Classical Validity and Entailment- Definitions
Let us have some formal definitions. Learn them, and notice how they are all interdefined in terms of one uniform, canonical vocabulary.
An argumentis a purported deduction[1] of a single proposition, designated the conclusion of the argument, from a (possibly empty)[2] set of propositions, designated the premisses of the argument.
An argument is valid iffthere is no possible world in which the premisses are all true and the conclusion false.
In a valid argument the (set of)[3] premisses entails the conclusion.
An argument is invalid iffit is not valid
i.e. iffthere is a possible world in which the premisses are all true and the conclusion is false
A necessary truth is a proposition true in all possible worlds
A necessary falsehood is a proposition false in all possible worlds
A contingent truth is a proposition true (in the actual world), but false in at least one possible world
A contingent falsehood is a proposition false in the actual world, but true in at least one possible world
A set[4] of propositions is consistent iff there is a possible world in which all its members are true.
A set of propositions is inconsistent iff it is not consistent, i.e. if there is no possible world in which its members are all true.
It is vital to deploy this vocabulary accurately. And not to mix it up. Notice that
Truth and Falsehood are properties of propositions, and not of anything else.
Consistency and Inconsistency are properties of sets of propositions and not of anything else. (Unless you are allowing yourself to be harmlessly sloppy – see footnote 3).
Validity and Invalidity are properties of arguments, and not of anything else.
-oOo-
Problems with the classical conceptions of validity and entailment
Make sure you have mastered this section. The distinctions drawn may seem elusive at first. Re-read, and rethink, until you have it straight.
It is a straightforward consequence of the definitions that any argument with inconsistent premisses is trivially valid. For if there is no possible world in which the premisses of an argument are all true, then a fortiori there is no possible world in which the premisses are all true and the conclusion false. So the argument [A] encoded by
Grannie strangled the cobra
Grannie didn’t strangle the cobra
John Redwood is a lizard
is trivially valid.
And likewise, it is a straightforward consequence of the definitions that any argument with a necessary truth as conclusion is trivially valid. For if there is no possible world in which the conclusion is false, then a fortiori there is no possible world in which the premisses are all true and the conclusion false. So the argument [B] encoded by
Lincoln is the capital of Nebraska
There are no pianos in Japan
2 + 2 = 4
is also trivially valid.
Now [A] and [B] may just strike you as bizarre. But there are parallel problems with arguments which are not thus weird at all. Consider argument [C], encoded by
Every even number is divisible by 2
48 is divisible by 2
48 is an even number
This looks like a poor piece of reasoning, being parallel to
Every Falklands islander is a British subject
Posh Spice is a British subject
Posh Spice is a Falklands islander
which takes you from true premisses to a false conclusion. But [C] is trivially a valid argument, because its conclusion is a necessary truth.
More generally, it looks as though any argument in mathematics, no matter how many mistaken steps it contains, will be valid, so long as the conclusion happens to be true. Any argument to the conclusion that is irrational will be a valid one.
Now some people, finding these results counterintuitive, will jump to the conclusion that there is something clearly wrong with our definitions of validity and entailment.
I understand the impetus, but this response is misplaced. The concepts of validity and entailment are trade-marked. Which means that their meaning is completely given by the definitions. So that it is just a fact, given the definitions, that arguments [A] and [B] are valid. And just a fact, given the definitions, that their premisses entail their conclusions. Just as it is a fact, given the definition of ‘bachelor’ that all bachelors are unmarried.
Any sense of discomfort over [A] and [B] turning out to be valid properly belongs elsewhere. Thus:-
We all have, in advance of any theorising about the matter, an intuitive sense of what counts as cogent[5] reasoning, proper deductive practice, a good argument, and so on. And what the formal logicians are trying to do, with their various definitions, is to capture this intuitive notion, and make it precise. And how successful they are will of course be an empirical question. So the right way to frame your worries about [A] and [B] is this:
“It looks as though the concept of validity will not give us a good, clear account of cogent reasoning. It doesn’t even distinguish good arguments from bad ones. [A] and [B] turn out to be (trivially) valid, but surely they are not good models of arguments. We just don’t reason like that. And all kinds of terrible arguments in mathematics will be certified as valid , so long as their conclusion happens to be true. So what’s so good about validity? Why is it worth studying?”
