Talk:Logical consequence/Archive (entailment)

Entailment

Can anyone explain what "model of" is supposed to mean? How do models apply to logic?

Truth Table Needed

I was directed to this page from LogicalImplication, and I was looking for a truth table. Can we put the truth table from the MaterialConditional page here? That would be helpful. Thanks. —Preceding unsigned comment added by 71.182.112.176 (talk) 17:30, 22 July 2009 (UTC)

That would not be desirable since then it would become too apparent that material -, logical -, syntactic - and semantic implication are really the same thing. — Preceding unsigned comment added by 83.134.176.120 (talk) 04:57, 8 May 2012 (UTC)

Examples

This entry is woefully in need of examples. There seems to be a contextual nuance that differentiates semantic from logical entailment, as well as from the meaning in pragmatism. This nuance isn't conveyed when the same words are used in all of the definitions. —BozoTheScary 16:11, 2 May 2006 (UTC)

I have some questions regarding this page, and I believe it has to do with your asking for more examples. First of all, what is a frikin “model”?? this thing is so vague and mysterious. And the “model theory” page doesn't help at all. People only talk about high-order logics and that damn “samantics”... Can't we have an explanation of all this in predicate logic?... Can't we frist explain things in a way that people can uderstand, and just then talk in that beautiful way that is so abstract and general?...

And regarding that two examples given, the first is so silly and meaningless. It's just a tiny syllogism. And it's damn small, There is no way for us to imagine when it would NOT hold. And then the second example requires the readers to understand what the heck is an “empty model”, and I could not, since I don't even imagine what a model is. -- 200.185.249.203 (talk) 23:43, 15 March 2008 (UTC)

Implication question

What is the general consensus? Premise: A implies B. Conclusion: Not A implies not B. (Or would it be "Not A does not imply not B"?) And from that first conclusion, can it then be said "Not B implies not A" or even "B implies A"? -- 24.153.226.102 18:10, 28 February 2006 (UTC)

I'm not sure that this is the right place for this particular discussion, but your conclusions are incorrect.

Given the premise "A implies B", we may validly conclude that "Not B implies not A". However, it does not generally follow that "Not A implies not B"; nor does it follow that "Not A does not imply not B", which is equivalent to "Not (Not A implies not B)".

One way to understand these assertions is to take as example propositions: A="Charlie's age is in {2,3}"; B="Charlie's age is in {2,3,4,5}". Then your premise: "A implies B" holds, since if Charlie's age is in {2,3}, it is also in {2,3,4,5}. Your required conclusion, "Not A implies not B", is equivalent to "Charlie's age is not in {2,3} implies it is not in {2,3,4,5}", which is clearly false if Charlie is aged 4 or 5.

You can use the same example to see that the valid conclusions are plausible (but not to prove them).

An easy way to picture these propositions is to draw the Venn diagrams of the sets of ages for which A and B are true, and also mark the sets not A and not B. In such pictures, we can identify proposition A with its truth conditions, that is, the set of conditions under which A is true, in this case the set of ages for Charlie {2,3}. The Venn diagram for "A implies B" is then equivalent to the Venn diagram for "A is a subset of B".yoyo 11:27, 25 September 2006 (UTC)

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A implies B

The modus ponens rule is: If A is true then B is true.
The modus tollens rule is: If B is false then A is false.
Those are the only two valid logical inferences of which I am aware in Propositional Logic. capitalist 03:39, 1 March 2006 (UTC)

Completeness and Godel's Theorem

I don't believe that the reference to completeness and Godel's theorem is correct. Godel's theorem can be applied to logics that satisfy a completeness theorem (e.g. axioms of arithmetic on first order logic). The incompleteness theorem does provide a proposition that can't be proven or disproven but is satisfied by the "natural" model. But there are other models for which the proposition does not hold.

Maybe a better example for an incomplete logic would be the second order logic. Second order logic does not satisfy a compactness theorem. I think that this means that a proof system based on finite proofs will not work.

Turtle59 06:31, 13 March 2006 (UTC)

Diagram

In the diagram, shouldn't B be a subset of A and not vice versa which is what the diagram is? --Eok20 01:49, 1 July 2006 (UTC)

You're perfectly right and I allowed myself to add a subtext to the diagram, as well as correct the conclusion drawn out of it, saying now: "every B has to be an A" instead of "every A is a B", as before. - Frank.Lenzer@uni-jena.de 12 October 2006

If this diagram is showing what I think it is then it is wrong according to the "correction" underneath. Assuming the diagram is showing models of A and B then it shows A entails B. The definition of entailment says A entails B if all models of A are also models of B. Therefore A has less models than B and also A's models are a subset of B's models. Assuming the diagram is showing models then this should be cleared up because it refers to "A" and "B" whereas further up the page these are defined as sets of sentences. - b 16th November 2006

The diagram (an Euler, not a Venn as originally labeled) is misleading because it suggests that B has non-A members. However, A can entail B when B has no non-A members. I'll boldly remove it. Jcblackmon (talk) 05:31, 15 December 2008 (UTC)

Terms for the arguments of an entailment

Are there specific words for the two arguments of an entailment? If we have that $A\models B$, is A the "entailer" or the "antecedent" or the "conditional" or what? Similarly, what is B called?

• A is conventionally referred to as the 'antecedent' and B the 'consequent'. Adam

What is a "deducibility relation"?

Also, what is the difference between the |= and |- symbols in implication? I'm having a hard time understanding this section... 169.232.78.24 (talk) —Preceding comment was added at 04:42, 11 June 2008 (UTC)

• |= and |- denote semantic and syntatic consequence respectively, and are symbols generally only used in model theory to relate sets of sentences to one another. Adam. — Preceding unsigned comment added by 82.46.105.58 (talk) 21:16, 14 July 2011 (UTC)

Dictionary Needed

Perhaps a little ramp is needed to map the peculiar notions so liberally assigned to conventional words in these articles on implication and entailment. Perhaps its an issue of tautological structure. A good clarifier I've found is the following: http://www.fallacyfiles.org/somernot.html —Preceding unsigned comment added by 72.136.100.3 (talk) 20:24, 6 September 2008 (UTC)

The phrase "just in case" used in this article means something quite different in ordinary english english (where it can only mean "in the unlikely event that"). Here it seems to mean "only when" and these words would be better as they mean the same both sides of the pond. 88.11.129.94 (talk) 20:35, 21 November 2008 (UTC)

"when and only when" would be closer, but restriction to variation in time is not intended. The phrase "just in case" as used in philosophy and logic is synonymous with "if and only if" - usually abbreviated to "iff" - so far as I am aware on both side of the pond, and in Australia other English speaking countries as well. We could replace "just in case" with "if and only if" of "iff", salve veritate if it make a clearer to the reader. As with other disciplines, philosophers and logicians form a community of language users, and it functions most smoothly if the same words are used in similar contexts. It includes a vocabulary or terms and phrases that have an agreed meaning in the community, enabling users to say and understand more precisely what is intended than by using 'ordinary' common usage, which is often less precise and ambiguous. The same is true in say physics, where terms such as "force" and "momentum" have specific meanings, more exact and from ordinary usage. It is better to use the 'technical' terms just because they have the more precise meaning. To rewrite legal, scientific, mathematical, philosophical, logical etc texts users common language would, by making then less precise, make them less not more comprehensible. Hope this helps. --Philogo 14:34, 24 November 2008 (UTC)

What happened to this Article?

