What is a Truth Table?
A truth table is a mathematical table that determines whether or not a compound assertion is true. Each statement in a truth table generator is usually represented by a letter or variable, such as p, q, or r. In addition, each statement has its own column in the truth table that contains all potential truth values.
Why Traditional Logic Neglect Truth Table?
One of the most common questions concerning conventional logic is why truth tables aren’t taught. Why does traditional logic neglect them, yet modern logic, the most popular type of reasoning encountered in high school and college, uses them?
Many students are exposed to a scattering of reasoning in high school math classes. They will most certainly come across simple truth tables here. Ludwig Wittgenstein, a philosopher, devised truth tables. He created them to go with the calculus that current analytic philosophers had turned logic into. The current system was a quick technique to determine the truth of complicated logical assertions.
Analysis
Consider the following statement:
There are seven days a week and twenty-four hours a day.
We would like to reduce this to its formal constituents as quickly as possible in a modern logic system. Assume the following scenario:
Q = There are 24 hours in a day
P = There are 7 days in a week
If we accomplished this, the statement would be represented as follows:
P and Q are two letters of the alphabet.
How can we tell if the statements P and Q are correct? In this example, P and Q, the elements of the statement determine this in current logic. P and Q are true only if both of the propositions they represent are true. In other words, if it is true to claim that there are seven days in a week and that there are twenty-four hours in a day.
If both of these propositions are untrue, the combined statement is also wrong.
We’d use truth tables to lay out all of P and Q’s truth possibilities so we could see when they’re true and when they’re false:
We don’t need to go to such lengths to ensure that a simple assertion like P and Q is correct. What if you had something like P and (Q) or (If R, then S)?
When statements become more complicated, truth tables can make calculating their truth value considerably easier.
So, if truth tables make determining the truth of assertions easier, why aren’t they taught classical logic?
– The first reason is that they are ineffective in genuine debates or discussions because most statements in common speech and even academic discourse never reach the level of complexity that would necessitate using a truth table to deduce them. They are helpful in specific scientific applications and computer programming (the most practical use of current logic), but they are rarely utilised or required outside of those domains.
– The philosophical differences between the two systems of logic are another reason for the absence of truth tables in classical logic. To put it bluntly, traditional logic does not believe in truth tables.
The concept of “truth functionality” explains why they are employed in one system but not the other. The term “truth functionality” simply indicates that the truth or falsity of a statement’s pieces will inform us whether the complete statement is true or false.
The truth of the assertion P and Q can be deduced from its constituent parts, P and Q. A “conjunctive proposition” is one that links antecedent P with consequent Q. Conjunctive propositions, according to traditional logicians, are the only ones whose truth can be “solved” in a truth table. In this sense, no other kind of logical propositions (P or Q, If P, then Q, and so on) are truth-functional. Their truth value cannot be determined only by the truth value of their pieces.
Examples
However, this does not deter proponents of modern reasoning from attempting to quantify such statements, even if this leads to weird anomalies. Consider the following scenario:
“If it rains, then my dog will get wet.”
We would use a truth table to “solve” for the truth of this sentence in current reasoning. For example, if we assume the following:
P = It rains
Q = My dog gets wet
Then the truth table will be –
Except when P is true and Q is false, this proposition is regarded as true in every situation (line 2). For example, assume my dog is an outside dog with no shelter from the rain. In that case, when it rains, my dog gets wet—both P and Q would be true, and hence (line 1) the complete phrase If it rains, my dog gets wet (If P, then Q) would be true.
Let’s pretend it was raining outside, but my dog was dry and warm in the garage. It would be correct to remark that it was raining in that situation but false to assert that my dog would be wet (line 2). My dog does not get wet when it rains. As a result, the statement is false.
However, what about the other two options? What happens if it doesn’t rain and my dog gets wet (maybe because I sprayed him with the hose), or if it rains, but he doesn’t get wet (lines 3 and 4)? In what sense does it make sense to assert that if it rains, my dog gets wet is true in these cases?
In reality, it’s completely illogical. For example, how could a statement conditional on something occurring be true or false if it does not?
A conditional statement asserts a necessary logical relationship between the rain and your wet dog under the conventional approach. But, on the other hand, modern logic can only keep its system clean and efficient if it can solve the truth value of the whole from the parts.
We try to shoehorn language into formal efficiency like a Procrustean bed. However, certain things are lost in the process, such as common sense.
The truth tables don’t make sense in any case, not simply when the antecedent is untrue. Consider the following statement:
The moon orbits the earth if ducks can swim.
In this situation, both the antecedent and the consequent are correct. As a result, the truth table indicates (line 1) that the entire sentence is correct. Is that true, though? What is the relationship between the fact that ducks can swim and that the moon orbits the earth? Although both the antecedent and the consequent are correct, there is no logical connection. Therefore, the statement is neither true nor false in the old system. It’s just pointless.
Parting words,
The traditional method captures the real-world meaning of conditional statements: asserting a required link between the antecedent (P) and the consequent (S) (Q). However, the only relationship recognised in current logic is the coincidental truth or falsity of the separate pieces.
We contend that the rain and the wet dog have a fundamental metaphysical relationship in the traditional view.
Modern logic was created by persons same like reflection calculator who were opposed to the concept of metaphysics in the first place. Traditional logicians disagree with current philosophers’ belief that language may be quantified. Language is distorted by attempts to measure it.