What is This?

This is a supplement to section 2.10 of the Logic Text. You should read that before you read this. Section 2.10 introduces truth tables, and explains how we can use truth tables to determine whether or not a sentence is a tautology, and whether or not an argument is tautologically valid.

Truth Functions

Each of our connectives can be thought of as expressing a truth-function. A truth-function is a function that takes one or more truth values as inputs, and outputs a truth value. So, for example, the rule for negation is:

• if the input, the truth value of $${\mathord{◻}}$$, is T, then the output, the truth value of $${\mathord{\sim}}{\mathord{◻}}$$, is F.
• if the input is F, then the output is T

We can summarize this with the following table:

$${\mathord{◻}}$$ $${\mathord{\sim}}{\mathord{◻}}$$
T F
F T

The first row of the table corresponds to our first bullet point: the truth value assigned to $${\mathord{◻}}$$ is T, so the truth value assigned to $${\mathord{\sim}}{\mathord{◻}}$$ is F. The second row corresponds to the second bullet point.

The following table summarizes the truth-functions expressed by each of our binary connectives:

$${\mathord{◻}}$$ $${\mathord{○}}$$ $${\mathord{◻}}{\mathord{\wedge}}{\mathord{○}}$$ $${\mathord{◻}}{\mathord{\vee}}{\mathord{○}}$$ $${\mathord{◻}}{\mathbin{\rightarrow}}{\mathord{○}}$$ $${\mathord{◻}}{\mathord{\leftrightarrow}}{\mathord{○}}$$
T T T T T T
T F F T F F
F T F T T F
F F F F T T

Again, each row represents a possible input—in the first row, we assign T to both $${\mathord{◻}}$$ and $${\mathord{○}}$$, for example—and each column represents the outputs given those inputs.

Formal Validity and Logical Truth

Validity is a property of arguments. Truth is a property of sentences.

Some sentences are true in virtue of their logical form. We call these sentences logical truths:

Logical Truth
A sentence is a logical truth just in case no sentence that has its logical form is false.

Some arguments are valid in virtue of their logical form. We call these arguments formally valid:

Formal Validity
An argument is formally valid just in case no argument with the same logical form has true premises but a false conclusion.

You cannot apply these definitions if you don’t have a theory of logical form. We now have a theory of sentential logical form—that is, logical form due to truth-functional sentential connectives, like ‘and’ and ‘or’ and ‘if…then’. So we can now apply these definitions.

We call a sentence that is a logical truth in virtue of its sentential logical form a tautology. We can show that a sentence is a tautology by showing that it is assigned T on every row of its truth table. That shows that no sentence with that sentential logical form can ever be false.

We call an argument that is formally valid in virtue of its sentential logical form tautologically valid. We can show that an argument is tautologically valid by showing that no row of its truth table assigns T to all of its premises, but F to its conclusion. That shows that no argument with thhis form can ever have true premises and a false conclusion.

Tautologies

Here are some tautologies:

• $$P{\mathbin{\rightarrow}}P$$
• $$P{\mathord{\vee}}{\mathord{\sim}}P$$
• $$P{\mathord{\leftrightarrow}}P$$
• $${\mathord{\sim}}(P{\mathord{\wedge}}{\mathord{\sim}}P)$$

Try plugging in sentences for $$P$$, and see if you can come up with any that strike you as false:

$$P{\mathbin{\rightarrow}}P$$
If , then .
$$P{\mathord{\vee}}{\mathord{\sim}}P$$
Either , or it is not the case that
$$P{\mathord{\leftrightarrow}}P$$
just in case
$${\mathord{\sim}}(P{\mathord{\wedge}}{\mathord{\sim}}P)$$
Both it is not the case that ( and it is not the case that )

To show that a sentence is a tautology, we construct its truth table, and show that the sentence is assigned T on every row of its truth table. For example, here is the truth table for $$P{\mathord{\vee}}{\mathord{\sim}}P$$:

$$P$$ $$P{\mathord{\vee}}{\mathord{\sim}}P$$
T ?
F ?

The truth table for a sentence first lists each sentence letter contained in the sentence—so here, just $$P$$—followed by the sentence itself. Each row of the truth table represents a possible assignment of truth values to the sentence letters. The truth table for the sentence must have one row for each possible assignment. Here there are two possible assignments, and so two rows: T to $$P$$, or F to $$P$$.

