Why is the biconditional true if both components are false?

I already know that a false statement implies anything. Because I ask only for intuition, please do not answer with formal proofs or use truth tables (which I already understand).

Source: p 333, A Concise Introduction to Logic (12 Ed, 2014), by Patrick J. Hurley

The truth table shows that the biconditional is true when its two components have the same truth value and that otherwise it is false. These results are required by the fact that p ≡ q is simply a shorter way of writing (p ⊃ q) ∧ (q ⊃ p). If p and q are either both true or both false, then p ⊃ q and q ⊃ p are both true, making their conjunction true. [...]

I understand the above by rewriting p ≡ q as (p ⊃ q) ∧ (q ⊃ p); but without relying on this expansion, how can you intuit this directly?

Epilogue: With the aid below, I finally comprehended my problem as summarised here. I had confused 'True' with 'Truth Value'. To wit, the Truth of ≡ is decided by the similarity or difference of the two Truth Values on either side of ≡, and is NOT decided by the truth values.
So even if both p and q are false, p and q have the same Truth Value (because they are both false), and so are logically equivalent.

• Well, it's not true that "a false statement implies anything"—just that a material conditional with a false antecedent can be true. – ChristopherE Dec 30 '15 at 2:10

The intuition is:

1. The biconditional X ≡ Y says "X and Y always have the same truth value."
2. Therefore either X and Y are both true; or X and Y are both false.
3. Another name for the biconditional is is equivalence, or logical equivalence. That name carries more of the intuition. X and Y are equivalent. Either both are true or both are false.
4. If they're both true, each implies the other because True implies True.
5. If they're both false, each implies the other because False implies False.

You can run these points from top to bottom or bottom to top to suit your intuition and whatever presentation is in the book.

• Thanks. I hope that you do not mind my addition of numbering. Can you please clarify 5? I do not understand: each implies the other because False implies False. – Greek - Area 51 Proposal Dec 29 '15 at 17:28
• @LePressentiment False implies False. You already mentioned that in your question so I took that as given. If X and Y are both false, then X implies Y and Y implies X because False implies False. – user4894 Dec 29 '15 at 18:01
• @LePressentiment - you started your question with "I already know that a false statement implies anything"; thus "a false statement implies anything false". – Mauro ALLEGRANZA Dec 29 '15 at 20:27
• @user4894 Thank you. I now understand 5, but how does 5 (ie: 2 false propositions) imply that the (material) biconditional is true? I still find strange deriving truth from 2 false propositions? – Greek - Area 51 Proposal Dec 29 '15 at 20:39
• @MauroALLEGRANZA Thank you. Now I understand 5. – Greek - Area 51 Proposal Dec 29 '15 at 20:39

In the simplified semantics for logical languages, meaning is "proxied" by the truth conditions for sentences.

There is a long tradition in modern logic regarding "extensionality". From Frege's Begriffsschrift onwards, mathematical logic has been concerned mainly with truth-functional contexts.

According to Frege, the reference (Bedeutung) of a proposition is its truth-value, either the True or the False. For Frege, complete propositions, like names, have objects as their Bedeutungen, and in particular, the truth-values the True or the False. In this way, he is able to transcribe sentential connectives such as “and” and “or,” etc., as truth functions in the strictest sense — functions that take truth-values as argument and yield truth-values as value.

According to Alfred North Whitehead & Bertrand Russell, Principia Mathematica to *56 (2nd ed - 1927), page 115 :

It is obvious that two propositions are equivalent when, and only when, both are true or both are false. Following Frege, we shall call the truthvalue of a proposition truth if it is true, and falsehood if it is false. Thus two propositions are equivalent when they have the same truth-value.

We shall give the name of a truth-function to a function f(p) whose argument is a proposition, and whose truth-value depends only upon the truth-value of its argument. All the functions of propositions with which we shall be specially concerned will be truth-functions, i.e. we shall have

p≡q.⊃.f(p)≡f(q).

The reason of this is, that the functions of propositions with which we deal are all built up by means of the primitive ideas. But it is not a universal characteristic of functions of propositions to be truth-functions. For example, "A believes p" may be true for one true value of p and false for another.

In "modern" terms, this is the so-called Replacement theorem; see S.C.Kleene, Introduction to Metamathematics (1952), page 116 :

If A and B are formulas, C_A is a formula constructed from a specified occurnce of A using only the [truth-functional connectives], and C_B results from C_A by replacing this occurrence of A by B, then A ↔ B ⊢ C_A ↔ C_B.

In a "universe" of extensional contexts, like that considered by the simplified semantics of logical languages, it is natural to equate the condition for two equivalent formulae having the same truth conditions with that of expressing "the same thing", i.e. "having the same meaning".

• Thank you for your response, but I am quite confused because it appears too complicated for me; I am only starting logic. Can you simplify it please? – Greek - Area 51 Proposal Dec 29 '15 at 17:18