8.1 Symbolizing Material Conditionals
This module introduces ways to symbolize conditional claims. Conditional claims play a role in many valid forms of argument.
Table of Contents
- 8.1 Symbolizing Material Conditionals
8.1.1 Why Material Conditionals?
Conditionals fill in the missing piece to make an argument valid.
Image Credit: Willi Heidelbach [CC BY 2.0 (https://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons
The Usefulness of Material Conditionals
Recall the truth table for a material conditional, where P => Q represents the claim that “If P, then Q”.
| P | Q | P => Q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Material conditionals are false when the antecedent is true and the consequent is false, and they are true otherwise. This is similar to the definition of a valid argument: an argument is invalid when it is possible for the premises to be true with the conclusion false, and valid otherwise. Because of this similarity, material conditionals are useful for representing the relationship that should hold in a valid argument between a reason or premise and a conclusion.
For instance, suppose that a person argues in favor of this conclusion for this reason:
Reason: Victims deserve to be avenged.
Conclusion: Capital punishment should be mandatory for murderers.
We could represent these as a premise and a conclusion:
1. Victims deserve to be avenged.
C. Capital punishment should be mandatory for murderers.
This is an invalid argument, obviously. It is very easy to make it valid by adding as a premise a material conditional which has the first premise in the antecedent, and the conclusion in the consequent:
1. Victims deserve to be avenged.
2. If victims deserve to be avenged, then capital punishment should be mandatory for murderers.
C. Capital punishment should be mandatory for murderers.
Symbolized, this would be:
1. V
2. V => M
C. M
The truth table would look like this:
| V | M | 1. V | 2. V => M | C. M |
| T | T | T | T | T |
| T | F | T | F | F |
| F | T | F | T | T |
| F | F | F | T | F |
There is only one row where both premises are true. In this row, the conclusion is true. So, the argument is valid. Of course, the argument might not be sound! In a debate about capital punishment, someone would need to offer some defense of the second premise, a reason to think that it is true, and to defend it against likely objections. The premise is at least valid, however.
Because of this, material conditionals are useful for representing reasons, and also for adding premises needed to make arguments valid.
8.1.2 Symbolizing Material Conditionals
Material conditionals are symbolized with a right-facing arrow.
Identifying Antecedent and Consequent
“Material Implication” is the relationship which holds between the antecedent and consequent of a conditional; in a sentence of the form If P then Q, P is the antecedent (‘first part’), and Q is the consequent (‘second part’). For instance, in “If it is raining, then it is wet outside”:
“It is raining” is the antecedent
“It is wet outside” is the consequent
Symbolizing material conditionals begins with identifying the antecedent and the consequent of a conditional sentence. In English, the order of a conditional is often reversed, so don’t rely on the order in English when symbolizing. Instead, consider what would make the sentence false. When a conditional is false, the antecedent is true and the consequent is false. For instance, suppose it were wet outside, but not raining. The sentence “if it is raining, then it is wet outside” is still true. Suppose instead that it were raining, but not wet outside. In that case, “If it is raining, then it is wet outside” would be false. So, this helps with identifying “It is raining” as the antecedent, and “It is wet outside” as the consequent.
Symbolizing Material Implication
There are two standard symbols traditionally used in logic for material implication:
Single arrow (→): P → Q
Horseshoe (⊃): P ⊃ Q
Unfortunately, neither of these symbols appears on a standard keyboard. So, two alternatives are used:
Greater than Sign ( > ): P > Q
Equals + Greater than ( => ): P => Q
For assignments in our class, we’ll use the => sign to symbolize material implication. Simply type the equals sign (=) followed by the ‘greater than’ (>) sign without a space in between. Outside of our class, => is typically used to represent a strict conditional, rather than a material conditional, so remember that we are only using => as a convenience for students using standard keyboards.
If… Then…
The easiest material conditionals to symbolize are those which already have the form if . . . then . . .
First, identify the antecedent (after the word if), and identify the consequent (after the word then):
If it is Tuesday, then there is a test.
Sometimes, in English, the word then is dropped after a comma:
If it is Tuesday, there is a test.
Still, the antecedent is “it is Tuesday”, and the consequent is “there is a test”. We’ll assign capital letters to each atomic sentence:
Let
I = It is Tuesday
T = There is a test
Now, we add the material implication symbol => between them, with a single space on either side:
I => T
If it is Tuesday then there is a test
… Only If
The phrase “only if” indicates that what follows it is the consequent. For example, consider:
You owe rent only if you lived in the apartment
Here, the sentence as a whole would be false in the case where you owe rent, but you didn’t live in the apartment. Otherwise, the sentence would be true. So, “You owe rent” is the antecedent, and “you lived in the apartment” is the consequent. We could rephrase this sentence as:
If you owe rent, then you lived in the apartment
So it is symbolized as:
Let
R = You owe rent
L = You lived in the apartment
R => L
If you owe rent then you lived in the apartment
You owe rent only if you lived in the apartment
English may reverse the order of these. For instance, we might say, “Only if you lived in the apartment do you owe rent.” Still, “You owe rent” is the antecedent, and “You lived in the apartment is the consequent.”
