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Introduction & Symbols
Introduction & Symbols
Logical Operators & Translation
Logical Operators & Translation
By way of introduction, let's say that propositional logic is the logic that evaluates propositions or statements. Readers should note that many systems also refer to propositional logic as sentential logic, or the logic of sentences.
By way of introduction, let's say that propositional logic is the logic that evaluates propositions or statements. Readers should note that many systems also refer to propositional logic as sentential logic, or the logic of sentences.
That is, propositional logic gives us the tools to evaluate, compare, and assess the truth value of statements, and arguments composed of statements. More precisely, propositional logic deals with whole or fundamental statements in a way that allows us to formally assess the properties of that language. Propositional logic, then, is our first foray into formal symbolic logic.
That is, propositional logic gives us the tools to evaluate, compare, and assess the truth value of statements, and arguments composed of statements. More precisely, propositional logic deals with whole or fundamental statements in a way that allows us to formally assess the properties of that language. Propositional logic, then, is our first foray into formal symbolic logic.
Propositional Logic: Symbols
Propositional Logic: Symbols
As was just explained, the focus of propositional logic starts with statements or propositions. In order to best assess the logical structure of propositions, it becomes necessary to translate them from ordinary language into an artificial “language” which will allow us to avoid any confusion or distraction by the argument's content.
As was just explained, the focus of propositional logic starts with statements or propositions. In order to best assess the logical structure of propositions, it becomes necessary to translate them from ordinary language into an artificial “language” which will allow us to avoid any confusion or distraction by the argument's content.
Amazingly, all propositions can be translated into this symbolic language. This is an important element of logic's applicability to coding and computer sciences. The tricky part is that the same proposition can be written in many different ways. Below are some helpful hints for translating commonly used phrases.
Amazingly, all propositions can be translated into this symbolic language. This is an important element of logic's applicability to coding and computer sciences. The tricky part is that the same proposition can be written in many different ways. Below are some helpful hints for translating commonly used phrases.
NOTE: These semantics will allow us to reproduce even the most complex statements in order to analyze the structure of arguments, and determine their validity. Readers should note that each set of symbols is merely a convention. Different books, disciplines, and systems use different sets of symbols. We trust that the symbols adopted here serve adequately. For simplicity, the remainder of these resources will prioritize the following operators: ( & ), ( ⟶ ), and ( ⟷ ) over and above ( • ), ( ⊃ ), and ( ≡ ), respectively.
NOTE: These semantics will allow us to reproduce even the most complex statements in order to analyze the structure of arguments, and determine their validity. Readers should note that each set of symbols is merely a convention. Different books, disciplines, and systems use different sets of symbols. We trust that the symbols adopted here serve adequately. For simplicity, the remainder of these resources will prioritize the following operators: ( & ), ( ⟶ ), and ( ⟷ ) over and above ( • ), ( ⊃ ), and ( ≡ ), respectively.
Translating Simple Statements
Translating Simple Statements
In order to apply the logical operators above, we first need to understand their role in relation to simple vs. compound statements. A simple statement has one subject and one predicate. Each simple statement should be symbolized with a single letter variable (P, Q, R, S, etc.), and if appropriate, that variable should reflect the subject of the statement. If however, there are multiple propositions about the same subject, it is best to use a variable that reflects the predicate of the statement to avoid false equivalency.
In order to apply the logical operators above, we first need to understand their role in relation to simple vs. compound statements. A simple statement has one subject and one predicate. Each simple statement should be symbolized with a single letter variable (P, Q, R, S, etc.), and if appropriate, that variable should reflect the subject of the statement. If however, there are multiple propositions about the same subject, it is best to use a variable that reflects the predicate of the statement to avoid false equivalency.
Notice that when dealing with simple statements like those above, no logical operators are needed.
Notice that when dealing with simple statements like those above, no logical operators are needed.
Translating Complex Statements
Translating Complex Statements
Logical operators (sometimes also referred to as 'connectives') are necessary when only dealing with compound statements. A compound statement combines at least one logical operator with one or more simple statements. The negation ( ~ ) operator can be attached to a single variable, however, all other operators should be used to connect two variables together.
Logical operators (sometimes also referred to as 'connectives') are necessary when only dealing with compound statements. A compound statement combines at least one logical operator with one or more simple statements. The negation ( ~ ) operator can be attached to a single variable, however, all other operators should be used to connect two variables together.