I found an old version of this article which seems much nicer, what happened? —Preceding unsigned comment added by 76.10.150.120 (talk) 20:17, 5 December 2008 (UTC)

I redirected Logical Impication to Entailment, see talk page for reasons. --Philogo 13:56, 9 December 2008 (UTC)

This was an error. Logical Implication is the *more common* term; entailment is the *less common* term (within mathematics). This is a good article for the model-theory-specific definition of entailment, but a terrible article for "logical implication", which can refer to the material conditional (implies operator), formal deduction within a formal system, or this. Entailment as a generic seems to be used primarily in old-school philosophy. The old article is better. Logical implication should perhaps be a 'master article' linking to its different technical meanings. 67.241.20.26 (talk) 20:08, 4 May 2009 (UTC)

I agree that the redirection is in error, and should be undone. My sense of the utility of wikipedia is that it is not meant to be a resource for elite specialists. Logical implication is a term embedded in everyday use which most laypersons and high school level students will be aware of. Entailment is a subtle and in-depth analysis of deeper logical meaning. It should be referenced from within the discussion of logical implication, and not the other way around. To keep the redirection with entailment as a more general category is illogical, not useful, incorrect, and imperious. Please revert it so that entailment, as a subject in Wikipedia, is a sub-topic. The article on entailment is excellent, interesting, and important, but as a category of discourse, it needs to be embedded in a way to allow those more familiar with "logical implication" to find that term.--Mykrafone (talk) 18:44, 20 July 2012 (UTC) — Preceding unsigned comment added by Mykrafone (talk • contribs) 18:38, 20 July 2012 (UTC)

The 'fact' tags are a reminder to myself to add the page numbers from Quine (or other source). Please feel free to beat me to it.

I came to this article without reading 'logical implication' and without knowledge of the previous history of that article. If someone wishes to redress that article, please be aware that my edits do not address that thread. --Ancheta Wis (talk) 06:07, 29 August 2009 (UTC)

Somehow, I would like to include Alfred Tarski's contribution to the use of models in logic in this article. Again, if you want to beat me to it, and contribute to this article first, please feel free to do so. --Ancheta Wis (talk) 16:49, 25 November 2009 (UTC)

⊧?

⊧ redirects to this article. What does it mean? The list of logic symbols calls it "is a model of" which does not provide any semantics for a novice like me. Thanks for your help, --Abdull (talk) 17:49, 8 September 2010 (UTC)

$\vDash$, or Double turnstile which can be read entails, is like the $\vdash$ Turnstile (symbol), which can be read yields. Alfred Tarski is famous for Model theory which includes first order logic. --Ancheta Wis (talk) 18:30, 8 September 2010 (UTC)

Duplication of content

Much of the content of this article (Entailment) is duplicated at Material conditional and Logical implication. The difference between these concepts should be clarified and material should be (re-)moved from each article, as appropriate. - dcljr (talk) 19:56, 15 January 2011 (UTC)

Note: Before making any changes, interested parties should probably read Talk:Material conditional and Talk:Logical implication, too. - dcljr (talk) 20:09, 15 January 2011 (UTC)

There is also logical consequence. Vesal (talk) 21:37, 15 January 2011 (UTC)

At first, I would leave material conditional alone. Is there any reason for having three separate articles on logical implication, logical consequence and entailment? Here are the lead sentences from each article:

• In logic, entailment (or logical implication) is a relation between sets of sentences and a sentence.
• In logic and mathematics, logical implication is a relation that, typically defined, holds between sets of formulae and a single formula.
• Logical consequence is a fundamental concept in logic. It is the relation that holds between a set of sentences (or propositions) and a sentence (proposition) when the former "entails" the latter.

There seems to be some agreement on the various talk pages that a merger should take place. The question is how to do it: which article should be the main article? Vesal (talk) 23:17, 15 January 2011 (UTC)

I would vote for "Entailment" as the best title for a merged article, which is not to say that the article so currently entiled is necessarily the best or main article. One they are merged that is history. The terms "Entailment", logical implication, Logical consequence are interlinked as indeed are logical truth, "logically necessary truth", "validity", "inconsistent" and "self-contradictory". Strawson, PF, Introduction to Logical Theory, Methuen 1952 (1963) gives an extended consideration to this, sections 13 , Entailment and insconsistency, p. 19 ; & sec 14 Logically necessary statments; Entailment and necessity" p 21Philogo (talk) 20:04, 16 January 2011 (UTC)

except for casting of meaning I fail to grasp any difference between any of logical implication, material implication, logical consequence, syntactic consequence, semantic consequence, metaconclusion, ... the first sentences on these pages stress that they are a relation between a set of sentences on one side and a sentence on the other... but saying that a set of sentences holds is saying that the sentence formed by conjunction of sentences holds. so they too are really relations between one sentence and another. They are all the same as material implication. All the mumbo jumbo about interpretations etc, are just extra premises or axioms suppressed from the lefthand set. its like saying to the reader instead of repeating really really obvious sentences on the left hand, consider it part of your logic and when we use the new implication symbol remember to conjunct these from now on for brevity suppressed sentences with the lefthand set of every semantic or whatever implication. If you try to produce a truth table for any other implication than material implication, you either get the exact same truth table as material implication, or sentences expressing your assumptions you hide from the lefthand set of sentences. This also explains the plethora of implication symbols: for a specific use of symbol, is a specific set of for brevity reasons suppressed axioms. Not specifying them is ambiguous, specifying them allows us to reduce them to material implication with a more verbose left hand set of sentences... The combination of fear for lack of generality (when the difference between these symbols is understood) or blind context-dependent parroting (when the difference between these symbols is not understood) is the major source for plethora of symbols for the same thing. This is just one of the many areas where axiom-wars are going on, instead of openly proclaiming axioms hide them in a symbol and get others to imply them whenever they derive a conclusion. — Preceding unsigned comment added by 83.134.176.120 (talk) 05:21, 8 May 2012 (UTC)

These variants of material implication are ultimate examples of memes: they reproduce by embedding their sentence within the conclusion symbol — Preceding unsigned comment added by 83.134.176.120 (talk) 05:30, 8 May 2012 (UTC)

Merger proposal

It has been suggested that logical implication and logical consequence be merged into this article.

Comment The main article should be Logical consequence and content should be merged to that name.Greg Bard (talk) 21:40, 17 January 2011 (UTC)

Support Entailment is the best title for a merged article (which is not to say that the article so currently entiled is necessarily the best or main article.) Philogo (talk) 00:03, 18 January 2011 (UTC)

Just to round it out, I'll support logical implication as the title for the merged article, without preference for the content. CRGreathouse (t | c) 01:40, 19 January 2011 (UTC)

Comment. I just checked how other people handle this, but there is no real consensus elsewhere either. SEP has an entry on logical consequence, while Simon Blackburn in the Oxford Dictionary of Philosophy follows the above proposal: "entailment" is the main entry and "logical implication" and "logical consequence" redirect to entailment ... Vesal (talk) 03:36, 19 January 2011 (UTC)

It looks like we're all agreed on the merge, so let's focus on making an article and worry about where to put it later. (Objections?) CRGreathouse (t | c) 09:20, 19 January 2011 (UTC)

Unfortunately one has to select a destination article (page) into which the other articles (pages) are merged; the destination article becomes the name of the merged article, and the merged-in articles are each replaced by a redirect to the merged article. It is possible however to rename the article post merge. If that's not clear it's all decribed in Help:Merging. Philogo (talk) 03:12, 20 January 2011 (UTC)

Philogo (talk) 18:54, 20 January 2011 (UTC)

Philogo (talk) 18:54, 20 January 2011 (UTC)

The merged in pages are at the bottom of the article: entailment#ex logical implication and entailment#ex logical consequence The article now needs editing to weave in the material therefrom pruning out duplicate materialPhilogo (talk) 18:54, 20 January 2011 (UTC)

Contents of the paras entailment#ex logical implication and entailment#ex logical consequence have been woven in, and the para deletedPhilogo (talk) 20:44, 20 January 2011 (UTC)

The lede has been re-written. The body of the article reauires editing to remove duplications, inconsistencies &c..Philogo (talk) 19:05, 26 January 2011 (UTC)

lede

Re opening

• In logic, entailment or logical implication is a relation between a set of sentences and a sentence (logical consequent).

Is it quite correct to imply thus that the term logical implication is applicable to the relationship between a set of sentences and a sentence rather than between a sentence and a sentence? The next sentnece in the lede (below, enphasis added) assumes not (as opposed to logical conseqnency) .