A good question. A very good question, as we will see later in term. All I will say for now is
It isn’t as bad as you think
Once you get clear about why logicians prize validity, you will see that it isn’t a bad first try at capturing our intuitive notion of a good deductive argument. And then we can do something to lessen the counterintuitive impact of the results given above.
The background motivation for the classical account of validity is the idea that good deductive reasoning is about truth-preservation. That the point of deductive reasoning is to stay in touch with the truth. So that if you begin with true premisses, and reason correctly, there is an absolute guarantee that your conclusions will be true.[6]
So it must be, it seems, at least part of whatever it is that constitutes good reasoning that it is impossible for the premisses to be true and the conclusion false. And that notion is precisely validity. So perhaps we can say that good deductive arguments, cogent pieces of reasoning, must at least be valid, leaving it open whether they perhaps need further properties as well. After all, the arguments that we do think are intuitively cogent seem to all turn out to be valid.[7] Here’s an intuitively cogent argument:
No gorilla worships rugby, but all the Welsh do. So no gorilla is Welsh.
and the concept of validity seems to be getting it right. The argument is indeed valid, and surely what makes it work is that although we can imagine worlds in which some gorillas do worship rugby, and we can imagine worlds in which some of the Welsh do not, we cannot coherently imagine a world which contains Welsh gorillas, and in which no gorilla worships rugby, and all the Welsh do. Logic rules such worlds impossible.
“That’s all very well, you may say. OK, the property of validity matches up some of the time with our intuitive grasp of good argument. But so what? It doesn’t match up some of the time as well. What about arguments like [A], [B] and [C] above? They are valid, but surely defective as arguments.”
Part of the logician’s response to this is given in the parallel document on Ex Falso Quodlibet, which discusses the legitimacy of argument [A] in great detail. But the main thrust is to be found elsewhere. I shall put it dramatically:-
Formal Logicians are not actually interested in valid arguments
This is because validity is a property of particular arguments, containing concrete propositions – things with actual truth-values - as premisses and conclusions. Arguments like
All frogs hate blenders
Any creature which hates blenders is technophobic
All frogs are technophobic
Formal logicians are not interested in that argument - who cares about frogs and blenders? Nor are they interested in this one
All zemindars are rich
All rich people are ruthless
All zemindars are ruthless
(Who cares about zemindars?) Nor are they even interested in this one
All logicians are emotionally stunted
All emotionally stunted people swoon to Coldplay
All logicians swoon to Coldplay
(Who cares about logicians?). What they are interested in is the pattern of reasoning which all these arguments share, or the form of argument, as we call it. That’s why they are called formal logicians. Formal Logic is the study of forms of reasoning. We study schematic arguments, things which are written down like this
All As are Bs
All Bs are Cs
All As are Cs
or like this
Some As are Bs
All Bs are Cs
Some As are Cs
or this Most As are Bs
Most Bs are Cs
Most As are Cs
or this Either P is true or Q is true
P is false
Q is true
As these are schematic, mere patterns of argument, our definition of validity does not apply to them. They do not contain concrete propositions with truth-values. (Try asking yourself whether ‘All As are Bs encodes a true or a false message). So we need to extend our definition to allow it to apply to forms of argument as well as to individual arguments. The idea will be, of course, that the merepresence of the pattern guarantees that you cannot move from true premisses to a false conclusion; that the pattern is truth-preserving. This notion is cunningly captured by:-
A form (of argument) is a valid form iff all its instances are valid arguments.
Which means: no matter how you flesh out the pattern with real properties or propositions, the resulting argument will be valid by the standard definition.
And since an invalid form of argument will be precisely one which is not a valid form, and therefore one which does not guarantee that you cannot move from true premisses to a false conclusion, we can likewise define:-
A form is an invalid form iff at least one of its instances is an argument with true premisses and a false conclusion
Resolving the Problems
Now we are in a position to at least reduce your anxieties concerning the propriety of classical concepts of validity and entailment. Look first at argument [C] above, which to refresh your memory I repeat:-
Every even number is divisible by 2
48 is divisible by 2
48 is an even number
The argument itself is valid. But that is of only passing interest. More important for logic is the form of argument. If you think carefully, you should be able to abstract the form[8] (e.g.)
