# If one element of a conjunction is false, is the whole statement false?

Take the statement: "Either Brown is in Barcelona or Jones owns a Ford." I know that if one element of the conjunction is true (Jones owns a Ford, for example), then the whole thing is true. Does the same thing apply for falsity? If one element is false, is the whole thing false?

• Welcome to Philosophy.SE. This is neither a conjunction (X and Y) nor a disjunction (X or Y); it's an exclusive disjunction (either X or Y), which is true if and only if exactly one of the disjuncts are true. Feb 25 '17 at 23:26
• Your statement is a disjunction (OR) not a conjunction (AND), and "either" might even suggest that your are referring to Exclusive OR. OR and AND are dual to each other, what applies for truth to one applies for falsity to the other. If one element of a disjunction is true then it is true, if one element of a conjunction is false then it is false. Feb 25 '17 at 23:30
• oh, I see now, the sentence is not a conjunction. For the actual sentence, no, you don't know that it is true if you only know one element is true and you also don't know it is false if you only know one element is false. Feb 25 '17 at 23:31
• Conjunction is true only when both conjuncts are true; otherwise is false. Feb 26 '17 at 8:01
• Disjunction (inclusive, that usually used in math logic with "or") is false only when both disjuncts are false; otherwise is true. Feb 26 '17 at 8:02

If one element of a conjunction is false, is the whole statement false?

Yes. A conjunction of two propositions is only true when BOTH propositions constituting the conjunction are true.

This is the truth table for conjunctions:

This is a technical point but instead of "element" the term you are asking about is the "operand". In a conjunction statement such as "φ ∧ ψ" there are three elements, two operands (φ and ψ) and the conjunction (or connective) operator (∧). Your question could also be phrased, "If one [statement, proposition, assertion, premise, etc.] in a conjunction..." (Note that the ampersand is not as common anymore, but along with ∧ and •, & is an acceptable symbol for the "and" operator of a conjunction.)

If one element of a [disjunction] is false, is the whole statement false?

No.

The answer is different if you mean an inclusive disjunction or an exclusive disjunction. In both cases they are false when both operands are false, but the exclusive disjunction is also false when both operands are true. (See the truth table below)

Take the statement: "Either Brown is in Barcelona or Jones owns a Ford."

First of all, that statement is not a conjunction statement, it is a disjunction. This is the truth table for disjunction, "inclusive" - a more common "or" statement" - is on the left and "exclusive" disjunction is on the right:
To be clear, the "aVb" column is "inclusive or" and the "a V̲ b" column is "exclusive or" (sometimes indicated by "xor").

I know that if one element of the conjunction(sic) is true (Jones owns a Ford, for example), then the whole thing is true.

Given your example is a disjunction, I am presuming you meant your question about disjunctions.

In an inclusive disjunction with only two operands, If only one operand is true, or, both operands are true, then the whole thing is true:

Only if no more than one operand of an exclusive disjunction with only two operands is true, is the exclusive disjunction statement true:

Does the same thing apply for falsity? If one element is false, is the whole thing false?

Yes for conjunctions.
No for disjunctions.

Although in terms of grammar your sentence uses a conjunction to join the two clauses, in terms of a type of logical complex sentence, it is a DISJUNCTION. This is because it uses the connective or, not the connective and.

As you rightly say, if one disjunct in the disjunction is true then the whole sentence is true. On that basis, therefore, the sentence must be true if one of the disjuncts is true and the other one is false. So, no, it is not the case that if one disjunct in a disjunction is false the sentence is necessarily false.