• If Γ is a set of one or more sentences and S1 is the conjunction of the elements of Γ and S2 is a sentence then Γ entails S2 (and S2 is called the logical consequent of Γ and S1 is said to logically imply S2 ) if and only if S1 and not-S2 are logically contradictory.

Cf eg http://planetmath.org/encyclopedia/LogicalImplication.html:

• Logical implication is an operation on two logical values, typically the values of two propositions, that produces a value of false just in case the first operand is true and the second operand is false.

(although that definition does not differentiate Logical implication amd material implication!) and http://whatis.techtarget.com/definition/0,,sid9_gci833443,00.html Philogo (talk) 16:23, 27 January 2011 (UTC)

Sorry if I made a mistake. I have no idea. As a mathematician I am not really interested in this kind of hair-splitting and find it hard to remember the fine distinctions some people make. I guess since lots of other mathematicians have been involved over the years and many probably had equally little interest in these fine distinctions, it should be possible to find a source for almost every usage – but obviously that doesn't mean we should claim non-standard usage as correct. Just correct me if you think I was wrong. Hans Adler 17:05, 27 January 2011 (UTC)

I did not mean to split hairs but establish a consistent use of terms. I have edited to emphasise the "redirect terms" elsewhere. The distinction between material implication and entailment/logical implication causes beginners a lot of confusion for some reason, that's why it is better to steer clear of that issue and use term "entailment" rather tham "logical implication" as the article's title.Philogo (talk) 18:12, 27 January 2011 (UTC)

Move to logical consequence

The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the move request was: not moved; there is no consensus or reason given for moving. ErikHaugen (talk | contribs) 09:33, 12 February 2011 (UTC)

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Entailment → Logical consequence — — Preceding unsigned comment added by Gregbard (talk • contribs) 16:20, 4 February 2011 (UTC)

No record of any discussion at: Wikipedia:Requested moves, possibly because the above edits were anonymous (unsigned). Philogo (talk) 01:43, 5 February 2011 (UTC)

You're sort of right. I'd never noticed before that simply adding unsigned doesn't fix it as the bot needs the "(UTC)" to detect the date and {{subst:unsigned2}} doesn't include this, so although I thought I fixed it I hadn't really. Dpmuk (talk) 02:04, 5 February 2011 (UTC)

No reasons given in a favour of unsigned proposal.Philogo (talk) 02:07, 5 February 2011 (UTC)

Oppose the proposed move/rename on grounds that no reasons are provided for the proposal and that time woud be better spent at this stage in editing the body of the article (whcih is in a dire state) most-merge. Philogo (talk) 19:41, 5 February 2011 (UTC)

Oppose the move. The sections on non-monotonic logic etc need more editors, in the meantime. --Ancheta Wis (talk) 09:17, 6 February 2011 (UTC)

The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

syntactic consequence

The article seeks to explain syntactic consequence as opposed to semantic consequence. We should ensure that it gets it right if it mentions the matter at all. There was quite a lot of discussion on the terms in talk:logical consequence. As I recall the consensus was that the terms semantic consequence and logical consequence were synonymous and NOT to be confused with syntactic consequence which was synonymous with derivablity. It would appear that entailment, semantic consequence, logical consequence are synonymous and the term logical implication is closely related if not also synonymous. None of them are synonymous with syntactic consequence. Philogo (talk) 20:40, 5 February 2011 (UTC)

My difficulty with a consensus that entailment, semantic consequence, logical consequence are synonymous is that only entailment is a large-enough word to encompass non-monotonic reasoning or defeasible reasoning. I am willing to concede that semantic consequence or logical consequence have the connotation that the sentences of the antecedent (i.e. Γ in the lede of the article) somehow encompass an inquiry into a topic. However, entailment, in its use in the English language, allows the possibility of provisional reasoning. Thus the existence of a consequent (i.e. S2 in the lede of the article) need not raise its status in a chain of reasoning, but rather enumerates a possibility to be attacked and possibly discarded. If there is a consensus that a logical form Γ ⊃ S2 encompasses some entailment then the agreement that there is consensus is only formal, which neglects some real-world factor which ought to be included in the semantics. --Ancheta Wis (talk) 17:33, 6 February 2011 (UTC)

That seems to raise some additional issues. (a) Are the terms entailment, semantic consequence, logical consequence synonymous and the term logical implication closely related if not also synonymous.(b) Is the term entailment restricted to deductive arguments in classical logic or does it extend to non-monotonic reasoning or defeasible reasoning? Perpaps each issue should have a separate thread to this one, which concerns syntactic consequence as opposed to semantic consequence. Philogo (talk) 19:58, 6 February 2011 (UTC)

We need to distinguish accounts of entailment from the actual thing. In philosophy, entailment is the relation between the premises and conclusion of a valid argument [1]. The entire question is to understand what this relationship exactly is. Most people take strict implication as a starting point, although relevance logic proponents dispute this [2][3]. But they all rely on modal operators, so one has to flesh out what necessity really means. According to the SEP entry, the most successful proposal is the appeal to formality. Once we accept that entailment is a virtue of the logical form of sentences, we can start talking about derivations in a formal system, and we can also talk about interpretations − we have notions of syntactic and semantic consequence. Most of us consider a statement to be true if it corresponds to how things are in the real world, so it is natural to prefer semantic consequence, but not everyone shares the underlying metaphysical assumptions! Semantic consequence and syntactic consequence are two different accounts of entailment, which the SEP entry calls the proof-centered and the model-centered approach. The proof-centered approach has notable proponents, so we cannot simply assume that semantic consequence is the correct account, much less a synonym, of entailment. Vesal (talk) 14:33, 7 February 2011 (UTC)

Are you content with the lede?Philogo (talk) 20:54, 7 February 2011 (UTC)

I think you've done a great job getting these articles in a decent shape. The lead is okay with me as we focus on the usage of the term in logic. Personally, though, I would prefer a more philosophical definition of entailment as the relation at the heart of valid arguments, rather than the current definition, which in my view is the dominant 20th century logical account of that relationship. Vesal (talk) 11:39, 10 February 2011 (UTC)

The article is about the usage of the term in logic hence the we focus on the usage of the term in logic. Can you indicate some sources for accounts of entailment other than the dominant 20th century logical account; in my limited reading the nature of entailment has been explained by means of logical form since Aristotle. If you mean there are philosphocal isses that arise notwithstanding the acuracy of the lede, then it is the custom to have a paragraph headed philosophical issues. HAvin merged three seate wrtice we need to complete the clean up first however, don;t you think? Re the relation at the heart of valid arguments. In defing terms im amy field we have in to decide which terms are to be treated as primitive; the fewere the primiteve terms the better, but where there is a choice it may be a matter of convenience. We could say that Γ/S is a valid argument only if Γ entails S, or Γ entails S only if Γ/S is a valid argument. Alternatively, as we do in this article, can treat contradiction as more primitive and say Γ entails S if and only if ¬(Γ ∧ ¬S) is a logical contradiction (and if Γ/S is a valid argument if and only if ¬(Γ ∧ ¬S) is a contradiction). If we can then define contradiction in terms of logical form and truth we are taking these last two terms as more primitive. It would not be unreasonable to treat truth as primitive and undefined, but logical form is susceptible to further analysis (in temrs of so called logical constants and logical variables). As I mentioned above our article logical form is not at all satisfactory (have a look!) so we cannot just refer the reader to that article by writing contradiction .. logical form q.v.. unless we intend to bring that article up to standard. There is a logic task force which should supposedly consider these matters (co-ordination and standrds of articles etc.) but alas it seems to have become dormant. Philogo (talk) 16:56, 10 February 2011 (UTC)

Philogo, if we were to use 'logical constants and logical variables' as a basis in the logical form article, then we are faced with 'left hand side' (lhs) and 'right hand side' (rhs) as primitive concepts. If we were to take lhs as 'that which denotes' (i.e., that which names something) and rhs as 'that which something' then we arrive at natural deduction etc. My philosophical position is that truth has got to be something we can arrive at (i.e., it belongs on the right-hand-side of any statement). Note that this assumes we parse expressions from left to right (an LR grammar).

