Logical consequence

The most widely prevailing view on how best to account for logical consequence is to appeal to formality. This is to say that whether statements follow from one another logically depends on the structure or logical form of the statements without regard to the contents of that form.

Syntactic accounts of logical consequence rely on schemes using inference rules. For instance, we can express the logical form of a valid argument as:

All X are Y

All Y are Z

Therefore, all X are Z.

This argument is formally valid, because every instance of arguments constructed using this scheme is valid.

This is in contrast to an argument like "Fred is Mike's brother's son. Therefore Fred is Mike's nephew." Since this argument depends on the meanings of the words "brother", "son", and "nephew", the statement "Fred is Mike's nephew" is a so-called material consequence of "Fred is Mike's brother's son", not a formal consequence. A formal consequence must be true in all cases, however this is an incomplete definition of formal consequence, since even the argument "P is Q's brother's son, therefore P is Q's nephew" is valid in all cases, but is not a formal argument.[1]

If you know 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.[1] Deductively valid arguments can be known to be so without recourse to experience, so they must be knowable a priori.[1] 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.[1]

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.[4]

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

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

Alternatively (and, most would say, equivalently):

${\displaystyle \Gamma }$ ${\displaystyle \vdash }$ ${\displaystyle A}$ is true if and only if it is impossible for all of the elements of ${\displaystyle \Gamma }$ 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:

${\displaystyle \Gamma }$ ${\displaystyle \vdash }$ ${\displaystyle A}$ is true if and only if there is no possible world at which all of the elements of ${\displaystyle \Gamma }$ 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't imagine a possible world where (a) all frogs are green; (b) Kermit is a frog; and (c) Kermit is not green.

• Anderson, A.R.; Belnap, N.D., Jr. (1975), Entailment, 1, Princeton, NJ: Princeton.
• Augusto, Luis M. (2017), Logical consequences. Theory and applications: An introduction. London: College Publications. Series: Mathematical logic and foundations.
• Barwise, Jon; Etchemendy, John (2008), Language, Proof and Logic, Stanford: CSLI Publications.
• Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.
• Davis, Martin, (editor) (1965), The Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems And Computable Functions, New York: Raven Press, ISBN 9780486432281CS1 maint: extra text: authors list (link). Papers include those by Gödel, Church, Rosser, Kleene, and Post.
• Dummett, Michael (1991), The Logical Basis of Metaphysics, Harvard University Press, ISBN 9780674537866.
• Edgington, Dorothy (2001), Conditionals, Blackwell in Lou Goble (ed.), The Blackwell Guide to Philosophical Logic.
• Edgington, Dorothy (2006), "Indicative Conditionals", Conditionals, Metaphysics Research Lab, Stanford University in Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy.
• Etchemendy, John (1990), The Concept of Logical Consequence, Harvard University Press.
• Goble, Lou, ed. (2001), The Blackwell Guide to Philosophical Logic, BlackwellCS1 maint: extra text: authors list (link).
• Hanson, William H (1997), "The concept of logical consequence", The Philosophical Review, 106 (3): 365–409, doi:10.2307/2998398, JSTOR 2998398 365–409.
• Hendricks, Vincent F. (2005), Thought 2 Talk: A Crash Course in Reflection and Expression, New York: Automatic Press / VIP, ISBN 978-87-991013-7-5
• Planchette, P. A. (2001), Logical Consequence in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.
• Quine, W.V. (1982), Methods of Logic, Cambridge, MA: Harvard University Press (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), (4th edition, 1982).
• Shapiro, Stewart (2002), Necessity, meaning, and rationality: the notion of logical consequence in D. Jacquette, ed., A Companion to Philosophical Logic. Blackwell.
• Tarski, Alfred (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.
• Ryszard Wójcicki (1988). Theory of Logical Calculi: Basic Theory of Consequence Operations. Springer. ISBN 978-90-277-2785-5.
• A paper on 'implication' from math.niu.edu, Implication
• A definition of 'implicant' AllWords

Wikimedia Commons has media related to Logical consequence.

• Beall, Jc; Restall, Greg (2013-11-19). "Logical Consequence". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy (Winter 2016 ed.).
• "Logical consequence". Internet Encyclopedia of Philosophy.
• Logical consequence at the Indiana Philosophy Ontology Project
• Logical consequence at PhilPapers
• "Implication", Encyclopedia of Mathematics, EMS Press, 2001 [1994]