HELPFUL HINTS
Propositional Logic: Translation
HELPFUL HINTS
Propositional Logic: Translation
Translating CONDITIONALS
Translating CONDITIONALS
Conditional statements ( ⟶ , ⊃ ) are composed of two parts: the antecedent (if) and the consequent (then). However, each part may not always be indicated by 'if' and 'then'. Sometimes other words are used, or they may be omitted altogether. Below are some common indicator words to determine which component is which:
Conditional statements ( ⟶ , ⊃ ) are composed of two parts: the antecedent (if) and the consequent (then). However, each part may not always be indicated by 'if' and 'then'. Sometimes other words are used, or they may be omitted altogether. Below are some common indicator words to determine which component is which:
Translating CONJUNCTIVES
Translating CONJUNCTIVES
When conjunctive statements ( & , • ) include a negation ( ∼ ), it can be difficult to determine whether the negation applies to a single variable, or the entire proposition. For example:
When conjunctive statements ( & , • ) include a negation ( ∼ ), it can be difficult to determine whether the negation applies to a single variable, or the entire proposition. For example:
Translating DISJUNCTIONS
Translating DISJUNCTIONS
Similarly, when disjunctive statements ( v ) include a negation ( ∼ ), it can be difficult to determine whether the negation applies to a single variable, or the entire proposition. For example:
Similarly, when disjunctive statements ( v ) include a negation ( ∼ ), it can be difficult to determine whether the negation applies to a single variable, or the entire proposition. For example:
Translating with Multiple Operators
Translating with Multiple Operators
Some statements might be so complex that they have multiple operators, in which case we will need additional punctuation, inducing parentheses ( ) and brackets [ ] , to separate the main operator from the secondary operator(s). If you have more than two simple statements, you will be in need of more than one operator. In which case we will need to begin by identifying the statement’s main operator (of which there is one) and secondary operator(s) (of which there can be many).
Some statements might be so complex that they have multiple operators, in which case we will need additional punctuation, inducing parentheses ( ) and brackets [ ] , to separate the main operator from the secondary operator(s). If you have more than two simple statements, you will be in need of more than one operator. In which case we will need to begin by identifying the statement’s main operator (of which there is one) and secondary operator(s) (of which there can be many).
Identifying the Main Operator
Identifying the Main Operator
A helpful trick for identifying the main operator is that it will most likely be near the punctuation which bestows the most accurate meaning on the proposition. The main operator will always go outside of the parentheses and brackets. It is also worth restating that the negation ( ~ ) operator can be attached to a single variable or on the outside of a statement in parentheses or brackets, while all other operators should only be used to connect two variables together. Finally, parentheses should be used first, with brackets after that. Compare the following examples:
A helpful trick for identifying the main operator is that it will most likely be near the punctuation which bestows the most accurate meaning on the proposition. The main operator will always go outside of the parentheses and brackets. It is also worth restating that the negation ( ~ ) operator can be attached to a single variable or on the outside of a statement in parentheses or brackets, while all other operators should only be used to connect two variables together. Finally, parentheses should be used first, with brackets after that. Compare the following examples:
How To Translate into Propositional Logic
How To Translate into Propositional Logic
IN 4 EASY STEPS: We will now translate a proposition one step at a time. The steps we will use should be applicable to translating any statement in propositional logic.
IN 4 EASY STEPS: We will now translate a proposition one step at a time. The steps we will use should be applicable to translating any statement in propositional logic.
STEP 1:
STATEMENT TYPE
STEP 1:
STATEMENT TYPE
First, we must determine whether or not we are dealing with a simple or compound statement. This will determine whether or not we will be using any logical operators. Consider the following example:
First, we must determine whether or not we are dealing with a simple or compound statement. This will determine whether or not we will be using any logical operators. Consider the following example:
This is a compound statement (and quite a complex one at that).
This is a compound statement (and quite a complex one at that).
STEP 2:
VARIABLE SELECTION
STEP 2:
VARIABLE SELECTION
Next we will assign variables. For each simple statement, select the most appropriate variable that best captures the meaning of that statement (either by reference to the statement's subject or predicate). A helpful trick is to list the variables in the order that they appear with space in between to add their connectives, as illustrated below:
Next we will assign variables. For each simple statement, select the most appropriate variable that best captures the meaning of that statement (either by reference to the statement's subject or predicate). A helpful trick is to list the variables in the order that they appear with space in between to add their connectives, as illustrated below:
After variables have been selected, they are best laid out in the order that they appear with space in between to insert operators and any other necessary notation.