Entailment, as a concept, appears to get its strength from the convention that the lhs is conjunctive and rhs is disjunctive; Gerhard Gentzen baked this into his sequent calculus. --Ancheta Wis (talk) 13:31, 12 February 2011 (UTC)

Regarding the logical form article, a 'logical constant' has much in common with a 'name', while 'logical variables' have much in common with the placeholders in an 'expression'. Please forgive me if I state the obvious. --Ancheta Wis (talk) 13:46, 12 February 2011 (UTC) upon re-reading I see that I was not using logical constant in the standard way. --Ancheta Wis (talk) 16:19, 12 February 2011 (UTC)

There are logical constants, eg ¬∨∀ non-logical constants, eg a,b,c,F,G,H and variables eg x,y,z. a,b,c are non-logical constants that stand for individual members of the domain, and are thus like names. See First-order logic#Syntax I am not familiar with this LHS and RHS stuff; is this from computer science? I am familiar with PROLOG which is based on and maps nicely to first-order logic.Philogo (talk) 17:04, 12 February 2011 (UTC)

It's actually from physics. The equation of general relativity, for example, has a left hand side which is mathematical in nature, and a right hand side which is physical in nature. But you are right, there are languages such as Thue which use the lhs/rhs concept. Even language C has the concept of lvalue. --Ancheta Wis (talk) 17:12, 12 February 2011 (UTC) --Ancheta Wis (talk) 17:12, 12 February 2011 (UTC)

I had never noticed that about the equations of general relativity; I'll check it out. Regarding definition, IMHO, terms are either (a) primitive or (b)derived terms and the derived terms are either (b1) defined from the primitives alone or (b2) from the primitive terms and/or the derived terms. What terms are selected as primitive is a matter of convenience. It is of pedagogical interest and hence to this Encyclopadia since it makes sence to explain the primitive terms and then defeine the dereived terms by refence to the primitive terms and/or other derived terms. As faras my Physics studies went, by means of dimensional analysis terms in physics could be derived take Mass, Length and Time as primitive. Thus F=M/A; A=L/T hence F=M/(L/T)=M.T.L^-1 . Simlalarly in Euclidean Geometry, the primitve temrs are/includue point, line and areas; derived temrs like figure are defined in terms of these primitives. In Logic we can take Truth as primitive. We can define entailment in terms of contradiction, and contradiction in terms of truth and logical form. It would be best if the aricles on logical form and contradiction were satisfacory so this articel could refer to them without expalaining them in this article but they are not so we cannot unless we sort them out. Philogo (talk) 19:36, 12 February 2011 (UTC)

Open world assumption

My personal interest in this page hinges on the Open world assumption. Is it feasible that this might also be included in the article as well? --Ancheta Wis (talk) 09:35, 6 February 2011 (UTC)

parallel discussion

An editor has raised objections ("I disapprove of this organization...") to the merging/editing of this article at Wikipedia talk:WikiProject Philosophy#Duplication of content and general confusion . I have suggested that such discussion should take place here on this article's talk page. Philogo (talk) 20:43, 6 February 2011 (UTC)

Logical Form

Unfortunately our article Logical Form is not a such quality that this article can rely on that article to explain matters (take a look), which is disappointing considering that it such a fundamental concept. The temptation of course is to say what needs to be said in this article but that of course sows the seeds of a future problem: if Logical Form is brought up to scratch then there will be duplication of content/possiblity of conflict between Entailment and Logical Form. (A similar problem exits/may develop (overlap/conflict) with the articles validity and contradiction. Philogo (talk) 15:18, 9 February 2011 (UTC)

Limitations

see new section Entailment#Limitations. Over to you, editors with expertise in the fields mentioned. Philogo (talk) 19:35, 12 February 2011 (UTC)

Special:WhatLinksHere/Main contention

Note that there are many links to there from articles, but that page points to a dab. Should it be redirected here? Incnis Mrsi (talk) 18:51, 8 March 2011 (UTC)

Pre-theoretical + article quality comment

This article seems to lack a discussion on the pre-theoretical notion, or existence thereof. See [4] for instance. Tijfo098 (talk) 10:58, 5 April 2011 (UTC)

The fact that it gives The Concept of Logical Consequence as reference without ever discussing its thesis makes this article funny if not suspect. Tijfo098 (talk) 11:08, 5 April 2011 (UTC)

It's just another example of our lack of writers with enough deep knowledge of philosophy to cover that concept. We end up with lots of articles on first-order logic because that's what the authors here were exposed to in school and thought was "logic consequence". This is true, unfortunately, even for some philosophically-minded editors here.

It would be nice to see this article essentially mirror the SEP article [5]. That article is a little shallow, because many topics are covered in depth in other articles on the SEP, but it hits the topics I have heard about that should be included here. But I am not particularly familiar with the philosophy literature, and I don't have it at hand, so that makes it hard for me to do much for articles like this one. — Carl (CBM · talk) 11:46, 5 April 2011 (UTC)

Tijfo098, I personally would appreciate some guidance on what you want to see: in the 326 revisions by 47 contributors over the past 8 months, there has been an attempt by us to address the needs of the readers of this article. As you can see from the diff above, the term 'pre-theoretical' was in the article 8 months ago, and was dropped during the last eight months. Perhaps you might add a note or citation on your interest. Typically a learned contributor will add concepts which are couched in specialist's language, but please don't let that stop you! The generalists among us can then use the contribution to build on, if not in this article, then perhaps in another. --As a non-specialist, 'pre-theoretical' seemed a harmless-enough word at the time, something on the order of 'intuitive', so out it went. --Ancheta Wis (talk) 13:47, 5 April 2011 (UTC)

I don't know who wrote the first three sections "Logical form" and "proof procedures", and "Relationship to other terms" but they're basically drivel—not even useful for teaching someone entailment in propositional calculus. The "Limitations" section (which is really about limitations of classical logic, not entailment) comes too soon. The "Explanation" section is what you'd expect in a discussion of entailemnt in FOL, which is fine by itself except for the title. "Accounts of logical consequence" is yet again on propostional logic, a bit better written that the first part of the article. Even if you want this to be Introduction to entailment, you should at least follow the structure of an introductory work, e.g. McKeon's IEP entry or his 2010 book ISBN 1433106450. Tijfo098 (talk) 16:27, 5 April 2011 (UTC)

Also, if you want to actually touch on the philosophical issues, Stewart Shapiro appears to be a good wp:secondary discussion, incl. Tarski vs. Etchemendy etc. (It's a defense of the model-theoretic consequence; it's hard to find someone writing on this without an opinion of his own.) Shapiro also wrote a similar piece (but I suspect with less math detail) in Jacquette's A Companion to Philosophical Logic (given "resource" in the article). So, it's not like there's no good material on this. Shapiro really likes writing about this, here's the preprint of some paper/chapter on the same issue: [6]. So, it's even free :-) Also, Etchemendy revisits the issue here (2008). There's an old draft of that on citeseerx, but it says "pls, don't quote". Tijfo098 (talk) 17:17, 5 April 2011 (UTC)

Tijfo098, thank you for replying; in defense of the current version of this article, it was not until the changes over the last 8 months were made that I finally was able to get something from the sequent calculus article, so there must be something useful in the current version. I will study the tips you have left to get what I can out of your contribution to this talk page. If it is all right with you I will post findings to this talk page, to see what you think. OK? This is with a view to adding to the current article, of course.

