Entailment

In logic, entailment is a relation between a set of sentences (e.g.,[1] meaningfully declarative sentences or truthbearers) and a sentence. Let Γ be a set of one or more sentences; let S1 be the conjunction of the elements of Γ, and let S2 be a sentence: then, Γ entails S2 if and only if S1 and not-S2 are logically inconsistent. S2 is called a tautological consequence of Γ; S1 is said to logically imply S2.

Two sentences are inconsistent if and only if they cannot both be true; they are logically inconsistent if and only if they are inconsistent as a result of their logical form.

Thus if

Γ = {“Roses are red”, “Violets are blue”},
S1 = “Roses are red and Violets are blue”

and

S2 = “Violets are Blue”

then Γ entails S2, because

“Roses are red and Violets are blue”

and

“Violets are not blue”

are logically inconsistent. 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.

On the other hand if

Γ = {“John is a bachelor”},
S1 = “John is a bachelor” and
S2 = “John is a man”

then Γ does not entail S2 because

S1, “John is a bachelor”

and

(not)S2, “John is not a man”

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.

Contents

Logical form

Main article: logical form

Given Γ is a set of one or more declarative sentences.

Since

and

it follows that

It is therefore of the first importance to clarify the term logical form and explain how the logically relevant form(s)[2] of a sentence can be established.

The logical form of sentences can be revealed by means of a formal language enabling the following definition of entailment. Roughly, if S1 and S2 are interpretations of two sentences θ and ψ in a formal language of classical logic, then S1 entails S2 if and only if not (θ and not ψ) is true under all interpretations.

More precisely, 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, Γ entails S2 if and only if not (S1 and not-S2) is a logical truth. S2 is called the 'logical consequent' of Γ. S1 is said to 'logically imply' S2.

Not (S1 and not-S2) is a logical truth if θ and Ψ are closed well-formed formulae (often denoted 'wff'), wffs (sentences) in a formal language L in classical logic, and I is an interpretation of L, and θ is true under I if and only if S1 and Ψ is true under I if and only if S2, and not (θ and not Ψ) is logically valid.

A closed wff Φ in L is 'logically valid' if and only if Φ is true under all interpretations of L. Hence

Thus if Γ = {“Roses are red”, “Violets are blue”}, S1=“Roses are red and Violets are blue” and S2 = “Violets are Blue” then Γ entails S2 because not(S1 and not-S2), “It is not the case that roses are red and violets are blue and violets are not blue” is a logical truth.

Not(S1 and not-S2) is a logical truth because there are two closed wfs, P&Q and Q in a formal language L in classical logic and there is an interpretation I of L, and P&Q is true under I if and only if roses are red and violets are blue, and Q is true under I if and only if violets are blue, and ¬((P&Q)&¬Q) is logically valid. ¬((P&Q)&¬Q)) is logically valid because it is true under all interpretations of L (note that ¬ means not).

It will be noted that, on these definitions, if (i) S1 is inconsistent (self-contradictory) or (ii) not-S2 is inconsistent (self-contradictory) then (S1 and not-S2) is inconsistent (not consistent) and hence S1 entails S2.

Proof procedures, and relationship between entailment and deduction

It is of considerable interest to be able to prove that Γ entails S2 and hence that Γ/S2 is a valid argument. Ideally, entailment and deduction would be extensionally equivalent. However, this is not always the case. In such a case, it is useful to break the equivalence down into its two parts:

A deductive system S is complete for a language L if and only if A \models_L X implies A \vdash_S X: that is, if all valid arguments are deducible (or provable), where \vdash_S denotes the deducibility relation for the system S. NB A \models_L X means that X is provable from A for a language L, and A \vdash_S X means that X is a semantic consequence of A for the system S.

A deductive system S is sound for a language L if and only if A \vdash_S X implies A \models_L X: that is, if no invalid arguments are provable.

