# Is “If…then” not a truly logical connective?

Since you can completely fill out a truth table for "not A", "A and B", and "A or B", but not for "if A then B", can it really be a logical connective?

The only thing that can be confirmed about the truth value of "if A then B" is that it is false when "A" is true and "B" is false, but for other values, "if A then B" cannot be confirmed without additional info.

• " but not for (if A then B)" ??? See Logical connective and see Truth table for "if p, then q". – Mauro ALLEGRANZA Apr 13 '19 at 16:51
• There is no logical connective that corresponds precisely to 'if ...then'. That's the way round the problem is. That phrase in English carries associations that are irrelevant to the strictly logical connective. 'P (false) implies Q (false)' is true for a truth-table conditional but that's completely out of line with how 'if ... then' is used in natural language. Both usages have their justification; what they lack is mutual correspondence. – Geoffrey Thomas Apr 13 '19 at 19:21
• You have to consider that the truth table def of "if..., then..." is specific for classical truth-functional logic, where bivalence holds. Thus, your raeson : "The only thing that can be confirmed about the truth value of "if A then B" is that it is false when "A" is true and "B" is false," is exactly the right one. If that is the case when the conditional is FALSE, by bivalence all the remaining conditions are those when the conditional is TRUE. There are no other possibilities. – Mauro ALLEGRANZA Apr 14 '19 at 12:06
• Bivalence and truth-functionality are separate things. Bivalence requires that the conditional have the value true or false, but you need the additional criterion of truth-functionality to require that the conditional will always be true in the cases other than A-true and B-false. Stalnaker's conditional is bivalent, but it is not truth-functional, since in the event that A is false its truth value depends on what holds in the closest possible world. – Bumble Apr 14 '19 at 16:25

You are quite right to say that in general conditionals "if A then B" are not truth functions and so cannot be defined by a truth table. The only exception is the classical material implication (or material conditional) which is defined by equivalence to "(not A) or B", or to "not(A and not B)". Material implication is the simplest of all the conditionals but it is only one of many. It is useful in mathematics and formal logic, but is not typical of ordinary English usage. Some other conditionals include connexive conditionals, relevant implication, conditional probabilities, strict implication, variably strict implication, counterfactual conditionals, intuitionistic implication, etc.

Also, some linguists have advanced the thesis that it is a mistake to think of conditionals as a sentential connective at all. Angelika Kratzer, for example, has this to say:

The history of the conditional is the story of a syntactic mistake. There is no two-place "if...then" connective in the logical forms for natural languages. If-clauses are devices for restricting the domain of operators.

• Modals and Conditionals, Oxford, (2012), p.106.

Unfortunately many elementary textbooks of logic introduce material implication and leave the reader with the impression that this is all there is to conditionals, which is highly misleading and does a great disservice. The logic of conditionals is an immense subject on which thousands of papers and scores of books have been published, and it is still being actively researched. If you are interested in finding out more, I can recommend

• Jonanthan Bennett, Conditionals: A Philosophical Guide, Oxford, (2003)
• Ernest Adams, The Logic of Conditionals, Reidel, (1975)
• David Lewis, Counterfactuals, Oxford, (1973)
• Nicholas Rescher, Conditionals, MIT Press (2007)
• David Sanford. If P then Q: Conditionals and the Foundations of Reasoning, Routledge (2003).
• Thank you for expanding my mind. I'd like to add that I also gained some insights from THIS ARTICLE – csp2018 Apr 14 '19 at 9:58

One can completely fill out the truth table for "if A then B" much like one can for "not A", "A and B", and "A or B": A logical connective is defined by Wikipedia as

In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a symbol or word used to connect two or more sentences (of either a formal or a natural language) in a grammatically valid way, such that the value of the compound sentence produced depends only on that of the original sentences and on the meaning of the connective.

Does "if A then B" fit that definition? The "if...then" connects two sentences "A" and "B" in a grammatically valid way. The meaning of the connective would be the "T" or "F" assigned to it in the truth table.

So, for this definition, "if A then B" would be a logical connective.

Stanford Truth Table Tool, http://web.stanford.edu/class/cs103/tools/truth-table-tool/

Wikipedia contributors. (2019, March 27). Logical connective. In Wikipedia, The Free Encyclopedia. Retrieved 16:44, April 13, 2019, from https://en.wikipedia.org/w/index.php?title=Logical_connective&oldid=889689402

• I can see that the truth tables are defined that way, but it seems like only way to confirm the truth value of (if A then B) from the truth values of A and B is when A is true and B is false, which makes (if A then B) false. How is it possible to confirm the truth value of (if A then B) for other values of A and B? – csp2018 Apr 13 '19 at 18:58
• @csp2018 If A is true and B is false then the conditional is false (F). That would be the third line in the table. Otherwise the conditional is true (T) as the table shows. This covers all the possibilities if one assumes that these propositions, A and B, have to be either true or false. – Frank Hubeny Apr 13 '19 at 19:20