I should add that I am studying M. Shanahan (1997) Solving the frame problem: A mathematical investigation of the common sense law of inertia Cambridge MA: MIT Press, with an eye on the nonmonotonic logic aspects, so my view is attuned to my investigation, and not to other aspects of the article, so I am likely deaf to issues others may care about. But fire away if you catch blatant misstatements. --Ancheta Wis (talk) 20:42, 5 April 2011 (UTC)

I have restored the 2009 paragraph "philosophical issues" which touched upon pre-theoretical notion. It is establisded prcatice in the Logic articles when dealing with a technical terms, to give a straight-forward account of the terms as generally used and then, if there are philosphocal isses, then to put these in a seperate paragraph. Ancheta's remark Typically a learned contributor will add concepts which are couched in specialist's language, but please don't let that stop you! is a little depressing as it stands. A good author, no matter how learned, would surely write in a way that is accessible to his (or her) intended audience. Therefore we should always bear in mind who that audience is. For an encyclopdia aricle we should assune that the reader knows nothing about the subject's article, althjough we might assume he or she has some knowledge of the suject's area. Thus an article on "Iron" mght be written as though the reader knew nothing about the stuff, but was aware of the concepts element, metal etc. — Philogos (talk) 05:35, 9 April 2011 (UTC)

Dump merge

I see that part of the wierd structure an repetition in this article stem from having pasted this in it. (Not that it makes that much difference). Dumping together a bunch of poor quality articles will not result in a better one. Tijfo098 (talk) 12:17, 7 April 2011 (UTC)

It was agreed to merge three articles. I did the merge, revised intro and drew attention to the fact that a clean up would be needed to remove duplication - see section Merger proposal above. It would appear that nobody has followed up with the pruning. I have just deleted one section.— Philogos (talk) 03:28, 8 April 2011 (UTC)

Consequence relation

That redirects here, but this article fails to define it, or to explain the basic point that the consequence relation for a logic is usually infinite, which is why you bother with proof systems. (At least propositional proof system is not a red link). Tijfo098 (talk) 12:02, 7 April 2011 (UTC)

see first sentence:-

• In logic, entailment is a relation between a set of sentences (meaningfully declarative sentences or truthbearers) and a sentence; if Γ is a set of one or more sentences and S1 is the conjunction of the elements of Γ and S2 is a sentence then Γ entails S2 (and S2 is called the logical consequent of Γ and S1 is said to logically imply S2 ) if and only if S1 and not-S2 are logically contradictory.

Is this insufficient? — Philogos (talk) 05:00, 9 April 2011 (UTC)

Tijfo098 and Philogos, would it be OK with you if we were to add the sentence above (" the consequence relation for a logic is usually infinite, which is why you bother with proof systems") below the section on Relationship to other terms? My problem is I don't have a citation. --Ancheta Wis (talk) 07:34, 1 May 2011 (UTC)

In "the consequence relation for a logic is usually infinite, which is why you bother with proof systems" I do not understnad what it means by saying "the consequence relation .. is .. infinite". I am not familiar with this use of the term "infinite". What does it mean to say a relation is "infinite"? What other relations are infinute and which are not? Which of the follwing are "infinite" and what is the test: above, bigger, between, identical to, hotter than, loves, hates, understands, successor? Where did you get the idea that "the consequence relation for a logic is usually infinite"? If only usually, when is it not? — Philogos (talk) 22:33, 15 May 2011 (UTC)

Tijfo098, if you have a citation it looks like that is the only way that sentence is getting into the article, as Philogo does not support it as it stands. --Ancheta Wis (talk) 23:00, 15 May 2011 (UTC)

Ancheta Wis: I don't understand the sentence - do you?— Philogos (talk) 00:13, 17 May 2011 (UTC)

Philogo, My intuitive reading of the sentence is that unproven statements require an infinite chain of 'becauses' -- which is the state of the vast majority of our statements -- that is, most of what we think we know is unproven or uncertain. This is Imre Lakatos' Proofs and Refutations: ...we have to give up finality, certainty, or both.... paraphrase of Lakatos (1976) footnote 1, p.64. --Ancheta Wis (talk) 13:09, 17 May 2011 (UTC)

I salute your intuitions; your reading is somewhat clearer than the proposed sentence. I take it that the point being made is that S1 entailing S2 does not make S2 necessarily true; it only guarantees that S2 is true if S1 is true. Thus an argument with S1 and premis and S2 as conclusion (i.e. S1/S2) would be VALID but not necessarily SOUND. This is a very basic and uncontroversial point in logic, and certaibly explained in the article argument if not in this article. However if S2 is enatialed by and empty set of premises, then /s2 cannot be unsound becuase its premises (none) cannot be false. The consulusion of such an argument would then be a logical truth, a type of necessary truth. Examples would be and arguments of the form /p or not P, /not (p and not p) etc. If you are worried by the fact that the validity of such arguments depends on assued rules on inferences then you would be amused to read Lews Carroll's Achiles and the Tortoise (if you ahve not already done so). Perhaps such matters might be includued in a section entiled "limitations of enatailment" which might refer to such matters whcih are more the subject of philosophy of logic rather than logic per se. But I think such matters shoud be thus deferred to the end, just as we need to do a bit of arithmetic before we discuss the nature of numbers (i.e. do mathematics before philosophy of mathermatcs_ — Philogos (talk) 23:15, 17 May 2011 (UTC)

The Lede

I wonder if other editors think that the lede has been improved by the recent changes shown here. Does the introduction of the symbolism clarify the meaning (or does it just restrict comprehension to those familiar with the notation, without adding any meaning)? Does the substitution of the term "proposition" for "sentence" make it clearer, and if not what is the poinst of the substitution? (the existence of propositions is controversial). Is the initial use of the term "logical proposition" helpful? (how is a logical proposition different from a "proposition" - the term subsequently used)?. — Philogos (talk) 22:40, 31 July 2011 (UTC)

My edit to the latest lede was intended to correct a misstatement and is not meant to express support for the edits.

If we are discussing substance rather than style, I agree that it is misleading to confuse proposition with sentence, and also misleading to segue from logical proposition to proposition.

If symbols are used freely to stand for words, it is a small step to also use mathematical notation. If the mathematical notation for conjunction and for negation is intended to compress the wording, then I have no argument with its use; however, in the interest of clarity, at least a link to the meanings of the notation is called for.

However, if we were to discuss style and purpose as well, we run up against the issue of scale. As John Ziman has said about mathematics, a small number of mathematical statements is transparent to our thought, but as the number of mathematical statements increase, the sheer bulk of the statements becomes opaque.

It may be the major editor for the latest lede was attempting to make the lede more transparent. A matter of style. In that case, rewording to use 'sentence' instead of 'proposition' would not hurt. --Ancheta Wis (talk) 14:12, 1 August 2011 (UTC)

Do you think it is now more transparent? Is "propostion being intended as a mere synonym for (meaningful declarative) sentence? I am wondering what the intention behind the substitution was. — Philogos (talk)

Now we are definitely talking about style; the previous lede was precise in its usage of 'sentence' (that which expresses a single thought, but English is not a precise language) versus 'proposition' (which carries the connotation 'logical proposition'). I personally favor 'English sentence' because it allows constructive proofs. Propositions can create false alternatives. --Ancheta Wis (talk) 18:44, 1 August 2011 (UTC)

I do not think that you would find a RS that required the comprising sentences to be English; why not Russian of Greek?— Philogos (talk) 19:17, 1 August 2011 (UTC)

Long second sentence

Ancheta: The second sentence was long, my bad, but your worthy to simplify it/break it up has altered the sense somewhat. Look at the dif [dir http://en.wikipedia.org/w/index.php?title=Entailment&action=historysubmit&diff=444068939&oldid=443068685%7C here] What this sentence is TRYING to say is:

• Let Γ be a set of one or more sentences;
• Let S1 be the conjunction of the elements of Γ
• Let S2 be a sentence:

_________________

• then:

1. Γ entails S2 AND
2. S2 is called the logical consequent of Γ AND
3. S1 is said to logically imply S2 )

IF AND ONLY IF

• S1 and not-S2 are logically inconsistent.