… If
The word “if” on its own is very different from the word “only if”. While “only if” indicates that what follows is the consequent, “if” by itself indicates that what follows is the antecedent. For instance, in “If you owe rent, you lived in the apartment”, “You owe rent” is the antecedent; whereas in “You owe rent if you lived in the apartment”, “You lived in the apartment is the antecedent. Consider:
You deserve a ticket if you were speeding.
This sentence would be false in the situation where you were speeding, but you still didn’t deserve a ticket. Here, “You were speeding” is the antecedent, and “you deserve a ticket” is the consequent. We symbolize it as:
Let
S = You were speeding
D = You deserve a ticket
S => D
If you were speeding then you deserve a ticket
You deserve a ticket if you were speeding
Here is a chart to remember the correct symbolization:
| if P then Q | P => Q |
| P only if Q | P => Q |
| P if Q | Q => P |
8.1.3 Symbolizing Material Conditionals with Negation
When symbolizing conditions with negation, pay attention to scope.
Scope of the Negation
Negation can apply to the antecedent of a conditional, the consequent of a conditional, or to the whole conditional. Before symbolizing, it is important to determine the “scope” of the negation, or what the negation applies to. For instance, there is a very important difference between:
(a) It is not the case that, if you pay more money, then you get better service.
and
(b) If it is not the case that you pay more money, then you get better service.
Sentence (a) is true when you pay more money and yet don’t get better service, and false otherwise. Sentence (b) is true when you don’t pay more money, and yet do get better service; it’s false if you don’t pay more money, and don’t get better service. Both of these are different from (c), which would be false when you pay more money, and you do get better service, and true otherwise:
(c) If you pay more money, then it is not the case that you get better service.
The difference between these sentences is the ‘scope’ of the negation. Sometimes ordinary English is ambiguous about the scope of a negation. For instance, consider this sentence:
If you make one late payment then you aren’t untrustworthy.
This sentence could be saying that just one late payment doesn’t make someone untrustworthy:
It is not the case that, if you make one late payment, then you aren’t untrustworthy
The sentence could also be saying that having merely one late payment actually makes someone trustworthy:
If you make one late payment, then you are non-untrustworthy (i.e., trustworthy).
Symbolizing conditionals forces us to get rid of ambiguity and make clear the scope of the conditional.
Negated Antecedent
When the antecedent of a conditional is negated, the negation sign ~ is placed next to the sentence in the antecedent, without any spaces. For instance:
Let
M = You pay more money
B = You get better service
~M => B
If you do not pay more money then you get better service.
or
Let
O = You make one late payment
T = You are trustworthy
~O => T
If you do not make one late payment then you are trustworthy
Negated Consequent
When the consequent a conditional is negated, the negation sign ~ is placed next to the sentence in the consequent, without any spaces. For instance, using the earlier dictionary:
O => ~T
If you make one late payment then you are not trustworthy
M => ~B
If you do pay more money then you do not get better service.
~O => ~T
If you do not make one late payment then you are not trustworthy
~M => ~B
If you do not pay more money then you do not get better service
Negated Conditional
When the whole conditional is negated, this means that the conditional as a whole is false: in other words, the antecedent is true, and the consequent is false. We put the conditional as a whole in parentheses, and then place the ~ sign in front of the parentheses, without any spaces. For instance, using the dictionary above:
~(O => T)
It is not the case that if you make one late payment then you are trustworthy
~(M => B)
It is not the case that if you pay more money then you get better service
~(O => ~T)
It is not the case that if you make one late payment then you are not trustworthy
~(M => ~B)
It is not the case that if you pay more money then you do not get better service.
~(~O => ~T)
It is not the case that if you do not make one late payment then you are not trustworthy
~(~M => ~B)
It is not the case that if you do not pay more money then you do not get better service
Summary
Here is a chart summarizing the scope of negation over a conditional:
| If P then not Q | P => ~Q |
| If not P then Q | ~P => Q |
| Not the case that If P then Q | ~(P => Q) |
8.1.4 Symbolizing Complex Material Conditionals
Biconditionals are symbolized with a double arrow.
