Implication vs. Entailment

In my recent post “The Perfect Goodness of God – Again” I used conditional derivation to prove a conditional statement, and took that to be sufficient to prove that the antecedent of the conditional statement entailed the consequent. Then I had second thoughts about that approach to proving an entailment.

Penance for my possible sin against logic is to look up the terms ‘implication’ and ‘entailment’ (as well as other related terms: ‘deduction’ and ‘validity’) in various dictionaries of philosophy and philosophy handbooks, and to think about the definitions and make comments on these terms.

I felt a little less stupid upon reading the following caution in The Philosopher’s Toolkit (hereafter TPT), authored by our own Julian Baggini, along with Peter Fosl:

The problem with the distinction as set out [above in the entry on Implication] is that it is all much, much messier than this. So much messier, in fact, that any attempt to tidy it up in a text such as this is bound to result in either an incongruously bloated entry or utter confusion. (TPT, p.148)

One of the most helpful comments on this topic, as well as one of the most wrong-headed comments, came from TPT. Baggini and Fosl give this very useful bit of advice:

The second lesson is that the simplistic distinction set out [here in this entry on Implication] is a decent rule of thumb. If you restrict your use of ‘entailment’ to valid deductions and your use of ‘implication’ to true conditionals, you won’t go far wrong. (TPT, p.149)

Another way to put this point is to say that valid deductions are paradigm cases of ‘entailment’ and ground the meaning of that term, while true conditional statements are paradigm cases of ‘implication’ and ground the meaning of that term. A conditional statement has the form ‘IF P, THEN Q’ where P and Q are themselves statements or propositions.

What struck me as a wrong-headed comment suggests that we avoid use of these terms:

The first [lesson to draw from this entry on Implication] is to avoid using the terms ‘implication’ and ‘entailment’ if an alternative, clearer way of expressing what you want to say is available. Talk about a ‘valid deduction’ or a ‘true conditional’, not about entailment and implication. (TPT, p.149)

In general, I have an immediate negative reaction whenever someone recommends abandoning the use of a word because of alleged vagueness or ambiguity or un-clarity of that word, especially if the word has often been used in discussions of important issues. Some people, for example, want us all to discard the words ‘liberal’ and ‘conservative’ because of alleged unclarity in these words.

Words should not be abandoned quickly and easily. We are married to our words and should not seek a divorce as soon as we experience a first quarrel or even a third or fourth quarrel. When a word has a history of use and the appearance of having contributed to important debates and discussions, then we ought to be very reluctant to throw in the towel on such a word. Often times the problem is that lots of stupid or ignorant people misuse a word (such as Rush Limbaugh misusing and abusing the word ‘liberal’). A word like ‘liberal’ should not be abandoned just because there is a crowd of ignorant yahoos who aren’t able to use the word correctly or intelligently.

Anyway, ‘implication’ and ‘entailment’ are used frequently in philosophical arguments and discussions, and it seems to me, are among the most important terms for use in philosophical arguments and discussions. The last thing we should do is abandon use of these words. There might well be issues of ambiguity or unclarity here. I’m not saying that the meaning of these words is clear and obvious. But if there are problems of unclarity with these words, then those of us who have an interest in philosophical issues ought to learn about these problems and make a sincere effort to disambiguate the ambiguities and sharpen up the vagueness, or at least be cautious and try to be aware of the potential for confusion and unclarity that is associated with these words.

The Oxford Guide to Philosophy (hereafter: OGP) presents the word ‘implication’ as being ambiguous between two main “uses for logic”:

1. Implication understood as a relation between a set of premises and a conclusion deducible from or a logical consequence of those premisses.
2. Implication understood as the relation between antecedent and consequent of a true conditional proposition.
(OGP, p.423)

We see here, as in TPT, the two primary phenomena to which the terms ‘implication’ and ‘entailment’ are related: deductive arguments and conditional statements. But here in the OGP, the term ‘implication’ covers both phenomena. So, we can see how this might be a bit confusing. Sometimes ‘implication’ is used to refer to the relation between antecedent and consequent in a true conditional statement, and other times it is used to refer to the relationship between the premises of a valid deductive argument and the conclusion of that argument.

To avoid confusion, it might be best to try to limit the use of ‘implication’ (at least in philosophical discussions) to the relation between antecedent and consequent in a true conditional statement, since the term ‘entailment’ can be used to cover the relationship between the premises of a valid deductive argument and the conclusion.

The meaning and logic of conditional statements is a matter of controversy in modern philosophy, so the meaning of the word ‘implication’ is ambiguous and problematic. On the other hand, deductive arguments and the methods for determining the validity of deductive arguments are less controversial, and, I believe, more intuitive, so the word ‘entailment’ appears to be less problematic than ‘implication’.