Put another way (S1 and not-S2 are logically inconsistent) $\leftrightarrow$ (Γ entails S2) $\wedge$ (S2 is called the logical consequent of Γ) $\wedge$ (S1 is said to logically imply S2) — Philogos (talk) 01:55, 11 August 2011 (UTC)

Philogo, sorry for the unintended clauses. To fix this, I moved the parenthetical clauses after the main sentence. When I was re-reading the update, it occurred to me that the concrete example (roses are red etc) might serve in a new section, Inconsistency, as explanation. I will just edit the article as an example, then revert myself to show what I meant. --Ancheta Wis (talk) 03:07, 11 August 2011 (UTC)

Ancheta: Might this be better/clearer:

In logic, entailment is a relation between a set of sentences (meaningfully declarative sentences or truthbearers) and another sentence. If a sentence, S1, is the conjunction of the elements of a set of one or more sentences, Γ, then, Γ entails another sentence, S2, if and only if S1 and not-S2 are logically inconsistent. If a set of one or more sentences, Γ, entails a sentence, S2, then S2 is called the logical consequent of the conjunction of the elements of Γ, S1,and the conjunction of the elements of Γ, S1, is said to logically imply S2.

— Philogos (talk) 12:32, 11 August 2011 (UTC)

Explanation needed

It seems that entailment is not the same as material implication. Please explain the difference succinctly!Soler97 (talk) 06:11, 25 September 2011 (UTC)

Philogos is better qualified than I am to answer you, but here is my take on the difference:

1. Entailment is a relation between sentences. (now restart reading from the article)
2. Material implication is a truth function between logical propositions. (Here is where Philogos can interject about the difficulties with basing entailment on logical propositions.)

One of the difficulties, it seems, is that we use sentences to talk to each other, and not propositions.

--Ancheta Wis (talk) 08:51, 25 September 2011 (UTC)

I am still confused. How is a sentence different from a proposition? Soler97 (talk) 04:31, 27 September 2011 (UTC)

This lies at the root of my interest in entailment (and I likely differ from Philogos here); a sentence differs from a proposition because a proposition is a position taken by a rhetor about the truth of a statement, namely its truth value. In boolean logic (which I take to mean the logic of most speakers), the truth-functionality of a relation (such as the material conditional) assumes that the arguments of that relation have already been evaluated. But there are other kinds of logic (I have learned I am a constructivist by studying this article, and its links.). To me, a sentence is a thoughtbearer, rather than that which Philogos formulated in the lede as truthbearer. But there are fields in which truth is not a given, but which is something to be sought; hence my interest.

More succinctly, "a sentence bears a single thought"; a proposition bears a truth value. --Ancheta Wis (talk) 10:05, 27 September 2011 (UTC)

I remain confused. Could you give an example of a sentence and of a proposition, where the two are very similar except for the sentence/proposition difference? I'm sure I am not the only person who would like to see an explanation of the difference between entailment and material implication. Once it is clearly spelled out it should be added to the article. Soler97 (talk) 23:14, 27 September 2011 (UTC)

Sure, but you are going to have to do some work digesting it into the form you request. I am providing the example:

My citation is from John L. Casti, (1998) The Cambridge Quintet: a work of scientific speculation ISBN 0-7382-0138-3 183 pages.

The book is written in the style of a shōsetsu: a speculation, involving a hypothetical dinner party hosted by C. P. Snow, to which he has invited J. B. S. Haldane, Ludwig Wittgenstein, Erwin Schrödinger, and Alan Turing. Snow's task is to write a report to a Government Ministry about the capabilities of the new 'thinking machines' of the late 1940s. Casti's speculation involved the construction of multiple thought experiments proposed (as sentences) by the respective scientists, which embody a series of questions, positions, and rebuttals of Alan Turing's propositions. Snow encapsulates this in Turing's question (p.166) as "is there any logical reason why we cannot envision technology advancing to the point where we could construct a computing machine that would be indistinguishable from a human being in its cognitive capabilities?". Wittgenstein's position (pp. 166-167) is "no, because human thinking is completely tied up with language ...". Wittgenstein p.167 states this entails participation in human life (which was inconceivable back at a time when computing machines were room-sized). Schrödinger p. 168 is forced to side with Turing, while Haldane p. 168 delivers the ancient Scottish verdict of not proven.

Each of the sentences with the exception of Haldane's judgement is a proposition. Even the statement involving Turing's question is a sentence with a question embedded. Haldane's verdict is a sentence, which cannot ever be a proposition. The speculation by Casti itself evades the form of a proposition, forcing the reader to evaluate the book for himself.

In contrast, a material conditional involves resolution of the arguments down to true or false. --Ancheta Wis (talk) 01:05, 28 September 2011 (UTC)

I thought of another example, in subjunctive mood. In this case, it is not necessarily true (irrealis). Furthermore the sentence does not have to be true:

• "May Governor Christie agree, that he run."

Now compare to the logical proposition:

• "Governor Christie will not agree to run".

My point is that an English sentence can have the capacity to express some things that a true/false up/down proposition cannot ever express, because a proposition will necessarily miss some subtleties which can be captured in the irrealis mood. --Ancheta Wis (talk) 02:31, 28 September 2011 (UTC)

Are you saying that some sentences are propositions but that others are not? Also, by "sentence" do you mean the ordinary meaning of this word in English grammar, or is it some technical sense that has not been specified? Soler97 (talk) 23:43, 28 September 2011 (UTC)

Please read the links, if you have questions: "a sentence bears a single thought"; a proposition bears a truth value. Yes, absolutely, a proposition is a sentence, but not vice versa. No, I am not trying to use specialized terminology, but I am assuming that you know the subjunctive mood, from school perhaps. --Ancheta Wis (talk) 00:04, 29 September 2011 (UTC)

If I understand you correctly, a sentence is just a grammatically correct utterance in a natural language. As such, it can express a wish, a question, a command, feeling a pain, a piece of sheer nonsense etc. This is very confusing, as the article on entailment is an article on logic, not linguistics. There is another use of "sentence" in mathematical logic ie http://en.wikipedia.org/wiki/Sentence_%28mathematical_logic%29 Are you certain this is not the one referred to in the entailment article? To talk about the logical consistency of questions, commands, utterances of pain etc seems nonsensical to me.

The article should be intelligible on its own. It should not require the lay reader to refer to links, except for definitions of technical terms. Soler97 (talk) 22:59, 29 September 2011 (UTC)

So black swans are nonsense? We ignore 'nonsense' at our peril. I agree that a truth-functional statement has been purified sufficiently to view as a declarative sentence. But to discount the sense which is entailed in a natural, linguistic statement is imprudent and unsound. For example, when a physician speaks with a patient, that physician indeed uses logic to solve the case, but other possibilities have to be considered, including the symptoms that present themselves with the patient's utterances of pain. --Ancheta Wis (talk) 00:56, 30 September 2011 (UTC)

By the way, the categories at the foot of this talk page show that the purview of this article includes linguistics. --Ancheta Wis (talk) 17:44, 30 September 2011 (UTC)

During my involvement with this article, in which I have observed various editors attempt clarification/exposition, I have not seen a linguistic assessment of the various related terms vis a vis entailment, such as the material conditional. It is now becoming clear to.me that you appear to posit your original question from the POV of hindsight, where truth values are givens rather than TBDs (to be determined)s. That would explain to me how you could posit that all mathematical sentences have truth values. Entailment takes a different position. In the process of entailment, an inquiry is taking place before us, during which we examine arguments (that is, evidence ) for their truth value. This is intrinsically non-rhetorical because an investigation is occurring, which is distinctly different from a truth-functional which takes a rhetorical stance true or false, but not TBD. In entailment we admit things can be TBD, and seek to evaluate or at least admit we don't know yet, or cannot know because, ... etc. You get the gist.

Now compare this to material conditional, where the arguments have known values. As you can see, an investigation must already have occurred. It is a situation where hindsight reigns over a done deal. --Ancheta Wis (talk) 15:39, 30 September 2011 (UTC)

Can you comment on this, please: The difference between material implication and logical implication is contextual. The first is a statement of logic, the second of metalogic. The difference between "p implies q" and "p is a proof of q" is that the first is a statement within formal logic, the second is a statement about it. Soler97 (talk) 01:10, 1 October 2011 (UTC)

Hmmm. A lot is left unsaid here.

Anyway, according to H.S. Wall (1963) Creative Mathematics Austin:University of Texas Press, "a proof is a succession of statements, each of which leads to the next."

