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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?

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    For reference, it looks like this example may have come from "forallx: Cambridge", which is released under CC BY 4.0: homepages.ucl.ac.uk/~uctytbu/forallxcam.pdf. See page 44. – Mike Hill May 4 at 20:56
  • Gensler's A to Z of Logic : " ENTAILMENT ( strict implication) : "A ( logically) entails ( or strictly implies) B" means " it is logically necessary that if A then B". - So yes, in ordinary philosophical discussions it is perfectly correct to say that "having 4 legs" entails "having more than 2 legs", or that " knowing that P" entails " P is true". It your question was a philosophical one , the formality condition is not required for entailment. – user37859 May 5 at 13:17
  • Mark Sainsbury , Logical Forms (An Introduction To Philosophical Logic) ( Blackwell) " in general A entails B iff it is logically impossible for A to be true yet B not true. " You will find logic hard " entals " You won't find logic easy" " . ( p. 68). – user37859 May 5 at 14:24
  • Sometimes less is more. – Ian MacDonald May 6 at 19:58
  • When we use Rudolf Carnap (1952) Meaning Postulates to formalize the semantic meanings then it is entailed. There are no missing premises because they are all included in the Meaning Postulates defining the details of the numerical relation between four objects and two objects. – polcott May 13 at 0:51
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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.

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    Finite integer arithmetic exists in the null. Loading the existence of 4 as a premise inherently hauls in integer arithmetic up to 4. However, the same is not true of infinite-field arithmetic, and without any other premises we can't actually say that numbers > 4 exist. (See modulo arithmetic for the most well-known counterexample.) – Joshua May 4 at 4:02
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    In modulo-4 arithmetic, 4 is equivalent to 0, thus the premise "4 > 2" is false. This is a rich source of bugs : not normally at 4 but often at 32768, 65536, 2^31 etc where computer arithmetic can overflow (often silently producing incorrect results). – Brian Drummond May 4 at 10:56
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    Yep. So if we had a world where legs were counted entirely by $\mathbb{Z}_4$ (e.g. suppose that magically, once a fifth leg is added, they all vanish somehow), then this would be a legless animal and so would not have more than two legs. – The_Sympathizer May 4 at 15:03
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    I do not agree. Multiple "premises" are missing, such as whether "Daisy" in the second sentence is the same Daisy as in the first one. Or the meaning of "has" is the same in both sentences. Or what we mean by "more". Or whether some time passed between both sentences, and Daisy lost or gained legs in the meantime. Or whether the sentences are not English, but another language with other meaning for these words. Logic does not exist in a vacuum. Assumptions are ALWAYS required, and integer arithmetic is a valid assumption here. – isilanes May 4 at 16:07
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    Modulo-4 arithmetic has no order structure. In ℤ/4ℤ, 4 > 2 is not false, but meaningless. – user3840170 May 4 at 16:42
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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:

  1. 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:

  1. Intuitively, the entailment “Daisy has four legs. So Daisy has more than two legs.” should hold.

  2. 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.

  3. But for atomic sentences A and B, the entailment “AB” 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.

  4. 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.

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  • The book I have is 2019 paper version and the phrasing is different, it states: "Now, it is obvious that ‘F’ does not entail ‘T’". But still, it does not explain why does it not entail, apart from stating the mere fact. My question is valid since the statement is question-begging. – bodhihammer May 5 at 8:20
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    @bodhihammer: Yes, I quite agree your question is reasonable and valid, and I didn’t mean to suggest otherwise — the presentation in the book is somewhat unclear. I’ve tried to clarify my answer, including addressing the point you mention of why the entailment from “F” to “T” doesn’t hold (I’ve called them “A” and “B” instead to avoid confusion with the common use of “F”/“T” for “false” and “true”). But the key point is that (in either phrasing) the book never claims the entailment about Daisy fails: it says the entailment “FT” fails for atomic F, T, which isn’t the same. – Peter LeFanu Lumsdaine May 5 at 9:06
  • "it says the entailment “F ⊨ T” fails for atomic F, T, which isn’t the same." which has the same meaning in my question, other than me using atomic P and C (i.e. P ⊨ C) rather than F and T. It is not obvious at all that it does not entail even using the atomic F and T given the phrasing in the book. There is obviously a lot of information missing in the paragraph. – bodhihammer May 5 at 10:09
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    @bodhihammer: The point I think you’re misunderstanding is that the entailment “PC” for atomic P, C, is not the same as the entailment about Daisy’s legs. In the question, you ask why the entailment about Daisy fails — but the entailment that fails is just PC. Atomic formulas are just atomic formulas, and can take any truth-value in interpretations, so PC fails since you can interpret P as true and C as false. The book suggests representing the statements about Daisy as atomic formulas; that doesn’t mean the atomic formulas are the statements about Daisy. – Peter LeFanu Lumsdaine May 5 at 13:18
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    @bodhihammer: You say “my question was mere why the entailment might fail”. But the entailment you ask about in your original question, the entailment “Daisy has four legs. So Daisy has more than two legs.” — this entailment does not fail! So your question is based on a misunderstanding of what the book claims, and it can’t be answered without clearing up the misunderstanding. It’s a very reasonable misunderstanding, since the book is not so clear in its explanation, but it still means your question starts from a mistaken claim, and so I have tried to answer by explaining the mistake. – Peter LeFanu Lumsdaine May 5 at 16:20
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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 )

