Material Implication and Equivalence

Please read, reconstruct for yourselves the following:

1. What it means to say P implies Q
2. What it means to say P is materially equivalent to Q
3. What is the difference between material implication and logical implication?
4. What is the difference between material bi-conditional and logical bi-conditional? Explain material equivalence and logical equivalence: compare and contrast, please!

Discuss how from a falsehood anything follows: how and why it is that a false antecedent can imply either a true or a false consequent, as opposed how and why a true antecedent cannot imply a false consequent.

• Explain why a false antecedent (P) can imply a false consequent (Q)!
• Explain why a false antecedent (P) can imply a true consequent (Q)!
• Explain why the antecedent (P) is materially equivalent to its consequent (Q) when P and Q are either both true together or both false together!

Given the material conditional P -> Q, P is referred to as "antecedent" and Q is referred to as “consequent” in this form (forward implication from P to Q). P -> Q means "P materially implies Q" which is stated as the following material conditional (if-then) statement: "If P, then Q".
For the material conditional: P -> Q, ‘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.

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 where Q is false as well as in the case where 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 I live forever (without end), then I am god" both P and Q are false, yet the implication holds (true).

Example 2 (with a true consequent): "If Alexander the Great invades North America, then I (will) speak some Greek" also holds (true). (I do speak a little Greek).

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 examine Example 3!

Let: P := You write a great post, and
Let: Q := I give you $10.

The following options exist:

• P is true and Q is true.
• P is true, and Q is false.
• P is false, and Q is true.
• 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. Ex falso sequitur quodlibet! For more info on false if-clause research the Principle of Explosion of Formal Logic.

Implication ("If-then"), Material Implication, Material Equivalence, Logical Implication, Logical Equivalence!

P -> Q means: P 'implies' Q, where 'implies' can be rendered into the following conditional form: "If P, then Q". The symbol (->) denotes "material implication", which sets up a sufficient condition between P and Q, such that P materially implies Q: that is, if P is the case (i.e., true), then Q follows from P. The material implication is the most logical sense of an implication; it is that implication is the "lowest common denominator of all sorts of implications (i.e., if-then statements). The material implication is that implication which is in common in all implication: the sufficiency of P for Q and equivalently to the necessity of Q for P. The material conditional: P ->Q (If P then Q') logically entails the following: P => Q: P is a sufficient condition for Q. Q <= P: Q is a necessary condition for P.

**Material vs. Logical Implication**
Given the material conditional P -> Q, P is referred to as "antecedent" and Q is referred to as consequent in this form (forward implication from P to Q). Given the material conditional (if-then) statement (P -> Q), which can be stated equivalently 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.
Original Implication: P -> Q; this means 'P is sufficient for Q'
Converse of Original: Q -> P; this means 'Q is sufficient for P'

An implication and its converse taken together establish material equivalence between P and Q; that is, the conjunction ("and") of a material conditional [P -> Q] and its converse [Q -> P] yields a material bi-conditional [P <=> Q], which reads "P if and only if Q", and the bidirectional implication connective/truth-function/operator (<=>) is called "material equivalence". The operator (<=) is also called 'iff' (which stands for 'if and only if'). The material equivalence (<=>) relation is a both necessary and a sufficient condition!

P is materially equivalent to Q iff P and Q materially imply one another!

The "Original" Implication ("forward implication):
(P -> Q) = ("If P, then Q") = ("Q if P") = (P only if Q), which sets up the sufficiency of P for Q: P => Q.
The Converse of the "Original" Implication ("reverse implication"):

(Q -> P) = ("If Q, then P") = ("P if Q") = ("Q only if P"), which sets up the sufficiency of Q for P: Q => P.
P is materially equivalent to Q 'if and only if' P is a sufficient condition for Q, and likewise, Q is a sufficient condition for P.

Therefore, (P <=> Q) = ('P if Q' AND 'P only if Q) = (P if and only if Q) (P <=> Q) = ('P is both a necessary and a sufficient condition for Q').

The Difference Between Logical and Material Equivalence (Logical Equivalence is a Subset of Material Equivalence!)

In the case of material equivalence (P <=> Q), P and Q must materially imply one-another; where the term "implies" is to be understood as setting up the sufficiency of the antecedent (P, "if-part" of conditional) for the consequent (Q, "then-part' conditional). Likewise, in the case of logical equivalence (P ≡ Q), P and Q must logically imply one-another; where "logically implies" (Log: P --> Q) means the antecedent (P) logically entails the consequent (Q), which can be restated as follows: P logically implies Q means that "the consequent Q is a logical consequence of the antecedent P".

Why is the material bi-conditional 'P if and only if Q' = (P <=> Q) = (P iff Q) = 'P is a necessary and sufficient condition for Q'; P and Q can only be true together or false together. It cannot be the case that exactly one of (P,Q) is true, and the other false: i.e.,:

The material bi-conditional (<=>) excludes (rules out) the following options:
"{'P is true' and 'Q is false'}, or
{'P is false' and 'Q is true'}.

and includes (rules in) the following options two options:
{'P is true' and 'Q is true'}
{'P is false' and 'Q is false'}

The material bi-conditional 'P iff Q' states that P and Q materially imply one another.

In the material bi-conditional: {‘P implies Q’ and ‘Q implies P’}, the resultant “material equivalence” (i.e., bi-conditional implication: ‘P and Q imply one another’) outputs a truth value of “truth” if and only if P and Q are either both true together or both false together. The material equivalence operator (<=>) which outputs a truth-value of 'truth' if and only if P and Q are both either true together or false together.

The material bi-conditional (material equivalence) is denoted by the following symbols: (<=>) = iff = xnor = exclusive joint denial; where joint denial is the "neither-nor" option that says, :"Neither P is true nor Q is true" = "nor". The joint denial of P and Q is 'P nor Q', which denies P is true and denies Q is true.
To deny a proposition P (i.e. "P is true") means to accept that its negation is true: that is to accept "P is false", rather than merely not accepting that 'P is true' (i.e., rejecting that P is true).

Material equivalence logically includes (rules in) the following two options: joint affirmation ("both-and") and joint denial ("neither-nor") options.:

P and Q are both true together means: ('P is true' AND 'Q is true').
P and Q are both false together means: ('P is false' AND 'Q is false'),
Otherwise "xnor" outputs 'falsity' (F) (i.e., logically excludes = rules out)!

The Difference Between Logical and Material Equivalence
(Logical Equivalence is a Subset of Material Equivalence!)

In the case of material equivalence (P <=> Q), P and Q must materially imply one-another; where the term "implies" is to be understood as setting up the sufficiency of the antecedent (P, "if-part" of conditional) for the consequent (Q, "then-part' conditional).

Likewise, in the case of logical equivalence (P ≡ Q), P and Q must logically imply one-another; where "logically implies" (Log: P --> Q) means the antecedent (P) logically entails the consequent (Q), which can be restated as follows: P logically implies Q means that "the consequent Q is a logical consequence of the antecedent P".