When I read your request above, I interpreted the p and q as single statements. How are single statements meant to develop into anything meaningful? If q is a goal, then at the very least, p denotes a starting point. I see no motivating machinery or structure. If p and q denote the names of sets of statements, or perhaps denote hyperlinks to strings, then a proof has succeeded when a rewriting system has succeeded in transforming p into q.

My thinking about this has been influenced by the Thue programming language, and I am assuming machinery of that type in this response to you. There is even a Javascript implementation. Perhaps this is the kind of context you refer to?

Otherwise Lakatos (1976) Proofs and Refutations is meaningful to me.

As it stands, I cannot accept the formulation you posit, without more information.

--Ancheta Wis (talk) 05:50, 1 October 2011 (UTC)

Thanks for your efforts, but I think that (a) each of us is unable to understand what the other is saying and that further attempts at clarification are not going to lead anywhere, and that (b) you have failed to give a clear criterion of the difference between material implication and entailment. From the point of view of the lay reader, this is a big hole in the article. Perhaps someone else is willing to try. I would insert the formulation above (with terms defined) into the article, but no doubt you would immediately delete it. Soler97 (talk) 21:57, 2 October 2011 (UTC)

Fellow editors, I am working on another article and would appreciate someone else picking up the ball here. Specifically we need more citations to back up the latest contribution, per protocol. Thanks, --Ancheta Wis (talk) 02:25, 10 October 2011 (UTC)

• I tagged the section Entailment vs material implication with a "no reference or sources tag" as being challenged in expectation of references.

The edit summary seeks improvement and not deletion of the added content. If there are no references I will be forced to delete the information as original research, which actually prohibits such edits. Editors wishing to contribute to Wikipedia must do so according to Wikipedia policies and guidelines. Submitting material that lacks references can be deleted.

• Policies and guidelines that support removal of content without a source or reference.

Adding information to Wikipedia: "Unsourced information may be challenged and removed, because on Wikipedia a lack of information is better than misleading or false information—Wikipedia's reputation as a trusted encyclopedia depends on the information in articles being verifiable and reliable."

Try to fix problems: "Fix problems if you can, flag or remove them if you can't.".

Burden of evidence: "You may remove any material lacking a reliable source that directly supports it (although an alternate procedure would be to add a citation needed tag).".

Wikipedia verifiability policy states, "Anything that requires but lacks a source may be removed,..."

Wikipedia is not a publisher of original thought and includes, "... please do not use Wikipedia for any of the following:"

• "Primary (original) research, such as proposing theories and solutions, original ideas, defining terms, coining new words, etc."

Wikipedia is not a textbook: "Wikipedia is an encyclopedic reference, not a textbook. The purpose of Wikipedia is to present facts, not to teach subject matter.".

If references can not be supplied please be advised that the NOR noticeboard states, "If the content in question includes unpublished facts, arguments, speculation, and ideas; and any unpublished analysis or synthesis of published material that serves to advance a position, it may not be published on Wikipedia.". Otr500 (talk) 08:27, 26 December 2011 (UTC)

User:Soler97, three months ago, attempted to add value to this article and this section. I attempted to help, but only now have I found a relevant citation, which entails a complete rewrite of the section. (Unfortunate pun, eh?)

• Russell, Stuart J.; Norvig, Peter (1995), Artificial Intelligence: A Modern Approach, Prentice-Hall, pp. 158–174, especially 158, 163, and 174, ISBN 0-13-103805-2

Perhaps the editors of this page might attempt a rewrite of this section. --Ancheta Wis (talk) 17:25, 26 December 2011 (UTC)

I am one of the "lay reader's" that has been referenced above but I do also edit Wikipedia. I have an issue with two sentences in the article.

• A)- "They are logically inconsistent because as a result of their logical form they cannot be both true, their logical forms being p and q and not-q." (in the lead), and
• B)- "...are not logically inconsistent because they can both be true as a result of their logical form, their logical form being p and not-q.".

This content is just dropped in out of the blue. There is no information or links to inform the reader of this content. Would someone connect the dots for clarity, and so readers will have a better idea of the contents? References would also be important? Otr500 (talk) 00:30, 26 December 2011 (UTC)

Thank you for your comment. There is a rule that parses A above into two parts: 1 (p and q) and 2 (not-q).

(The rule is that NOT binds more tightly than AND).

In order for 1 (p and q) to be true, both its sub-parts, p, q, have to be true simultaneously. Thus 1 (p and q) contradicts 2 (not-q) being true. Thus A is inconsistent.

Similarly, B parses as a single form 1 (p and not-q). Thus B is not inconsistent.

Note that not inconsistent requires the law of the excluded middle be true, which is not the case for intuitionistic logic.

Hope this helps. --Ancheta Wis (talk) 18:01, 26 December 2011 (UTC)

Entailment corresponds exactly to tautological consequence, not logical consequence?

Doesn't S1 entail S2 if and only if S2 is a tautological consequence of S1? That's what I got out of Barwise and Etchemendy's Language, Proof and Logic (2008), pages 110-113. This article as wrriten seems to imply that S1 entails S2 if and only if S2 is a logical consequence of S1. There are logical consequences that aren't necessarily tautological consequences. Isn't the introductory example where "John is a bachelor" does not entail "John is a man" an example of "John is a man" not being a tautological consequence of "John is a bachelor," but still a logical consequence? It seems as if this article makes several references to logical consequence when it should actually be referencing tautological consequence. Entailment should correspond exactly to tautological consequence, not logical consequence. Hanlon1755 (talk) 07:51, 31 December 2011 (UTC)

"Logical consequence" is correct. -- 202.124.74.208 (talk) 12:42, 17 January 2012 (UTC)

You are ridiculous! You haven't explained yourself at all! I have concerns about that claim, and you haven't addressed them one bit! Hanlon1755 (talk) 18:18, 17 January 2012 (UTC)

WP:NPA. Stop edit warring. No one owes you an explanation for why you're wrong.—Machine Elf ¹⁷³⁵ 19:01, 17 January 2012 (UTC)

Greetings Hanlon, I think the fact that logical consequence was wrongfully merged with entailment is the cause of the some of these problems. Would you agree with me that logical consequence should be its own article, consistent with the category of the same name? I find that in the literature that I read, the fundamental concept that they are dealing with is logical consequence, and all the other terms are attempts to model it in different ways.Greg Bard (talk) 19:05, 17 January 2012 (UTC)

Yes, I do agree with you that logical consequence should be its own article. But I believe that article already exists as Conditional statement (logic). It is my understanding, after reading Barwise and Etchemendy, that the notion of logical consequence matches that of implication, which is expressed as the conditional statement. Tautological consequence seems to match that of entailment, given the example in the lead of entailment. Whatever the case may be though, there is nothing on Wikipedia about tautological consequence. A contribution of some sort: an article modification or creation, does seem warranted. Hanlon1755 (talk) 19:30, 17 January 2012 (UTC)

A statement B is a logical consequence of a statement A if and only if the material conditional A→B is a logical validity. The same holds for tautological consequence and tautological validity. "Entailment" as a term is not used much in modern mathematical logic, at least. I do see it in Curry's Foundations of Mathematical Logic (1963) where he uses "entailment" to mean the natural-language implication, compared to the truth functional material implication. — Carl (CBM · talk) 20:08, 17 January 2012 (UTC)

At some point I will propose that we move it back. I will let you know for sure. Any help in this regard would be appreciated. A conditional statement is a particular kind of instance of a logical consequence, but it is not synonymous. (I reverted your re-redirecting of l.c., so I hope you can forgive that). I think "tautological consequence" deserves to at least be a section in an article l.c., and perhaps can stand on its own as an article too. I am certainly looking forward to your contributions. Greg Bard (talk) 20:14, 17 January 2012 (UTC)

It is clearly stated in Barwise and Etchemendy that tautological consequence is "a strict form of logical consequence" (110), but that "Not every logical consequence of a sentence is a tautological consequence of that sentence" (111). Seeing this I have created a separate article for tautological consequence. Hanlon1755 (talk) 20:24, 17 January 2012 (UTC)

