**Material Implication, Logical Equivalencies, Converse, Inverse, Contrapositive**

**Given the material conditional P -> Q**, P is termed "antecedent" (condition) and Q is termed "consequent" (consequence), where P -> Q reads "P implies Q" and means that P is a sufficient condition for Q which in its turn means that Q is a necessary condition for P.

If the antecedent **(P) is true**, then the material implication (->) holds only if the consequent (Q) is also true. That is, a true antecedent/premise/condition (P) can only imply a true /consequent/conclusion/consequence (Q).

If the antecedent **(P) is true** and the consequent (Q) is false, then the implication does not hold (true). That is, a true antecedent cannot imply a false consequent: truth cannot imply falsity. This is the only option for which the material implication does not hold, i.e., the operator/connective (->) outputs false if and only if the antecedent (P) is true and the consequent (Q) is false.

If the antecedent **(P) is false**, then P materially implies Q, regardless of **whether Q is true or false**. *From falsity anything follows*. That is, a false antecedent (P) implies the consequent (Q) in the case when Q is false as well as when Q is true. For more information, please look up the "principle of explosion" which states in Latin: "Ex falso sequitur quodlibet" = "From falsity follows anything".

**Example 1**. (with a false consequent): "If 2+2 = 5, then I am god" both P and Q are false, yet the implication holds (true).

**Example 2**. (with a true consequent): "If Julius Caesar invades North America, I speak some Latin" also holds true. (I do speak a little Latin).

**Example 3**.: "If you write a great post, then I('ll) give you $10." This conditional constitutes a promise. Let us see for which truth value combinations of P and Q, the promise (implication) holds (true).

**Let us analyze Example (3)**:

- Let: P := You write a great post, and
- Let: Q := I give you $10.
- Case 1. P is true and Q is true.
- Case 2. P is true, and Q is false.
- Case 3. P is false, and Q is true.
- Case 4. P is false, and Q is false.

Suppose **case 1** is the case: "You write a great post, and I give you $10." Does the implication hold or have I broken my promise? I have fulfilled my promise in response to your great post. The implication holds (true).

Suppose **case 2** is the case: "You write a great post, but I do not give you $10". Does the implication hold or have I broken my promise? In fact, I have broken my promise, because I have not fulfilled the consequent of the conditional, given a true antecedent. Therefore, for this option, the implication does not hold (true), i.e., the implication outputs a truth value of false.

Suppose **case 3** is the case: "You do not write a great post, and I give you $10." Have I broken my promise? My promise was predicated on your writing a great post, and it says nothing about what should happen if the antecedent were false. My promise (implication) merely states what should happen if the antecedent were true. If you do not write a good post, but I nonetheless give you $10, then, strictly speaking, I have not violated my promise. Therefore, the implication holds (true), with a false antecedent (P) and a true consequent (Q).

Suppose **case 4** is the case: "You do not write a great post, and I do not give you $10". Here, even though both antecedent (P) and consequent (Q) are both false, the implication nonetheless holds (true). A falsehood can imply a falsehood, because from falsity anything follows.

P -> Q means "P materially implies Q" which is stated as the following material conditional (if-then) statement: "If P, then Q". The material conditional P -> Q implies that P is a sufficient condition for Q.

Given the material conditional (if-then) statement (P -> Q), the operator/connective (->) is called material implication, which sets up a material conditional "If P (is the case), then Q (follows)", which can equivalently stated as "Q if P", which in its turn is equivalent to stating "P only if Q", which implies that P is a sufficient condition for Q, which is represented as follows: P => Q.

Moreover, the sufficiency of P for Q is logically equivalent to the necessity of Q for P: [P => Q] <= logically equivalent to => [Q <= P].

Consider the following four options:

- P => Q: P is sufficient for Q.
- Q <= P: Q is necessary for P.
- P <= Q: P is necessary for Q.
- Q => P: Q is sufficient for P.

Henceforth, let the symbol (≡) denote logical equivalence, and (≡|≡) denote logical non-equivalence.

- Options (1) and (2) are logically equivalent: (1) ≡ (2)
- Options (3) and (4) are logically equivalent. (3) ≡ (4)

Consider the original conditional (P -> Q) with the (forward) material implication (->), connecting P to Q, so that P is set up as a sufficient condition for Q.

Let us refer to P -> Q as the "original" material conditional:

A1. Original material conditional: (P -> Q), with a "forward implication", i.e., from P to Q.

A2. Converse of Original: (Q -> P), with a "reverse implication", i.e., from Q to P. reverse of "forward".

A3. Inverse of Original: (~P -> ~Q), with a "forward implication" and with both antecedent (P) and consequent (Q) negated.

A4. Contrapositive of Original: (~Q -> ~P), with a "reverse implication" and with both antecedent (P) and consequent (Q) negated; therefore, contrapositive = the inverse of the original in reverse; that is, contrapositive (of original) = converse of inverse (of original).
Note that:

- The contrapositive (A4) is logically equivalent to the original (A1):
A1 ≡ A4.
- The converse (A2) is logically equivalent to the inverse (A3): A2 ≡
A3.

The inverse (A3) can be derived by the contraposition of the converse (A2): by reversing the direction of the implication in (A2) and negating the P and Q. Note, that just as the contraposition of the original yields the contrapositive (of the original), where the contrapositive is logically equivalent to the original, so too does the contraposition of the converse yield the inverse (of the original), which is logically equivalent to the converse. Therefore, the contraposition of a conditional (C) validly yields another conditional (C*) that is logically equivalent to the conditional (C): C ≡ C*.

The third sentence is false because it has a true antecedent and a false consequent. – Eliran – 2016-05-10T00:53:02.593

@virmaior from my inexperienced perspective my question is different than the one you linked. The linked question seems to be asking more generally about what makes an implication true or false, while my question is asking for the justification of one specific scenario that logicians claim produces a true implication. – IgnorantCuriosity – 2016-05-10T01:47:43.463

Okay... there should be several questions on that already (though I didn't find one in the brief interruption between revising a manuscript for publication)... The lengthy question I suggested this duplicates

doesexplain what's going on here, but it might be too hard to follow. Though, if you're reading Tarski ... – virmaior – 2016-05-10T02:18:29.023@virmaior the Tarski book I'm reading is a very approachable introduction to the subject written for someone with no background knowledge. – IgnorantCuriosity – 2016-05-10T02:48:00.607