One question, that I have recently stumbled upon, is about how ‘implication’ relates to ‘entailment’. One thing that implication and entailment have in common is the characteristic of transitivity:

1. A implies B.
2. B implies C.
Therefore
3. A implies C.

4. IF A, THEN B.
5. IF B, THEN C.
Therefore
6. IF A, THEN C.

7. A entails B.
8. B entails C.
Therefore
9. A entails C.

Both entailments and implications can “transfer” through a sequence or chain.

The Cambridge Dictionary of Philosophy (2nd edition, hereafter: CDP) also indicates that ‘implication’ can be used to refer to valid deductive inference:

A number of statements together imply Q if their joint truth ensures the truth of Q. An argument is deductively valid when its premises imply its conclusion. Expressions of the following forms are often interchanged one for the other: ‘P implies Q’, ‘Q follows from P’, and ‘P entails Q’… (CDP, p.419)

Later in the same entry (on Implication), it is pointed out that ‘implication’ is also used of conditional statements.

In Talking Philosophy: A Wordbook (hereafter: TPW), A.W. Sparkes defines ‘implication’ in the way that makes it equivalent to entailment or valid deductive inference:

In the technical vocabulary of philosophy, implication is a relation between propositions. A proposition p implies q IFF it would be self-contradictory…to assert p and deny q. (TPW, p.76)

A couple of pages later, we read the entry on ‘entailment’:

Roughly speaking, ‘entail’ is synonymous with ‘imply’. (TPW, p.78)

One of the most interesting discussions can be found in Antony Flew’s A Dictionary of Philosophy (revised 2nd edition, hereafter: FDP). First Flew notes the problematic nature of the terms ‘implication’ and ‘entailment’:

A family of closely related notions, attempts to provide an adequate account of which have occupied many volumes. Problems arise when one seeks to determine relationships within the family, and, as a result, clear and agreed definitions are not available. (FDP, p.164)

Next, Flew lays out the sort of reasoning about these terms that I had been tempted to follow:

To assert the conditional statement ‘If A, then B’ is thought to be equivalent to saying that A implies B, and this in turn is often taken to mean that B is deducible from A. But if B is deducible from A, then to reason A, therefore B is to argue validly, which means that B follows from A or that A entails B. This train of connections might lead one to suppose that all these claims about the relation between A and B are just different ways of saying the same thing. (FDP, p. 164-165)

Flew cautions, however, that “there are other considerations that show that this certainly cannot be said without qualification.” (FDP, p.165)

Since it appears that ‘entailment’ is the clearer of the two terms, we should try to nail down the meaning of that half of the pair of terms first.

Flew distinguishes specifically between a material conditional and entailment:

The material conditional ‘A -> B ’  is true if as a matter of fact it is not the case that A is true and B is false, whereas for ‘A, therefore B’ to be a valid inference it must be impossible for B to be false when A is true…. Since it is generally accepted that ‘A’ entails ‘B’ iff ‘A, therefore B’ is a valid inference, this means that to say that ‘A’ entails ‘B’ is to say that ‘A -> B ’ is not merely true, but is necessarily true. (FDP, p. 165)
So, one could define ‘entailment’ in relation to the sort of implication involved in a material conditional statement.  [Note: Flew uses the horseshoe symbol here, but I don't know how to get that symbol to show up on a Patheos blog post.]

Since the term ‘entailment’ is tied to the notion of a valid deductive argument, and since ‘entailment’ seems to be the clearest term of the pair of related terms ‘entailment’ and ‘implication’, it makes sense to begin by defining or clarifying the concept of a ‘valid deductive argument’. There are a few different ways to do this:

deduction. A valid argument in which it is impossible to assert the premises and to deny the conclusion without thereby contradicting oneself. (FDP, p.85)

Deduction. An argument is deductive if it draws a conclusion from certain premises on the grounds that to deny the conclusion would be to contradict the premises. (A Dictionary of Philosophy, 2nd edition, by A.R. Lacey, p.52-53)

1.2 Deduction … It is the most rigorous form of argumentation there is, since in deduction, the move from premises to conclusions is such that if the premises are true, then the conclusion must also be true. (TPT, p.6)

deduction. A species of argument or inference where from a given set of premisses the conclusion must follow. For example, from the premisses P1, P2 the conclusion P1 and P2 is deducible. The set consisting of the premises and the negation of the conclusion is inconsistent. (OGP, p.194)

validity and truth. … If the argument ‘P1… Pn, therefore C’ is valid, it must be impossible for C to be false when P1… Pn are all true. (FDP, p. 364)

Valid. An inference of an argument is valid if its conclusion follows deductively from its premises. The premises may be false, but if they are true, the conclusions must be true. (A Dictionary of Philosophy, 2nd edition, by A.R. Lacey, p.260)