Many introductory textbooks (e.g. Mendelson's "Introduction to Mathematical Logic") that introduce first-order logic, include a complete and sound inference system for the first-order logic. In contrast, second-order logic — which allows quantification over predicates — does not have a complete and sound inference system with respect to a full Henkin (or standard) semantics.

Proof of entailment

Since

a proof that not (θ and not Ψ) is logically valid would be a proof that Γ entails S2.

It can be easily demonstrated, for example by means of a truth-table, that ¬((P & Q) & ¬Q) is a tautology and hence true under all interpretations and hence logically valid. Moreover, if T is a consistent theory in L and ¬(θ ∧ ¬Ψ) is a theorem in T (written ⊢T¬(θ ∧ ¬Ψ)) then ¬(θ ∧ ¬Ψ) is logically valid and, consequently, all interpretations of ¬(θ ∧ ¬Ψ) are logical truths, including not(S1 and not-S2). Hence Γ entails S2 if ⊢T¬(θ ∧ ¬Ψ) and T is consistent.

Relationship to other terms

Entailment is one of a number of inter-related terms of logical appraisal. Its relationship to other such terms includes the following see e.g. Strawson (1952)[3] Section 13, 'Entailment and Inconsistency', pp 19 et seq) where S1 and S2 are sentences, or S1 is the conjunction of all the sentences in some set of sentences Γ, S1 entails S2 if and only if:

  1. S1 and not-S2 are inconsistent (not consistent)
  2. (S1 and not-S2) is logically impossible
  3. Not(S1 and not-S2) is logically true
  4. Not(S1 and not-S2) is necessarily true
  5. (S1 and not-S2) is logically false
  6. S1 is a sufficient condition for S2
  7. S2 is a necessary condition for S1
  8. It would be self-contradictory to affirm S1 and deny S2
  9. An argument whose premise is S1 and conclusion is S2 is a valid argument
  10. If S1 and S2 are interpretations of two sentences θ and ψ in a formal language of classical logic (elementary logic), then S1 entails S2 if and only if not (θ and ¬ψ) is true in all interpretations.

Syntactic consequence

A formula A is a syntactic consequence[4][5][6][7] within some formal system FS of a set Г of formulas if there is a formal proof in FS of A from the set Г.

\Gamma \vdash_{\mathrm FS} A

Syntactic consequence does not depend on any interpretation of the formal system.[8]

Semantic consequence

A formula A is a semantic consequence of a set of statements Г

\Gamma \models A,

if and only if no interpretation \mathcal{I} makes all members of Г true and A false.[9] Or, in other words, the set of the interpretations that make all members of Г true is a subset of the set of the interpretations that make A true.

Entailment vs strict implication

The difference between strict implication and entailment is that entailment is a property of argument forms. The strict implication, on the other hand, is independent of argument forms.[10] For example, as demonstrated earlier, Γ = {"John is a bachelor."} does not entail "John is a man," because the argument forms of the sentences do not provide an entailment relation. On the other hand, the strict implication "John is a bachelor implies John is a man" is true; bachelors are men by definition, so there is no possible case in which John is a bachelor, but John is not a man.[11]

Entailment vs material implication

The difference between material implication and entailment is that they apply in different contexts. The first is a statement of logic, the second of metalogic. If p and q are two sentences then 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. Entailment is a concept of proof theory, [discuss] whereas material implication is the mechanics of a proof.

Limitations

Entailment is one form but not the only form of inference. Inductive reasoning is another. Scientific method involves inferences that are not solely entailment. Entailment does not encompass non-monotonic reasoning or defeasible reasoning. See also

Modal accounts

Modal accounts of logical consequence are variations on the following basic idea:

Alternatively (and, most would say, equivalently):

Such accounts are called "modal" because they appeal to the modal notions of necessity and (im)possibility. 'It is necessary that' is often cashed out as a universal quantifier over possible worlds, so that the accounts above translate as:

Consider the modal account in terms of the argument given as an example above:

All frogs are green.
Kermit is a frog.
Therefore, Kermit is green.