We calculate the truth value for the sentence on the first row as follows. First, we write the truth value assigned to the sentence letter under each of its occurrences:

$$P$$ $$P$$ $${\mathord{\vee}}$$ $${\mathord{\sim}}$$ $$P$$
T T T
F

Next, we note that $${\mathord{\sim}}P$$ is F when $$P$$ is T:

$$P$$ $$P$$ $${\mathord{\vee}}$$ $${\mathord{\sim}}$$ $$P$$
T T F T
F

Finally, we note that the disjunction is T when its first disjunct is T and its second disjunct is F (I’ve put the F in boldface to indicate that it is the truth-value of the whole sentence):

$$P$$ $$P$$ $${\mathord{\vee}}$$ $${\mathord{\sim}}$$ $$P$$
T T T F T
F

We calculate the second row in similar fashion. First, we note that $$P$$ is assigned $$F$$:

$$P$$ $$P$$ $${\mathord{\vee}}$$ $${\mathord{\sim}}$$ $$P$$
T T T F T
F F F

Next, we calculate the truth value of $${\mathord{\sim}}P$$:

$$P$$ $$P$$ $${\mathord{\vee}}$$ $${\mathord{\sim}}$$ $$P$$
T T T F T
F F T F

Finally, we calculate the truth value of the disjunction:

$$P$$ $$P$$ $${\mathord{\vee}}$$ $${\mathord{\sim}}$$ $$P$$
T T T F T
F F T T F

Since the sentence is true on all the rows of its truth table, it is a tautology.

Here are some sentences that are not tautologies. Try finding English sentences that make them false:

$$P{\mathord{\wedge}}{\mathord{\sim}}Q$$ Both and it is not the case that .

$$P{\mathbin{\rightarrow}}(P{\mathord{\wedge}}Q)$$ If , then both and .

Here is the truth table for the first of these:

$$P$$ $$Q$$ $$P{\mathord{\wedge}}{\mathord{\sim}}Q$$
T T ?
T F ?
F T ?
F F ?

Again, we calculate the truth value of the sentence for each row.

First, we write down the assigned truth value beneath each occurrence of a sentence letter:

$$P$$ $$Q$$ $$P$$ $${\mathord{\wedge}}$$ $${\mathord{\sim}}$$ $$Q$$
T T T T
T F
F T
F F

Next, we calculate the truth value of $${\mathord{\sim}}Q$$:

$$P$$ $$Q$$ $$P$$ $${\mathord{\wedge}}$$ $${\mathord{\sim}}$$ $$Q$$
T T T F T
T F
F T
F F

Finally, we calculate the truth value of the conjunction. It is F, because the right conjunct is F:

$$P$$ $$Q$$ $$P$$ $${\mathord{\wedge}}$$ $${\mathord{\sim}}$$ $$Q$$
T T T F F T
T F
F T
F F

Since the sentence is assigned F on at least one row of its truth table, it is not a logical truth. Can you find English sentences that are both true, that make this sentence false?

We don’t need to continue: we have shown that the sentence is not a tautology. But let’s complete the table anyway:

$$P$$ $$Q$$ $$P$$ $${\mathord{\wedge}}$$ $${\mathord{\sim}}$$ $$Q$$
T T T F F T
T F T T T F
F T F F F T
F F F F T F

This table shows that the sentence is false if both $$P$$ and $$Q$$ are true (row 1), that it is true if $$P$$ is true and $$Q$$ is false (row 2), and it is true if $$P$$ is false and $$Q$$ is true (row 3), or if $$P$$ is false and $$Q$$ is false (row 4).

Test this result by plugging in English sentences with those truth values into the form, and thinking about the truth value of the resulting complex English sentence:

Both and it is not the case that .

Tautological Validity

Sentences are true or false. Arguments are valid or invalid. So sentences are the sorts of things that can be logically true, and so tautologies. Arguments are the sorts of things that can be logically valid, and so tautologically valid.

Here are some tautologically valid arguments:

• $$P{\mathbin{\rightarrow}}Q\ .\ P {\therefore\ }Q$$
• $$P{\mathbin{\rightarrow}}Q\ .\ {\mathord{\sim}}Q {\therefore\ }{\mathord{\sim}}P$$
• $$P{\mathord{\wedge}}Q {\therefore\ }P$$
• $$P{\mathord{\vee}}Q\ .\ {\mathord{\sim}}Q {\therefore\ }P$$
• $$P{\mathord{\wedge}}Q\ .\ Q{\mathbin{\rightarrow}}R {\therefore\ }R$$

You might recognize the first two as instances of MP and MT, respectively.

We show that an argument is tautologically valid by constructing its truth-table, and showing that there are no rows of the table that assign T to all of the premises, but F to the conclusion.

Here is the truth-table for our first example above:

$$P$$ $$Q$$ $$P$$ $${\mathbin{\rightarrow}}$$ $$Q$$ . $$P$$ . $$Q$$
T T T T T . T . T
T F T F F . T . F
F T F T T . F . T
F F F T F . F . F

Consider this table row by row.

• On the first row, the premises are all true, but so is the conclusion. So this is not a row that has all true premises and a false conclusion.
• On the second row, the first premise is false. So this is not a row that has all true premises and a false conclusion.
• On the third row, the second premise is false. So this is not a row that has all true premises and a false conclusion.
• On the fourth row, the second premise is false. So this is not a row that has all true premises and a false conclusion.

What this tells us is that there is no way that all the premises could be true but the conclusion false. So the argument is tautologically valid.