Propositional Logic - Conditionals

Conditionals

Conditionals are somewhat harder to pin down compared to the rest of the operators. Conditionals do not have nearly as an intuitive analogue in natural language. The use of a conditional mostly seems to be a stand in for a type of valuation function where the antecedent is false, or the consequent is true.

This is where we get the notion that the conditional’s antecedent is the sufficient condition (can be true or false given the consequent is true), and the consequent is the necessary condition (cannot be true given the consequent is true).

For a conditional we can state:

$ v(P→Q)=T iff v(P)=F or v(Q)=T $

It will become clearer what this will mean in the following truth table:

P

Q

P→Q

T

T

T

T

F

F

F

T

T

F

F

T

Table 8

We should give an example of what a conditional would look like in natural language to illustrate certain facts. The following proposition is an example of a conditional “If the cheese is green, then the moon is white”. The antecedent is a sufficient condition because it need not be true for the conditional to be true, but the consequent must be true if the conditional is true. However, the implication is strange because in more meaningful propositions where the implication is true. Take the proposition, “If the water has spilled, then the floor is wet.” This proposition shows that it must necessarily follow that the floor is wet if the water has spilled. However, many conceptions of explanation are ontologically causal and therefore we would expect that the last two possible worlds in the truth table for the conditional would be false. IE: if the water had not spilled, then it must be true that the floor is not wet. This concept of explanation is closer to using the conjunction.

Therefore, we call the last two possible worlds of the conditional vacuously true.

Vacuous truth is shown in the highlighted part of the truth table, and these valuations are important to understand the next point.

We can also think about the 

Paradoxes of the Material Implication

Given the concept of vacuous truth, we can introduce paradoxes of the material implication. These are counter-intuitive results of the implication:

  1. The conditional is true if the antecedent is false, independent of the value of the consequent. So, the following proposition, “If the shape is a square, then the shape is a rectangle” is true if the shape is not a square but is or is not a rectangle.
  2. The conditional is true if the consequent is true, independent of the value of the antecedent. So, the proposition above is true if the shape is a rectangle, but the shape is or is not a square.

Although this is not a paradox in the formal sense per say (defined as $ p∧¬p $). The oddness of the translation of the implication becomes a strong argument to prefer more complicated systems of symbolic logic when trying to capture ideas in natural language on a more granular scale.

Natural Language

There are many ways to translation that an implication into natural language. It is good to try to memorize what these interpretations are:

  1. If P, then Q.
  2. P implies Q.
  3. Q if P.
  4. Q whenever P.
  5. P is sufficient for Q.
  6. Q is necessary for P.
  7. P only if Q.

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