1.4 Validity and soundness…Validity is a property of well-formed deductive arguments, which, to recap, are defined as arguments where the conclusion is in some sense (actually, hypothetically, etc.) presented as following from the premises necessarily (see 1.2). A valid deductive argument is one for which the conclusion follows from the premises in that way. (TPT, p.12) …If there is any conceivable way possible for the premises of an argument to be true but its conclusion simultaneously to be false, then it is an invalid argument. (TPT, p.13)

valid, …An argument is valid if it is impossible for the premises all to be true and, at the same time, the conclusion false. (CDP, p.948)

validity.  In logic, validity is most commonly attributed to either:
1. Deductive arguments, which are such that if the premises are true the conclusion must be true. … Any argument is valid if and only if the set consisting of its premises and the negation of its conclusion is inconsistent.
2. Propositions which are semantically valid, i.e. are true under any alternative interpretation of the non-logical words. (OGP, p.940)

We see that deductive validity, and thus entailment, is defined in relation to the following concepts: impossibility, contradiction, necessarily, inconsistent, ‘must be true’, ‘must follow’, ‘no conceivable way possible’. The concepts of impossibility and necessarily are ambiguous between logical impossibility and necessity and causal or empirical impossibility and necessity. The expressions ‘must be true’ and ‘must follow’ have a similar ambiguity. The concepts of contradiction and inconsistency, however, are more clearly and directly related to logic. So, my initial preference is to define ‘deductively valid argument’ and ‘entailment’ in terms of contradiction.

  • Keith Augustine

    Hi Bradley,

    I recently settled upon using these terms somewhat differently than you when deciding how I wanted to characterize an observational consequence O of a hypothesis H, as used in such reasoning as hypothetico-deductive method. What I settled upon was that “entailment” should refer to a deductive consequence (e.g., H entails O), whereas implication should refer to an inductive consequence (e.g., H implies O). In the latter case “inference” might be better, but obviously you cannot say H infers O (though you can say O can be inferred from H). To be clear I first used the phrase “deductive entailment” and “inductive implication” before using these terms these way. And when I wanted to signify a consequence of H that could be either deductively or inductively derived, I settled upon simply calling such a consequence a “prediction” (understood in an atemporal way, since logical relations are not temporal).

    • Bradley Bowen

      Your use of ‘implication’ to refer to inductive consequences fits well with the idea that ‘implication’ refers to the logical relationship in true conditional statements, because although true conditional statements include cases in which the antecedent entails the consequent (e.g. “If this is a triangle, then this has three sides.”) we also generally accept as true conditional statements in which the logical relationship between antecedent and consequent is a weaker one, such as a causal relationship (e.g. “If you fiip the switch, then the light will turn on.”).

  • opsarchangel

    great theatre of ruin

    what’s the harm of little idi*ts?

    monstrous.com/forum/index.php?topic=13908.0

    …..

    • Chris

      Looks like Dennis Markuze again….

  • Jason Thibodeau

    Bradley,

    You didn’t say how your research affects your claims about entailment in the “The Perfect Goodness of God–Again” post. In particular, I would be interested to hear whether you have revised your claim that “Zebras have stripes” does not entail “All triangles have three sides.”

    I think that what you have provided above shows that this claim is incorrect. That is, in fact “Zebras have stripes” DOES entail “All triangles have three sides.”

    You quote Flew saying, “it is generally accepted that ‘A’ entails ‘B’ iff ‘A, therefore B’ is a valid inference, this means that to say that ‘A’ entails ‘B’ is to say that ‘A -> B ’ is not merely true, but is necessarily true.”

    Since “All triangles have three sides” is necessarily true, the conditional statement “If zebras have stripes, then all triangles have three sides” is also necessarily true. And the inference from “zebras have stripes” to “all triangles have three sides” is a valid one (because it is not possible for the premise “zebras have stripes”, to be true and the conclusion, “All triangles have three sides,” to be false, for the trivial reason that it is not possible for the conclusion to be false).

    That point aside, there are a couple of other points worth making. If you can derive a conditional statement without any premises, then the statement is a tautology and, because of this, the antecedent does entail the consequent (that is, it is impossible for the antecedent to be true and the consequent false).

    More to the point of your post on the goodness of God. I believe the following is also true: If, from a single premise (without any additional premises), you can derive a conditional statement , then you have in fact shown that the premise entails the consequent. This is because, in this case, the premise must entail the antecedent (since the antecedent must have been derived from the premise or else be a tautology), AND if the premise entails the antecedent and the premise entails the conditional statement (which it must since you have derived the conditional statement from the premise), the premise entails the consequent.

    I’m not as confident about this last point, but, if it is true, your original claim is just fine.

    • Bradley Bowen

      Jason –

      Thank you for your thoughtful comments. This subject is not very sexy, and I have not managed to make it very interesting, so I greatly appreciate the fact that you not only read it, but also thought about the post and have made some intelligent comments too.