After variables have been selected, they are best laid out in the order that they appear with space in between to insert operators and any other necessary notation.
STEP 3:
OPERATOR SELECTION
STEP 3:
OPERATOR SELECTION
Next, we will plug in the appropriate operators. For each compound statement, determine which logical operator will be used to connect each constituent simple statement. Be sure to also capture any negations in the statement.
Next, we will plug in the appropriate operators. For each compound statement, determine which logical operator will be used to connect each constituent simple statement. Be sure to also capture any negations in the statement.
Notice here that two conditionals, a conjunction, and a negation have been added to the statement. However, we have a problem with our second conditional statement: the last two variables of the statement ('W' and 'M') actually come after the second “if”, so they should actually be placed as the antecedent of the second conditional operator while the 'H' variable should be the consequent, as illustrated below:
Notice here that two conditionals, a conjunction, and a negation have been added to the statement. However, we have a problem with our second conditional statement: the last two variables of the statement ('W' and 'M') actually come after the second “if”, so they should actually be placed as the antecedent of the second conditional operator while the 'H' variable should be the consequent, as illustrated below:
Here the variables have been rearranged to correctly reflect the semantic meaning of the sentence. This is important since, as we saw earlier, conditional statements sometimes have the consequent listed prior to the antecedent.
Here the variables have been rearranged to correctly reflect the semantic meaning of the sentence. This is important since, as we saw earlier, conditional statements sometimes have the consequent listed prior to the antecedent.
STEP 4:
ISOLATING MAIN OPERATOR
STEP 4:
ISOLATING MAIN OPERATOR
Finally, we will need to isolate the main and secondary operators, unless we are dealing with a simple statement. For complex compound statements, identify the main logical operator and place parentheses (and if needed, brackets) around each constituent statement. Be sure to identify the main logical operator and ensure that it is placed outside all parentheses and brackets, with the most secondary operators placed within the parentheses.
Finally, we will need to isolate the main and secondary operators, unless we are dealing with a simple statement. For complex compound statements, identify the main logical operator and place parentheses (and if needed, brackets) around each constituent statement. Be sure to identify the main logical operator and ensure that it is placed outside all parentheses and brackets, with the most secondary operators placed within the parentheses.
Recall that the main logical operator will most often be placed near the semantically meaningful punctuation of the statement. Remember to begin separating the main logical operator from the secondary operators with parentheses, and then move on to brackets.
Recall that the main logical operator will most often be placed near the semantically meaningful punctuation of the statement. Remember to begin separating the main logical operator from the secondary operators with parentheses, and then move on to brackets.
Notice here that we have begun by identifying the rst conditional as the main logical operator since it appears closest to the punctuation in the statement (the comma). This means that the variable 'R' is the antecedent, and the consequent appears to be the entirety of the remaining statement. However, we now know that the three remaining variables on the right side of the main logical operator cannot all be in parentheses (since parentheses can connect two variables at most), so they will be placed in brackets, as illustrated below:
Notice here that we have begun by identifying the rst conditional as the main logical operator since it appears closest to the punctuation in the statement (the comma). This means that the variable 'R' is the antecedent, and the consequent appears to be the entirety of the remaining statement. However, we now know that the three remaining variables on the right side of the main logical operator cannot all be in parentheses (since parentheses can connect two variables at most), so they will be placed in brackets, as illustrated below:
As was just reiterated, we also know that what remains in the brackets will need to be broken up further, since operators can at most connect two variables. This means that we will need to add parentheses around the most secondary compound statement inside the brackets. This change in notation is illustrated below:
As was just reiterated, we also know that what remains in the brackets will need to be broken up further, since operators can at most connect two variables. This means that we will need to add parentheses around the most secondary compound statement inside the brackets. This change in notation is illustrated below:
Notice here that since the variables 'W' and 'not M' form the antecedent of this secondary conditional statement, they will be placed together within the parentheses. As the consequent of this secondary conditional statement, the variable 'H' will go outside of the parentheses, but remain inside of the brackets.
Notice here that since the variables 'W' and 'not M' form the antecedent of this secondary conditional statement, they will be placed together within the parentheses. As the consequent of this secondary conditional statement, the variable 'H' will go outside of the parentheses, but remain inside of the brackets.
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