Chapter 8 Flashcards
Propositional logic
¬ Propositional logic is more broadly applicable than categorical logic. Its use is limited to assessing sufficiency and support in some cases of deductive reasoning. This means that propositional logic can’t be applied to inductive arguments, arguments from analogy, or to certain types of deductive arguments which can’t be adequately described using the tools discussed in the chapter.
¬ On its own, propositional logic can’t show you whether or not a deductive argument is successful; to know that you would also need to evaluate the satisfactoriness of the premises. When evaluating the satisfactoriness of premises or when looking at inductive arguments and arguments from analogy, we must rely on the tools discussed in earlier chapters.
Define propositional logic
¬ a type of symbolic logic that deals with the relationships between propositions using the basic logical connectives: ‘and’, ‘or’, ‘not’, and ‘if…then’.
Logical connective: a word or symbol that relates one statement to another or to itself
Logical connectives
a word or symbol that relates one statement to another or to itself. Not, and, v, and are all logical connectives.
Variable
a capital letter used in propositional logic to represent a proposition
Negation
a statement of the form “not P”. A negation is true if and only if the statement it negates is false.
Antecedent
the first proposition in an “if… then” statement, immediately following the “if”. In the conditional statement, “If P then Q”, P is the antecedent
Consequent
the second proposition in an “if… then” conditional statement, immediately following the “then”. In the conditional statement, “If P, then Q”. Q is the consequent.
Complex statements
A statement that includes more than one logical connective
Component statement
A statement that includes a logical connective and is embedded within a complex statement
Basic truth tables for “and”, “or”, “not”, and “if… then”
o Every proposition is either true or false.
1. Write down your variables in the order that they appear. Then put a column T,F, T, F immediately below the variable
2. Move to the next variable on the left and fill that column with T’s and F’s. In the case there’s only two variables you put down two T’s and two F’s
3. If there are more than two variables, you can employ a similar procedure.
Determine number of rows you’ll need using the formula 2n, where n=the number of variables.
creating long truth tables
- Assign a variable to each of the claims in the argument
- Translate each statement of the argument using variables and logical connectives
- Create the top line of a truth table by plotting various components of the argument. Includes variables, premises, and at least one conclusion. Give each variable its own column, and if any of the premises or conclusion(s) are complex statements, then also give each of the component statements (including negations) its own column
- Fill in each row and column of the truth table. Be sure to begin with the possible truth values of the individual variables, then fill in the truth values of any component statements, and then finally the premises and the conclusion.
- Once you have filled in the entire truth table, check to see if there is any combination where the premises are all true and the conclusion is false. If there is, the argument is invalid since it’s logically possible for the conclusion to be true and the premises false. If there is not, then the argument is valid since it’s impossible for the premises to be true and the conclusion false.
Short table method
- Determine the argument’s variables; translate its statements into variables and logical connectives; and fill in the top row of a truth table with variables, component statements, premises, and the conclusion. This is the same as steps 1 to 3 of the long truth table method, except that you can start out with only one row in your truth table
- Assume the conclusion is false (put F in the column for the conclusion). Then set the truth values of the variables and component statements that are included in the conclusion accordingly. (If the conclusion is S v P, make both S and P false so that the whole disjunction is false). If several different combinations of truth values would make the conclusion false, add additional rows to represent each combination.
- The truth values you have already filled in might dictate the truth values of other component statements and/or premises in the chart. If this is the case, fill those in.
- For each of the rows you have already created, assume the premises are true. Then see if you can fill in the truth values of the variables and component statements that make up those premises. If there is more than one way you can do this for a given row, you will need to add a row to the truth table to show each variation. If the truth values you have determined so far affect the truth values of any other component statements or premises, try to fill in those truth values consistently. At any point in this process, you may discover a contradiction—you many end up needing to set the truth value of a variable or component statement to T when it has already been filled in as F (or vice versa) in Steps 2 or 3. If this happens circle the contradictory truth value on that row to indicate that it is not a case in which the premises are true and the conclusion false.
- Stop when either (a) you have completed a row in which the premises are true without finding any contradictions or (b) you have filled out a row for every possible set of truth values in which the premises are all true and the conclusion is false and you have found a contradiction in every one. If you stop at (a) then it is possible for the premises to be true and the conclusion false, so the argument is invalid. If you stop at (b) then the argument is deductively valid.