Embedded Material Conditionals
It is not unusual for conditionals to be embedded in the antecedent or consequent of other conditionals. For instance, a company might promise that, if you sign up for their waiting list, then if they get an open slot then they will call you. Here, “If they get an open slot, then they will call you” is embedded within the consequent of “If you sign up for their waiting list . . .” The company says:
Let
O = They get an open slot
C = They will call you
S = You sign up
S => (O => C)
If you sign up, then if they get an open slot, then they will call you.
You think about this for a while. Then, you decide that if it is true that, if you sign up, then if they get an open slot, then they will call you . . . well then, in that case, you will sign up. Now, the conditional sentence earlier is in the antecedent of another conditional, one where your decision to sign up is in the consequent:
(S => (O => C)) => S
If it is the case that if you sign up, then if they get an open slot, then they will call you… then you sign up.
To give another example, here is a conditional within the consequent of another conditional:
If you eat the fruit, then if you die, then you should not have eaten the fruit.
This is different from a conditional within the antecedent of another conditional:
If it is the case that if you eat the fruit then you die, then you should not have eaten the fruit.
Let’s symbolize:
Let
F = You eat the fruit
D = You die
S = You should have eaten the fruit.
Now, compare:
F => (D => ~S)
If you eat the fruit, then if you die, then you should not have eaten the fruit.
(F => D) => ~S
If it is the case that, if you eat the fruit, then you die… then you should not have eaten the fruit.
Complex Material Conditionals
Let’s symbolize a complex material conditional to get a sense of how material conditionals can be used to represent legal or contractual obligations in real life. Suppose that the terms of a lease agreement state the following:
If your payment is late, then if you fail to pay within 5 days of the due date, then you will receive a written notice and unless you pay the full balance plus a late fee within 10 days, your service will be cancelled.
Let’s break down those terms and conditions slightly:
If your payment is late, then
If you do not pay within 5 days of the due date, then
You will receive a written notice
and
Unless
You pay a late fee, and
Pay the full balance within 10 days
…
Your service will be cancelled.
Symbolizing the terms and conditions can help make clear exactly what is or is not being claimed. Let’s first create a dictionary:
Let
L = Your payment is late
D = You pay within 5 days of the due date
W = You will receive a written notice
F = You pay a late fee
B = You pay the full balance within ten days
C = Your service will be cancelled
Now, to symbolize, we need to recognize that this sentence contains many logical connectives embedded within the scope of other logical connectives. Let’s start with the smallest pieces of the puzzle and work our way up, remembering to use parentheses.
F & B
You pay a late fee and you pay the full balance within ten days
Remember that the word “unless” indicates a disjunction, not a conditional:
(F & B) v C
Unless you pay a late fee and you pay the full balance within ten days, or your service will be cancelled
W & ((F & B) v C)
You will receive a written notice, and unless you pay a late fee and you pay the full balance within ten days, or your service will be cancelled
Now, we face our first conditional. Notice that the antecedent is negated: “if you do not pay within 5 days of the due date…”
~D => (W & ((F & B) v C))
If you do not pay within 5 days of the due date, then you will receive a written notice, and unless you pay a late fee and you pay the full balance within ten days, your service will be cancelled.
Lastly, we handle the main conditional of the sentence. Here, the consequent of the conditional is itself a conditional. We place the whole conditional in the consequent within parentheses:
L => (~D => (W & ((F & B) v C)))
If your payment is late, then if you do not pay within 5 days of the due date, then you will receive a written notice, and unless you pay a late fee and you pay the full balance within ten days, or your service will be cancelled.
Material Biconditionals
Earlier we introduced Material Biconditionals, as a conjunction of material conditionals. For instance:
You deserve the penalty if and only if you did the crime.
is equivalent to:
You deserve the penalty if you did the crime.
plus:
You deserve the penalty only if you did the crime.
Which means:
If you did the crime, then you deserve the penalty, and if you deserve the penalty, then you did the crime.
We can symbolize biconditionals in two ways. First, most simply, we can symbolize them as a conjunction of two conditionals:
(P => Q) & (Q => P)
For simplicity, however, the double-arrow symbol can be used:
P <=> Q
In standard logic, the double arrow looks like this: P ↔ Q. Likewise, it is standard to use the triple bar (≡)to represent material conditionals, and ⇔ to represent strict conditionals. However, as a convenience for our class we will use the <=> sign to represent material biconditionals, and instead use ≡ to represent the “strict biconditional”, or logical equivalence.
| P if and only if Q | (P => Q) & (Q => P) |
| P if and only if Q | P <=> Q |
Submodule 8.1 Quiz
Licenses and Attributions
Key Sources:
- Watson, Jeffrey (2019). Introduction to Logic. Licensed under: (CC BY-SA).
Next Page: 8.2. Truth Tables for Material Conditionals