The conclusion is a logical consequence of the premises because we can't imagine a possible world where (a) all frogs are green; (b) Kermit is a frog; and (c) Kermit is not green.

Modal-formal accounts

Modal-formal accounts of logical consequence combine the modal and formal accounts above, yielding variations on the following basic idea:

Most logicians would probably agree that logical consequence, as we intuitively understand it, has both a modal and a formal aspect, and that some version of the modal/formal account is therefore closest to being correct.

Warrant-based accounts

The accounts considered above are all "truth-preservational," in that they all assume that the characteristic feature of a good inference is that it never allows one to move from true premises to an untrue conclusion. As an alternative, some have proposed "warrant-preservational" accounts, according to which the characteristic feature of a good inference is that it never allows one to move from justifiably assertible premises to a conclusion that is not justifiably assertible. This is (roughly) the account favored by intuitionists such as Michael Dummett.

Non-monotonic logical consequence

Main article: Non-monotonic logic

The accounts discussed above all yield monotonic consequence relations, i.e. ones such that if A is a consequence of Γ, then A is a consequence of any superset of Γ. It is also possible to specify non-monotonic consequence relations to capture the idea that, e.g., 'Tweety can fly' is a logical consequence of

{Birds can typically fly, Tweety is a bird}

but not of

{Birds can typically fly, Tweety is a bird, Tweety is a penguin}.

For more on this, see belief revision#Non-monotonic inference relation.

Philosophical issues

It is impossible to state rigorously the definition of 'logical implication' as it is understood pretheoretically, but many have taken the Tarskian model-theoretic account as a replacement for it. Some, e.g. Etchemendy 1990, have argued that they do not coincide, not even if they happen to be co-extensional (which Etchemendy believes they are not). This debate has received some recent attention. See "The Blackwell Guide to Philosophical Logic",[13] for a good introduction to it.

\begin{matrix} p \rightarrow q & \quad & \quad & p \Rightarrow q \\ \mbox{not}\ p \ \mbox{or}\ q & \quad & \quad & p \ \mbox{implies}\ q \end{matrix}

See also

Notes

  1. ^ It should be noted that entailment can be employed in sentences which are not necessarily truthbearers, but also to express contingencies in alethic modalities: imagined situations, evocative situations, or in situations involving necessity, duty, or the law.
  2. ^ We should not assume that a sentence has just one logical form. If T1 ="If Greeks are men and men are mortal then Greeks are men", we could say that the logical form of T1 is "IF P and Q THEN P", and thus the logical form of ."Greeks are men" is "P". If T2 is ="If Greeks are men and men are mortal then Greeks are mortal", we could say that the logical from of T2 is "If all A is B and all B is C then all A is C", and thus the logical form of "Greeks are men" is "All A is B". Hence we cannot say that THE logical form of "Greeks are men" is "P" nor "All A is B".
  3. ^ Strawson, P.F. (1952). Introduction to Logical Theory. Methuen. 
  4. ^ Dummett, Michael (1993) Frege: philosophy of language Harvard University Press, p.82ff
  5. ^ Lear, Jonathan (1986) Aristotle and Logical Theory Cambridge University Press, 136p.
  6. ^ Creath, Richard, and Friedman, Michael (2007) The Cambridge companion to Carnap Cambridge University Press, 371p.
  7. ^ FOLDOC: "syntactic consequence"
  8. ^ Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971, p. 75.
  9. ^ The Cambridge Dictionary of Philosophy
  10. ^ Larson et al. 2007, pp. 79-80
  11. ^ Larson et al. 2007, pp. 79-80
  12. ^ See for example Davis 1965
  13. ^ Goble 2001 Chapter 6

Resources

External links

Negation \neg · XOR \oplus  · Biconditional \leftrightarrow  · Statement