it so that it only refers to cases of immediate entailment that fit a particular logical form. As for ourselves, we will use the term as we initially presented it, and thus will consider all of the above examples to be cases of immediate entailment.

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Written by Geoff Anders

May 27, 2011 at 2:16 pm

Posted in Uncategorized

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