      Based on your comments, you probably know as much or more about logic as I do, so my responses to your questions and comments may not be particularly enlightening, for you. But since you have taken the time to read and comment, I owe you a response.

      Does ‘Zebras have stripes’ entail ‘All triangles have three sides’? I suppose it depends on what you mean by ‘entail’. The word ‘entail’ can be tied to a particular system of logic, and get its meaning as a technical term in relation to that system of logic. In that case, I have no objection to using the word in a way that it is true that ‘Zebras have stripes’ entails ‘All triangles have three sides’.

      However, one can ask a deeper question about such a system of logic, namely, does it fully and accurately capture the meaning of ‘entails’ in ordinary language, when the word is not a technical term whose meaning is tied to a particular system of logic.

      The same sort of question has been prominent in relation to the meaning of conditional or “If…then…” statements. Conditional statements are often represented as material conditionals in propositional logic, but it is fairly clear that a material conditional does not fully and accurately capture the meaning of ordinary conditional statements, or at least there are many conditional statements that seem to have meanings that are not fully and accurately captured by material conditionals.

      My personal sense is that ‘Zebras have stripes’ does NOT entail ‘All triangles have three sides’ because the first statement has no logical connection with the second statement, and to claim that one statement entails another means or implies that there is a logical connection or a connection of relevance between the two statements.

      This seems parallel to the paradoxes of material conditionals. I would hesitate to say that the following conditional statement is true: ‘If zebras have stripes, then all triangles have three sides’. The corresponding material conditional is, of course, true, but that just suggests that material conditionals fail to fully and accurately capture the meaning of some conditional statements.

      I will give more responses to your comments later.

      • Jason Thibodeau

        Bradley said,

        “My personal sense is that ‘Zebras have stripes’ does NOT entail ‘All triangles have three sides’ because the first statement has no logical connection with the second statement, and to claim that one statement entails another means or implies that there is a logical connection or a connection of relevance between the two statements.”

        Well, I suppose I disagree. Undoubtedly this discussion hinges on the meaning of “logical connection.” But I don’t see why “Zebras have stripes” has no logical connection to “All triangles have three sides.” In fact there is a logical connection: the first statement entails the second.

        Let’s talk about tautologies since I am more comfortable here than with non-tautological necessary truths. Consider the tautology, If it is raining, then it is raining.” The statement “Zebras have stripes” has a logical connection to “If it is raining, then it is raining.” since it entails it. It is true that every statement entails “If it is raining, then it is raining” but so what? That just means that every statement has a logical connection to it.

        My point is that entailment is a logical connection. You would agree that some instances of entailment (e.g. that “God exists” entails “It is false that God does not exist”) involve genuine logical connections. But you think that other cases of entailment (like the one above) don’t involve genuine logical connections. But how do we distinguish the cases of entailment that involve genuine logical connections from the cases of entailment which lack a genuine logical connection. In all instances of entailment it is impossible for the conclusion to be false if the premise is true. That sounds like a genuine logical connection.

        So, if you think that, in some sense, there is no logical connection between “Zebras have stripes” and “Triangles have three sides” or “If it is raining, then it is raining”, then we need an understanding of ‘logical connection’ that explains why the species of entailment in which a statement entails a logically necessary truth does not consist of a logical connection.

        I’m not saying you are wrong, I am just saying that I can’t come up with a good way of making the distinction that you want.

    • Bradley Bowen

      Jason said:

      That point aside, there are a couple of other points worth making. If you can derive a conditional statement without any premises, then the statement is a tautology and, because of this, the antecedent does entail the consequent (that is, it is impossible for the antecedent to be true and the consequent false).

      ==============
      Response:
      I agree that the antecedent would entail the consequent, based on the definition of ‘enatail’ that you give.

      It is a separate question whether the antecedent would entail the consequent if we are using the word ‘entail’ not as a technical term, but as meaning what this word normally means (in modern philosophical discussions).

    • Bradley Bowen

      Jason said:

      More to the point of your post on the goodness of God. I believe the following is also true: If, from a single premise (without any additional premises), you can derive a conditional statement, then you have in fact shown that the premise entails the consequent.

      ===================
      Response:

      This seems incorrect to me.

      One way to prove a conditional statement (in typical propositional logic systems) is to prove the truth of the consequent:

      P
      therefore
      Q -> P

      In this case, since the premise P entails the consequent P, your claim would hold.

      However, one can also prove a conditional by proving the falsehood of the antecedent:

      P
      therefore
      ~P -> Q

      In this case, since the premise P does NOT entail the consequent Q, your claim would not hold.

      • Jason Thibodeau

        Yes. Thanks. As I said, I was less than fully confident of the claim, but I couldn’t put my finger on why.


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