Logical consequence

Logical consequence, arguably the most fundamental concept in logic, is the relation that holds between a set of sentences (or propositions) and a sentence (proposition) when the latter "follows from" the former. For example, Kermit is green is a logical consequence of All frogs are green and Kermit is a frog.

A formally specified logical consequence relation may be characterized model-theoretically or proof-theoretically (or both).

Logical consequence can also be expressed as a function from sets of sentences to sets of sentences (Tarski's preferred formulation), or as a relation between two sets of sentences (multiple-conclusion logic).

Accounts of logical consequence

This section describes some common accounts of logical consequence.

Γ will represent an arbitrary set of premises and A an arbitrary conclusion. Γ/A will denote the logical argument having Γ as its (set of) premises and A as its conclusion. Γ $\vdash$ A will mean that A is a logical consequence of Γ.

Modal accounts

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

Γ $\vdash$ A just in case it is necessary that if all of the elements of Γ are true, then A is true.

Alternatively (and, most would say, equivalently):

Γ $\vdash$ A just in case it is impossible for all of the elements of Γ to be true and A false.

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:

Γ $\vdash$ A just in case there is no possible world at which all of the elements of Γ are true and 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't imagine a possible world where (a) all frogs are green; (b) Kermit is a frog; and (c) Kermit is not green.

Formal accounts

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

Γ $\vdash$ A just in case no argument with the same logical form as Γ/A has true premises and a false conclusion.

Two common variations on this basic idea are:

Γ $\vdash$ A just in case no uniform substitution of the nonlogical terms in Γ/A yields an argument with true premises and a false conclusion.
Γ $\vdash$ A just in case there is no way of interpreting the nonlogical terms in Γ/A that yields an argument with true premises and a false conclusion.

Let us again consider the argument:

All frogs are green.

Kermit is a frog.

Therefore, Kermit is green.

Formal account (1) says that the conclusion is a logical consequence of the premises because no matter how we uniformly replace the nonlogical terms (frog, green, Kermit) in the argument, we do not get true premises and a false conclusion. Consider for example:

All skyscrapers are tall.

The Empire State Building is a skyscraper.

Therefore, the Empire State Building is tall.

All squares are rectangles.

All rectangles are quadrilaterals.

Therefore, a square is a quadrilateral.

All matter has mass.

Coffee tables are matter.

Therefore, coffee tables have mass.

All birds have feathers.

Penguins are birds.

Therefore, penguins have feathers.

We can make up arguments of this form all day, but we will never come up with one that has true premises and a false conclusion. The argument is deductively valid by virtue of its logical form, which might be characterized with the following template (in which F, G, and a are meaningless placeholders):

All Fs are Gs.

a is an F.

Therefore, a is a G.

Formal account (2) says that the conclusion of the "Kermit" argument is a logical consequence of the premises because no matter how we interpret the nonlogical terms (frog, green, Kermit) in the argument, we do not get true premises and a false conclusion. Suppose, for example, we interpret frog to mean plumber, green to mean shy, and Kermit to mean Madonna (the singer). Then the argument has two false premises (for not all plumbers are shy, and Madonna is not a plumber) and a false conclusion (for Madonna is not shy). We can come up with as many interpretations of frog, green, and Kermit as we like, but this will never result in an argument with true premises and a false conclusion.

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

Γ $\vdash$ A just in case it is impossible for an argument with the same logical form as Γ/A to have true premises and a false conclusion.

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

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 the article on non-monotonic logic.

Resources

Michael Dummett, 1991. The Logical Basis of Metaphysics. Harvard University Press.
John Etchemendy, 1990. The Concept of Logical Consequence. Harvard University Press.
Hanson, William H., 1997, "The concept of logical consequence," The Philosophical Review 106: 365-409.
Vincent F. Hendricks, 2005. Thought 2 Talk: A Crash Course in Reflection and Expression. New York: Automatic Press / VIP. ISBN 87-991013-7-8
Planchette, P. A., 2001, "Logical Consequence," in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.
Stewart Shapiro, 2002, "Necessity, meaning, and rationality: the notion of logical consequence" In D. Jacquette, ed., A Companion to Philosophical Logic. Blackwell.
Alfred Tarski, 1936, "On the concept of logical consequence." Reprinted in Tarski, A., 1983. Logic, Semantics, Metamathematics, 2nd ed. Oxford University Press. Originally published in Polish and German.

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