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Defining arrow

In this video, York student Tommy shows how we define the fourth connective of our formal language, 'arrow'.
Our next truth table is for arrow, also called material implication. This connects two sentences so we’ll use the two sentences P and Q, then we’ll show how arrow is going to work. First we need to write the four ways things can be with P and Q. Now the rough idea behind P arrow Q is that where P is true, Q is true. Now for the first row, P is true and Q is true so P arrow Q is going to be true; on the second row P is true but Q is false so P arrow Q is going to be false. On the 3rd row P is false. The idea is that where P is true, Q is true.
If P is false, we don’t have a break to that idea, so it’s going to be true. This also applies to the 4th row. So it is going to be true there too. So the truth table for P arrow Q is that it’s going to be true in every single case except for where the antecedent is true and the consequent is false. That is, just where it’s wrong to say that, where P, Q.

This video shows how we define the meaning and logical powers of arrow using a truth-table. The remainder of this step introduces further discussion of this connective.

Key points about arrow

First, arrow, like ampersand and vel, is a two-place connective. We can add a rule to the grammar of our language which exactly parallels the rules for ampersand and vel:

  • (R5) If ‘α’ and ‘β’ are wffs, then ‘(α (rightarrow) β)’ is a wff

Secondly, unlike ampersand and vel, the order of clauses in an arrow sentence makes a difference to its truth-conditions.

With ampersand and vel, the order of clauses does not make a difference to truth-conditions: ‘(P & Q)’ has the same truth-conditions as ‘(Q & P)’; and ‘(~P (vee) Q)’ has the same truth-conditions as ‘(Q (vee) ~P’).

Things are different with arrow: ‘(P (rightarrow) Q)’ has different truth-conditions from ‘(Q (rightarrow) P)’.

For this reason, we give different names to the clauses in an arrow sentence. The clause before the arrow is called the antecedent. The clause after the arrow is called the consequent.


Arrow claims are a kind of conditional claim (‘conditionals’, for short). You can think of this in the following way: where the condition specified in the antecedent applies, things are as the consequent says. Conditionals include ‘If … then … ’-claims in English, like ‘If it’s raining, we’re getting wet’ and claims of other forms, such as ‘You must be over 18 to buy alcohol’ and ‘Using a screen late at night results in poor sleep’.

Thirdly, arrow (like ampersand, tilde, and vel) is a truth-functional sentence connective. The truth-value of an arrow sentence is fixed in all cases by the truth-value of the sentences plugged into it.

An arrow sentence is only false when its antecedent is true and its consequent is false. In all other cases, it’s true, as shown in its defining truth-table (Fig. 1).

Defining truth-table for arrow

Figure 1. The truth-table defining arrow

Arrow-claims are called material conditionals, because their truth or falsity is fixed simply by how things stand with the truth-values of their antecedent and consequent. It’s fixed by the material facts relating to their antecedent and consequent.

Finally, and importantly, there seem to be parallels between arrow and at least some uses of ‘If … then … ’ as a sentence-connective in English. But whether arrow and ‘If … then … ’ have the same truth-table—even in the most favourable cases—is a controversial issue, as we’ll see. Despite this controversy, we’ll also see that we can use arrow to help us think in a systematic way about arguments expressed in English using ‘If … then … ’.

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Logic: The Language of Truth

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