An important step in representing statements is using appropriate language. For this we will use letters to make it easier to represent the statements.

The first step in this post is to define simple propositions. We define a simple proposition as something that has meaning on its own. For example:

  • “There is a white bear”
  • 10 is greater than 3

Now, an example of a non-simple proposition is “pants”, we have no statement regarding this word.

Simple propositions will be represented by letters p, q, r… Compound propositions will be composed of simple propositions, joined by connectives.

Symbols Definition
$$\sim$$ Denial
$$\wedge$$ e
$$\vee$$ or
$$\Rightarrow$$ implies
$$\Leftrightarrow$$ if and only if

Given the propositions p and q, then

  • \(p\wedge q\) is too.

  • \(p \Rightarrow q\) is too.

  • \(\sim q\) is too.

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Truth Table

A simple way to define a connective is through the truth table. Let’s look at some ways to represent a truth table.

The symbol \(\sim\) is a negation connective, that is, we want the opposite of a statement. For example \(\sim p\) is true if p is false and vice versa, more specifically \(\sim p\) is true (if and only if) p is false.

$$p$$ $$\sim p$$
V F
F V

The symbol \(\wedge\) is a connective that both propositions must be true.

$$p$$ $$q$$ $$p \wedge q$$
V V V
V F F
F V F
F F F

The previous truth table can be understood as:

  • The first two columns are filled in as the possible values ​​for the two propositions.
  • The last column corresponds to the results with the connective between the two propositions

  • We note that \(p \wedge q\) is true, if both are true.

The truth table for connective or (\(\vee\))

$$p$$ $$q$$ $$p \vee q$$
V V V
V F V
F V V
F F F

The previous truth table can be understood as:

  • The first two columns are filled in as the possible values ​​for the two propositions.
  • The last column corresponds to the results with the connective between the two propositions

  • We note that \(p \vee q\) is true if at least one proposition is true.

If and only if

The truth table for conceptual or (\(\Leftrightarrow\))

$$p$$ $$q$$ $$p \Leftrightarrow q$$
V V V
V F F
F V F
F F V

The previous truth table can be understood as:

  • The first two columns are filled in as the possible values ​​for the two propositions.
  • The last column corresponds to the results with the connective between the two propositions

  • We note that \(p \Leftrightarrow q\) is true if \(p\) and \(q\) are identical.

Simple implication

The truth table for connective or (\(\Leftrightarrow\))

$$p$$ $$q$$ $$p \Rightarrow q$$
V V V
V F F
F V V
F F V

The previous truth table can be understood as:

  • The first two columns are filled in as the possible values ​​for the two propositions.
  • The last column corresponds to the results with the connective between the two propositions

  • We note that \(p \Rightarrow q\) is false, only when \(p\) is true and \(q\) is false.

Implication for absurdity

The truth table for connective or (\(\Rightarrow\))

$$p$$ $$q$$ $$p \Rightarrow q$$ $$(\sim p) \Rightarrow (\sim q)$$
V V V V
V F F F
F V V V
F F V V

The previous truth table can be understood as:

  • The last two columns are identical
  • Shows that denying the thesis implies non-hypothesis
  • It is an example of proof by absurdity

This post is everything we need to work with connectors through the truth table. The next step is to define the quantifiers.