Every A has property D
n has property D
n is an A
And the important point is that this is an invalid form of argument. Which assertion we prove by actually producing an argument of that form with true premisses and a false conclusion. For example:-
Every man has a mother
Margaret Hilda Thatcher has a mother
Margaret Hilda Thatcher is a man
So here is what we can say about argument [C]. The argument itself is trivially valid, I suppose. But we are not interested in that fact. The reasoning was bad. Indeed, the pattern or form of the argument was invalid.
And isn’t that precisely what you thought wrong with [C]?
Now let’s look at [B]. Here it is again:-
Lincoln is the capital of Nebraska
There are no pianos in Japan
2 + 2 = 4
A valid argument? Certainly, but who cares about that? It is merely a consequence of the adventitious fact that 2 + 2 = 4 is a necessary truth. The reasoning is of course bizarre in the extreme. For here is the form of the argument:
P is true
Q is true
R is true
And that’s as hopeless as you can get. No connection in reasoning whatsoever between premisses and conclusion. Anyone arguing according to this schema would be certifiable. And isn’t that precisely what you thought wrong with [B]?
And now to bow out, having defended the classical concept of validity to the best of my ability.
“But what about argument [A]? “ I hear you say. [A] is different to [B] and [C], for there is a debate over whether or not the form is an acceptable pattern of reasoning. The form of is
P is true
P is not true
Q is true
and that is certainly a valid form (check it out!). So an appeal to form cannot allay any doubts you may have. This one is an important test case, for it is here that the concept of validityreally comes into conflict with your intuitions concerning cogency. Almost all practising logicians think it is an acceptable form. A few brave souls, and Wolfgang, think it is not.
I refer you to Ex Falso Quodlibet. But go there only when you are sure you understand the material so far.
-oOo-
[1] I was alerted to the importance of this element of the definition by a question in Monday’s class. Unless we insist, as part of the definition, that an argument is a purported deduction, there is nothing there to distinguish an argument form a mere set of propositions, whereupon the crucial distinction between an argument (which can only be valid or invalid) and a set of propositions (which can only be consistent or inconsistent) would collapse. Many thanks to my sharp-eyed interlocutor. If she reminds me who she is, I will put her name in lights, as being much more perspicacious on the matter than the university lecturer. And me, before she spoke.
[2] For technical reasons we allow the case where something is asserted as conclusion on no premisses at all. Don’t worry about it. It just makes the later development rather smoother. And it corresponds to the intuition that some propositions are guaranteed by Logic without further ado, and do not need to be deduced from something else. I insist, for instance that either Grannie is more venomous than the cobra, or she isn’t. But I don’t need premisses to get me there.
[3] As defined, entailment is strictly a relation which holds between a set of propositions and a single proposition. But we often allow ourselves to say that one proposition entails another. Which is slightly sloppy, because strictly we ought to be talking about the set containing that proposition. But as no harm can come from the habit, it doesn’t matter.
[4] The same kind of point as in the previous footnote. Logicians often speak of a single proposition (rather than a set) as being consistent or inconsistent. This is a bit sloppy, because they should really be talking about the set containing that single proposition. But again no harm can come from it, so we don’t mind. In fact we prefer not having to be thus picky every time we speak
[5] I would say sound reasoning for preference, but many logicians have trademarked sound , and defined it in terms of validity. And I want a word of the vernacular, which we can use to describe the intuitive concept. So I’m afraid we are stuck with cogent, unless someone can think of an improvement.
[6] As opposed to inductive reasoning – the kind practised by Sherlock Holmes - where the truth of any premisses does not logically guarantee the truth of the conclusion reached, but merely makes it probable.
[7] Later, in Week 4, we will see reason to doubt even this. But for now, I’m keeping my cards close to my chest.
[8] Logicians like to have a single, canonical form for all the different ways in which English might put things together, so they will tend to write instead something like
All As are Ds
n is A
n is D
It doesn’t matter, at this stage, how you specify forms, as long as you know how to read them.