Philosophy Stack Exchange is a question and answer site for those interested in the study of the fundamental nature of knowledge, reality, and existence. It only takes a minute to sign up.
Sign up to join this community
I have been going through one of the books and I found an example of conclusion C that is not entailed by a premise P.
Daisy has four legs. So daisy has more than two legs.
Surely though, the premise entails the conclusion as it's an analytical truth. The definition of 4 entails more than 2. What am I missing?
Exactly; in the above argument, the premise: "four is more than two" is missing.
In logic, the validity of an argument depends on its logical form.
The fact that 4 is greater than 2 is not a "logical fact" but an arithmetical one: it depends on the axioms of arithmetic.
Added, May 5th
Thanks to @Mike Hill for having found the correct reference.
See forall x : Calgary Remix, page 82: the example is aimed to show the limit of propositional logic.
Specifically, using propositional logic, the argument must be formalized with:
D, therefore E,
and thus it is not valid, because there is no way (within the limits of propositional logic) to express the semantic "link" between the two premises.
This question is based from a misunderstanding of the example that it quotes. In context, the book does not claim that the entailment “Daisy has four legs; so Daisy has more than two legs” fails. Rather, it notes that a certain very crude encoding of this entailment fails. This is given to illustrate that TFL (truth-functional logic, aka propositional logic) is not able to adequately encode such statements and arguments.
Specifically, as @Mike Hill points out in comments, this comes from the book forall x : Cambridge (P.D. Magnus, Tim Button, 2018). In context, it comes from Sec 11.5, The limits of these tests:
It is important to understand the limits of our achievement. I shall illustrate these limits with three examples.
First, consider the argument:
- Daisy has four legs. So Daisy has more than two legs.
To symbolise this argument in TFL [truth-functional logic], we would have to use two different atomic sentences – perhaps ‘F’ and ‘T’ – for the premise and the conclusion respectively. Now, it is obvious that ‘F’ does not tautologically entail ‘T’. But the English argument is surely valid!
[…two more examples…]
In different ways, these three examples highlight some of the limits of working with a language (like TFL) that can only handle truth-functional connectives.
To break this down a bit more clearly:
Intuitively, the entailment “Daisy has four legs. So Daisy has more than two legs.” should hold.
The language TFL only contains propositional connectives like “and” and “implies”, and atomic propositions. It doesn’t include anything to express predicates like “has 2 legs”, let alone numbers like “2” and “4” or the relations between them . So TFL can’t encode the internal structure of “Daisy has four legs” or “Daisy has more than two legs” — the best you can do is represent them by atomic statements A and B.
But for atomic sentences A and B, the entailment “A ⊨ B” fails (since you can just take a model where A is true and B is false). The point is they’re now just atomic sentences, i.e. meaningless symbols which can stand for any statement: they don’t contain any details about legs or numbers.
So encoding “Daisy has four legs” and “Daisy has more than two legs” as atomic sentences doesn’t express enough of the meaning of those sentences to show why the entailment between them holds. This illustrates a limitation of the language of TFL.
More generally: If the sentences about Daisy’s legs were encoded faithfully, in a way that properly expressed all the concepts involved, then yes, the entailment should hold. But if they’re inadequately encoded, in a way that discards or ignores too much of the intended meaning, then the entailment between their encodings may fail to hold.
So this is a little controversial. I'd say there's usual agreement that the statement "Daisy has four legs" needs to be translated into logical form before you can treat it as a statement suitable for deriving non-trivial conclusions. The question is whether one should translate this using numbers as mathematical objects, or whether one should translate number terms as proper logical particles.
For example, you could interpret the statement like this -
"There are objects a, b, c and d, such that Daisy possesses them, none of them are identical, and they are legs [and, for any other object e that Daisy possesses that is a leg, e is identical to either a, b, c or d]"
You could also interpret the statement that Daisy has more than two legs as a paraphrase that Daisy has at least three legs -
"There are objects a, b and c, such that Daisy possesses them, none of them are identical, and they are legs."
This paraphrase does allow the entailment to go through, using no further rules of logic than are contained in first order classical reasoning. The question is, is this really what we mean?
This uncertainty is enough itself for many to conclude that it's not a simple matter of fact that "two" does in fact connote this method of logical translation. Nonetheless, Logicism is a real project in the Philosophy of Mathematics that, despite numerous well-documented obstacles, continues to see occasional revisiting (see for example Neil Tennant's SEP article on Logicism and Neologicism )
Both @Ray LittleRock and @Mauro ALLEGRANZA's answers contain the two halves of the puzzle; my post here is simply to synthesize them.
