Logical consequence

A priori property

If it is known that ${\displaystyle Q}$ follows logically from ${\displaystyle P}$, then no information about the possible interpretations of ${\displaystyle P}$ or ${\displaystyle Q}$ will affect that knowledge. Our knowledge that ${\displaystyle Q}$ is a logical consequence of ${\displaystyle P}$ cannot be influenced by empirical knowledge. Deductively valid arguments can be known to be so without recourse to experience, so they must be knowable a priori. However, formality alone does not guarantee that logical consequence is not influenced by empirical knowledge. So the a priori property of logical consequence is considered to be independent of formality.

Proofs and models

The two prevailing techniques for providing accounts of logical consequence involve expressing the concept in terms of proofs and via models. The study of the syntactic consequence (of a logic) is called (its) proof theory whereas the study of (its) semantic consequence is called (its) model theory.

Syntactic consequence

A formula ${\displaystyle A}$ is a syntactic consequence within some formal system ${\displaystyle \mathcal{F}\mathcal{S}}$ of a set ${\displaystyle \mathrm{\Gamma }}$ of formulas if there is a formal proof in ${\displaystyle \mathcal{F}\mathcal{S}}$ of ${\displaystyle A}$ from the set ${\displaystyle \mathrm{\Gamma }}$. This is denoted ${\displaystyle \mathrm{\Gamma }{⊢}_{\mathcal{F}\mathcal{S}}A}$. The turnstile symbol ${\displaystyle ⊢}$ was originally introduced by Frege in 1879, but its current use only dates back to Rosser and Kleene (1934–1935).

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

Semantic consequence

A formula ${\displaystyle A}$ is a semantic consequence within some formal system ${\displaystyle \mathcal{F}\mathcal{S}}$ of a set of statements ${\displaystyle \mathrm{\Gamma }}$ if and only if there is no model ${\displaystyle \mathcal{I}}$ in which all members of ${\displaystyle \mathrm{\Gamma }}$ are true and ${\displaystyle A}$ is false. This is denoted ${\displaystyle \mathrm{\Gamma }{\models }_{\mathcal{F}\mathcal{S}}A}$. Or, in other words, the set of the interpretations that make all members of ${\displaystyle \mathrm{\Gamma }}$ true is a subset of the set of the interpretations that make ${\displaystyle A}$ true.

Modal accounts

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

${\displaystyle \mathrm{\Gamma }}$ ${\displaystyle ⊢}$ ${\displaystyle A}$ is true if and only if it is necessary that if all of the elements of ${\displaystyle \mathrm{\Gamma }}$ are true, then ${\displaystyle A}$ is true.

Alternatively (and, most would say, equivalently):

${\displaystyle \mathrm{\Gamma }}$ ${\displaystyle ⊢}$ ${\displaystyle A}$ is true if and only if it is impossible for all of the elements of ${\displaystyle \mathrm{\Gamma }}$ to be true and ${\displaystyle A}$ false.

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

${\displaystyle \mathrm{\Gamma }}$ ${\displaystyle ⊢}$ ${\displaystyle A}$ is true if and only if there is no possible world at which all of the elements of ${\displaystyle \mathrm{\Gamma }}$ are true and ${\displaystyle A}$ is false (untrue).

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 not 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:

${\displaystyle \mathrm{\Gamma }}$ ${\displaystyle ⊢}$ ${\displaystyle A}$ if and only if it is impossible for an argument with the same logical form as ${\displaystyle \mathrm{\Gamma }}$/${\displaystyle A}$ to have true premises and a false conclusion.

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.

Non-monotonic logical consequence

The accounts discussed above all yield monotonic consequence relations, i.e. ones such that if ${\displaystyle A}$ is a consequence of ${\displaystyle \mathrm{\Gamma }}$, then ${\displaystyle A}$ is a consequence of any superset of ${\displaystyle \mathrm{\Gamma }}$. 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}.

Resources

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• A paper on 'implication' from math.niu.edu, Implication Archived 2014-10-21 at the Wayback Machine
• A definition of 'implicant' AllWords