I think it's a wonderful start of an article. I invite your correspondence on these and any other matters.Greg Bard (talk) 20:43, 17 January 2012 (UTC)

There are very easy examples of this. For example $\lnot \exists xP(x)$ is a logical consequence of $\forall x\lnot P(x)$, but not a tautological consequence. There is more about this at tautology (logic). — Carl (CBM · talk) 20:46, 17 January 2012 (UTC)

Actually, within predicate calculus that is a tautological consequence in the sense that the phrase is being used here, and the proof is quite elementary. -- 202.124.73.16 (talk) 23:26, 17 January 2012 (UTC)

By "tautological consequence" I mean "substitution instance of a tautology from propositional logic". Unless "tautological consequence" is supposed to be a synonym for "logical consequence", there are many things that are provable in first-order logic but not tautological consequences, since by the completeness theorem provability corresponds to logical consequence in first-order logic. — Carl (CBM · talk) 02:52, 18 January 2012 (UTC)

Given that definition, you are correct, but there is an extended definition of tautological consequence for predicate calculus/first-order logic (see e.g. here), and on that extended definition you would not be correct. It would certainly be a mistake for Wikipedia to assume your definition is the only one. -- 202.124.75.156 (talk) 05:30, 19 January 2012 (UTC)

The definition you linked is the same one I am referring to, which is the same one described in "Tautologies versus validities in first-order logic" in tautology (logic). Under that definition $((\forall x)\lnot P(x))\to \lnot (\exists x)P(x)$ is a substitution instance of $A\to B$, which is not a tautology. The book you linked has a similar example on page 390. — Carl (CBM · talk) 12:27, 19 January 2012 (UTC)

That certainly is a widely used definition of tautology, but so is the use of "tautology" to mean "valid in every model"[7]. -- 202.124.73.25 (talk) 22:33, 19 January 2012 (UTC)

Actually, the usage of "tautology" to mean "first-order validity" is not very common at all. As is true with all terminology in the field there must be someone who does it differently, but the literature overall is pretty consistent on that point, if you look across the entire literature. — Carl (CBM · talk) 22:40, 19 January 2012 (UTC)

My thanks to User:Incnis Mrsi for reverting my inadvertent change to this page, due to an inadvertent touch on a slow tablet. Sorry. --Ancheta Wis (talk) 09:44, 20 January 2012 (UTC)

I don't think anyone here suggested that "tautology" meant "first-order validity," but rather that "X is tautology" is used to mean $\models X$ across a wide range of different logical systems. The definition given in Tautology (logic) is "a tautology is a formula which is true in every possible interpretation." This follows e.g. Ayer in Language, Truth, and Logic. See also completeness. Using "tautology" to refer only to what is deducible from propositional-calculus inference rules is probably the rarer usage, though occurring in several elementary texts. -- 202.124.75.182 (talk) 06:41, 22 January 2012 (UTC)

Lead example mistakenly refers to logical consequence when it should refer to tautological consequence?

It appears that the lead example involving John, being a bachelor, and being a man, is actually an example not of logical consequence as stated, but of tautological consequence. That is, if Γ = {“John is a bachelor”}, S1 = “John is a bachelor” and S2 = “John is a man,” then S2 is not a tautological consequence of Γ. S2 is still, however, a logical consequence of Γ. Comments, concerns? Hanlon1755 (talk) 00:44, 22 January 2012 (UTC)

If we translate "John is a bachelor" into first-order logic as B(J) and "John is a man" as M(J), then the latter is not entailed by the former unless we also assume a non-logical axiom such as $(\forall x)(B(x)\to M(x))$. I think the text currently in the lede is meant to point out that "John is a bachelor" does not entail "John is a man".

The deeper problem is that there are many kinds of "entailment" i.e. "logical consequence", just as there are many distinct logics. A better place to start is the SEP article at [8]. — Carl (CBM · talk) 00:56, 22 January 2012 (UTC)

Slow moving content fork debate

I understand that Logical consequence was merged into this page. Above, the proposal to rename this page "Logical consequence" was rejected. In the past two months, there have been repeated attempts to restore Logical consequence as a separate article, with repeated reversals of this move. This looks to me like an attempt to content fork? Should this issue be reopened and resolved one way or another so the extended reverting can come to an end? Good Ol’factory ^(talk) 02:49, 29 February 2012 (UTC)

I have reverted that change because of your project O.G. This was moved without a significant discussion, and it was a bad move. Logical consequence is one of the most fundamental concepts in logic. It appears by that name in books and journals, and yet people do what they do based on their experience and we get these kinds of messes. Greg Bard (talk) 03:07, 29 February 2012 (UTC)

I'm not sure what "your project O.G." means above. But the problem with what you say is this—you proposed that the article be moved to "Logical consequence", and that proposal was rejected above. So you can't just say "it was a bad decision" and do what you want anyway. That's why I have suggested a re-opening of the issue, so both sides can be heard. In the meantime, it helps if you stop forcing the article to the form you want so that others who want to comment can get a handle on the situation. Good Ol’factory ^(talk) 03:27, 29 February 2012 (UTC)

I'm going to make a request at ANI that you be stopped from moving philosophy categories. I would like to work with you, but you don't listen, and you seem to get insulted at the idea that you might not know what you are doing. Again, please stop. Greg Bard (talk) 03:33, 29 February 2012 (UTC)

WTH? I have never moved a philosophy category unilaterally. Every move has been done via proposal by me and then by consensus agreement—completely in line with procedure. I don't see what that issue has to do with the one here, apart from both are philosophy-related. General comments like this—if they need to be made at all—should be placed on my user talk page, not on an article talk page. Good Ol’factory ^(talk) 03:35, 29 February 2012 (UTC)

See response at G.O.'s talk page. --gb

Move back to logical consequence

The following discussion is an archived discussion of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the proposal was moved. --BDD (talk) 16:37, 31 October 2012 (UTC) (non-admin closure)

Entailment → Logical consequence –

A while back, the article Logical consequence was moved to Entailment. I subsequently proposed to move it back. However, it was not. I have always found this to be a very troubling development. Recently, I have been adding reference resources to articles and categories consistent with those resources. This is one that is not consistent (and I said it at the time.) Please take a look at SEP, InPho, PhilPapers, and IEP, none of which has an article on "entailment" independent of "logical consequence." The article itself is a bit scattered, and this has been a big stumbling block for me to improving it. It is only one of the most important concepts in logic. Please support this move, as it is consistent with the scholarly literature on the subject, and Wikipedia is the odd resource out in this regard. Greg Bard (talk) 07:55, 24 October 2012 (UTC)

• It looks like that was a merge, not a move. I see not much progress has been made on this article's content. I don't have an opinion for now as to what title is more common. Tijfo098 (talk) 08:33, 24 October 2012 (UTC)
• Who would have thought there could be so much drama over a mathematico-logical operator? My intuition is that "logical consequence" is a better name for the article. Greg Bard is correct that it's more common within the discipline. "Logical consequence" is furthermore much more unambiguous (one might even say it is "less ambiguous") than "entailment", which can mean a bunch of other stuff. The argument for "entailment" is that it more clearly relates to the operation as opposed to the product. IMO "logical implication" might be the best title of all for these reasons. Incidentally it seems like this article is kind of running off the rails, based on some special definition for "entailment" that is different than what (from my perspective) "logical consequence" (or logical implication) means; in which case maybe they should be split again? Forgive me if I'm totally off-base here, this article is so weird I'm beginning to doubt that I even understand what it's actually 'about'. Peace, groupuscule (talk) 08:56, 24 October 2012 (UTC)
• Support the move based on discussion with Greg Bard. If these pages are going to be merged, they should live at the most recognizable version. If "entailment" is a different/special thing involving the relationship between sentences and operators this can be clarified in a section of the page. Convinced that "logical consequence" can be used instead of "logical implication" based on what's popular in the current discourse, even if personally I'd prefer the latter. Also not seeing a lot of energy from anyone else working on the page who feels it should stay at "entailment". groupuscule (talk) 09:21, 28 October 2012 (UTC)

The above discussion is preserved as an archive of the proposal. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.