There are two forms of logical consequence. One of these - and the one being alluded to - is the syntactic consequence: it is typically denoted by $\vdash$. Such a consequence basically means "take the two sentences as purely formalistic strings of meaningfree symbols and, then, see if there's a way you can perform logically allowed operations on the premise sentence to transform it into the conclusion sentence". That is, when thinking about syntactic consequences, you have to take "Daisy", "four", "two", "legs" all as having no meaning other than just being symbols - if it helps to see what is wrong, substitute "four" with some more obviously meaningless like, say, "xyigu", and "more than two" with "xzvjzx":
Clearly, in this case, there is no way you can perform the exchange. There is no purely formal-logical rule that lets you exchange meaningless symbol "xyigu" and meaningless symbol "xzvjzx", and hence so also likewise meaningless symbol "4" and meaningless symbol "> 2". You could transform it into, say, "Daisy has four legs or has more than two legs" (disjunction introduction), "It is not true that it is not true that Daisy has four legs" (double negative introduction), but not much else - certainly not intersubstitution of the two symbols. That is not a logical rule - which is what @Mauro ALLEGRANZA is saying.
Thus, for it to be a syntactic consequence, you would need there to be another premise included - more raw material, so to speak, to work on with logical rules: namely, $4 > 2$ or, "'xyigu' implies 'xzvjzx'". If you do that, then you can indeed use formal rules to go from now the pair of premises (so both would go on the left of the $\vdash$, typically separated with a comma, or grouped with a brace), toward the conclusion.
The other notion of consequence, however, which is what gets you hung up, is that of semantic consequence: a consequence that follows from the meanings, or "interpretations of", the terms. And in that case, yes, you can indeed make the deduction: the needed copremise, "4 is greater than 2", without specifying it as an explicit premise, comes from when we consider "4" to be a meaningful symbol (and not just a meaningless string like "xyigu") whose meaning is a certain number, and so likewise for the symbol "2", and then these two numbers have the property that 4 > 2, and then the conclusion comes heeling. You see, the extra premise gets pulled in from the meanings - hence semantic (meaning-related) consequence. For what it's worth, this kind of consequences is denoted $\models$.
Entailment can be taken in a broad sense and in a narrower one.
One thing that is uncontroversial is that ' Daisie has four legs"' does not entail ' Daisie can run' for it is possible to have four legs and, for some reason ( having broken legs, having a disease affecting muscles, etc), not being able to run.
So, the minimal condition in order proposition P to entail proposition Q is that :
if P is true , Q cannot be false,
that is
necessarily ( if P is true, Q is true).
Note : asserting that P entails Q is stronger than asserting that the material conditional ( P --> Q) is true. "2+2 = 4" materially implies "Churchill smoke cigars" ( since both sentences are actually true) , but the first sentence does not entail the second, for there is a possible case in which the first is true while the second is false , namely the possible case in which Churchill did not smoke cigars.
P entails Q means ( synonymously ) : it is logically necessary that if P is true, Q is also true.
In this broad sense of entailment or logical implication, it is perfectly correct to say that having four legs entails having more than two legs.
In the same way , " Peter is a pianist" entails " Peter is a musician", or " Mary is John's wife" entails " Mary is a woman".
"individual d is Predicate_1"
and the second would reduce to
"individual d is Predicate_2"
in such a way that you wouldn't be able to conclude from the first sentence to the second. ( The reason is that there is a possible interpretation - in fact infinitely many possible interpretations - of 'd' , ' Predicate_1 ' and ' Predicate_2 ' that make the first sentence true and the second false).
EXAMPLE :
As I said there are infinitely many interpretations in which the first sentence is true and the second false. You only have to find an individual that actually satisfies a first predicate and that does not satisfy a second one. Having an actual case in which the premise is false and the conclusion false is enough to prove that it is possible to have such a case. And nothing else is required to give a couterexample.
Let's say that ' d' denotes the Earth
Let's say that 'Predicate_1' is ' x is a planet'
Let's say that 'Predicate _2' is : ' x has 2 natural satellites'
In that case, sentence (1) becomes : The Earth is a planet.
And sentence (2) becomes : The Earth has 2 natural satellites.
The premise is true and the conclusion is false. This is enough to prove that the form of the reasoning is not truth preserving. When a form isvalid, it has no possible instance with true premises and a false conclusion.
Compare this with the following form :
(1) ' d ' is Predicate _1
(2) For all x, if x is predcate_1 then x is Predicate_2
(3) Therefore, 'd' is Predicate_2.
in that case, is it possible to find an interpretation ( even arbitrary or fanciful) ) that makes the premises true and the conclusion false?
P entails Q iff (a) necessarily , (b) in virtue of the sole forms of propositions P and Q, (c) if P is true, Q is true.
that is
the form of the sentences rules out the possibility of having an interpretation in which the first sentence is true and the second false.
Note : Entailment in this narrow and strong sense can be seen as the converse of the relation of logical consequence. There are 3 excellent entries by Mc Keon on this topic in the Internet Encyclopedia Of Philosophy (https://www.iep.utm.edu/logcon/#H2)
Also, on the distinction between material and formal consequence ( by Restall, SEP) : https://plato.stanford.edu/entries/logical-consequence/#FormMateCons