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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":

  • "Daisy has xyigu legs" $\stackrel{?}{\vdash}$ "Daisy has xzvjzx legs".

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 $\models$.

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    Hmm. Something about this doesn't sit well with me. I think you're reading extralinguistic structure into Semantic consequence; all models in which Daisy has four legs are models in which Daisy has more than two legs, but the question in Semantic Consequence relations is whether all models that make "Daisy has four legs" true also make "Daisy has more than two legs" true. This seems to be assuming something about the logical treatment of number terms, rather than hanging on sem/syn distinctions. – Paul Ross May 4 at 16:17
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    The begining of this answer is absolutely on the money — this is indeed about which of the terms involved are specified enough to get meaning, as opposed to which parts are treated just as meaningless symbols. The “Daisy has xyigu legs” vs. “Daisy has xzvjzx legs” illustration is great. However, the second half is a bit misleading: this is nothing to do with the distinction between semantic and syntactic consequence — if 2 and 4 are encoded as xyigu and xzvjxz, then either syntactically or semantically, the connection is missing. [cond’t] – Peter LeFanu Lumsdaine May 4 at 21:47
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    So I wouldn’t say that the relation between more than 2 and 4 is “pulled in from the meanings”; rahter, it’s that this relation gets thrown away if one chooses to encode them as meaningless symbols xyigu and xzvjxz. – Peter LeFanu Lumsdaine May 4 at 21:48
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    Just a note that MathJax isn't enabled on this SE, so you would have to find another way to replace them. – Andrew T. May 5 at 11:55
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  • 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.

  • So entailment requires necessity, and more precisely, logical necessity. We may therefore say that, in the broad sense :

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".

  • Now there is a sense of entailment or logical implicaton that is narrower . It relates to the idea of a reasoning that is valid " vi formae" , in virtue of its form alone. In his sense " Daisy has 4 legs" does not entail " Daisy has more that two legs". Suppose you did not understand english, except its logical vocabulary ( 'for some', 'for all', 'is', 'not' ) , in that case the first sentence would reduce for you to its form

"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?


  • We may therefore define ths strong sense of entailment as

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

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    As a heads up, I think you copied the wrong link in there. Perhaps you could try again? – Paul Ross May 3 at 14:17
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    What would be "an interpretation in which the first sentence is true and the second false" for the example given in the question? This answer could be greatly improved by focusing less on the definition of entailment and more on the example given in the question and how the definition of entailment applies to it. – NotThatGuy May 3 at 22:14
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    While this is good as an explanation of how entailment (logical consequence) works, I don't really see how it helps with the question because a falsifying example ("an interpretation in which...") would have to exist where "Daisy has four legs" and does not have more than two legs. So something's missing here. @Mauro ALLEGRANZA's, however, I find is much more on point (and was my first thought as well). – The_Sympathizer May 4 at 14:58
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    "Suppose you did not understand English" : 'semantics'? (@The_Sympathizer, I was about to upvote your answer...) OP: "What am I missing?" - philosophers are difficult for the sake of argument. – Mazura May 4 at 17:37
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    @The Sympathizer. - There is no counterexample in the intended interpretation . But to be formally valid , it must be true in any possible interpretation that the conclusion is not false when the premises are true. – user37859 May 4 at 19:19

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