Truth tables or truth value table, is a table that shows the truth value of a compound statement, for each combination of truth values that can be assigned to its components.
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- 1 Origin
- 2 What are the tables of truth
- 3 The tables of truth
- 1 Trivalent truth tables
- 2 Construction of Tables of Truth
- 3 Other divergent truth tables
- 4 Sources
It was developed by Charles Sanders Peirce in the 1880s , but the most popular format is the one introduced by Ludwig Wittgenstein in his Tractatus logico-philosophicus, published in 1921 .
The table of “truth values” is used in the field of logic , to obtain the truth (V) or falsehood (F), truth values of an expression or a proposition. They also serve to determine if a certain inference scheme is formally valid as an argument, concluding that this is a tautology (we speak of a tautology when all the values in the mentioned table are “V” or is true ).
What are the tables of truth
The truth tables are, on the one hand, one of the simplest and best-known methods of formal logic, but at the same time also one of the most powerful and clear. To understand truth tables well is, to a large extent, to understand formal logic itself well.
Fundamentally, a truth table is a device for demonstrating certain logical and semantic properties of natural language statements or formulas of the language of propositional calculus:
- . Sin are tautological, contradictory or contingent
- . What are your truth conditions
- . What is its inferential role, that is, what are its logical conclusions and from what other propositions it follows logically.
The tables of truth
These tables can be constructed by interpreting logical signs such as: no, or, and, if … then, yes and only if. The interpretation corresponds to the meaning that these operations have within reasoning. A correspondence can be established between the results of these tables and the mathematical logical deduction . Consequently, truth tables are a decision method to check whether or not a proposition is a theorem. For the construction of the table, the value 1 (one) will be assigned to a true statement and 0 (zero) to a false statement.
- Negation: The truth value of negation is the opposite of the denied proposition.
- The conjunctionserves to indicate that two conditions are met simultaneously, for example:
The function is increasing and is defined for positive numbers, we use For the conjunction p ^ q to be true the two expressions that intervene must be true and only in that case as indicated by their truth table.
- Disjunction: Disjunction is only false if its two components are false.
With the disjunction as opposed to the conjunction, two expressions are represented that affirm that one of the two is true, so it is enough that one of them is true for the expression p ∨ q to be true.
- Conditional: The conditional is only false when the antecedent is true and the consequent is false. You cannot follow falsehood from the truth.
- Biconditional: The biconditional is only true if its components have the same truth value.
A proposition that is true for any truth value of its components is called tautology . Therefore, the last column of your truth table will consist of only ones. Contradiction is the negation of a tautology, therefore it is a false proposition whatever the truth value of its components. The last column of the truth table of a contradiction will only be made up of zeros.
Trivalent truth tables
Traditional truth tables can be rewritten by leaving empty boxes where the truth value of the atomic formula is irrelevant, for example, the disjunction table:
The first two lines indicate that no matter what the truth value of one of the disjuncts is, as long as the other is true, the disjunction will be true. In the same way, the conjunction table could be abbreviated as follows:
The last two lines indicate that no matter what the truth value of one of the disjuncts is, as long as the other is false, the conjunction will be false. The advantage of this type of tables is that they allow them to be extended very naturally to allow a third value of truth that is neither true nor false. It will be called “I” by “undetermined”. The abbreviated table of classical disjunction can now be used to develop a truth table (not abbreviated) for trivalent disjunction. First step: identify the different nine possibilities of combinations for two variables
Step Two: Use the first two lines of the abbreviated table to determine the truth value of the rows with at least one true argument:
Step Three: Since the last line of the abbreviated table is also the last line of the new table, the same true value corresponds to it: false.
Fourth step: Finally, how are the lines that are true or false according to the original table, the lines that still do not have truth value, since they are neither true (but would have remained as such in the second step) or false ( since they were not like this in the third step), they must be indeterminate!
In some cases, this truth table appears, not in three columns, but in one box. This has the advantage of making the pattern that emerges from the table clearer. Following the same steps, the conjunction table is obtained:
Construction of Tables of Truth
Algorithm to construct a truth table of a formula in proposition logic.
- Write the formula with a numberabove each operator that indicates its hierarchy. Positive integers are written in order, where the number 1 corresponds to the highest-ranking operator. When two operators have the same hierarchy, the lower number is assigned to the one on the left.
- Construct the syntax treestarting with the formula at the root and using in each case the least hierarchy operator. That is, from the largest number to the smallest.
- Number the branches of the tree in sequence starting from the leavesto the root, with the only condition that a branch can be numbered until the children are numbered. To start with the numbering of the sheets it is a good idea to do it in alphabetical order, so everyone gets the lines of the table in the same order to be able to compare results.
- Write the table headings the formulas following the numbering given to the branches in the syntax tree.
- Assign to the atoms, the leaves of the tree, all the possible truth values according to the established order. Of course, the order is arbitrary, but since the number of permutations is n !, it is convenient to establish an order to be able to easily compare results.
- Assign truth value to each of the remaining columns according to the operator indicated in the syntax tree using the truth table. It is convenient to memorize the tables of the operators, at the beginning they can have a summary with all the tables while they are memorized.
- The last column, corresponding to the original formula, is the one that indicates the possible truth values of the formula for each case.
Other divergent truth tables
In addition to the multipurpose and intentional tables, there are many other truth tables. For example, there are truth tables in which the lines branch into two or more sub-lines and are useful for what in logic we call supervaluations. There are also tables with values and more than 2n lines, how is this possible? Because, unlike traditional tables, in these tables the order of the lines does matter, in such a way that repeated lines count as different lines. Finally, there are also two-dimensional tables, originally used in certain intentional logics, but popularized thanks to the work of Robert Stalnaker and others.