Deductive validity refers to the fact that the premises provided upon which conclusions are being made are strong and can only lead towards one conclusion, if the premises are taken to be true. In fact, the solidity of a deductive argument significantly depends on the validity of the premises, if the premises are not valid, most likely the conclusion made from the premise is not valid as well. Key to constructing a deductively valid argument is the ability to include universal syllogism, predicate instantiation, as well as an affirmation of the antecedent. Euclid’s argument does meet the three conditions described above. The assertion that what equal the same are equal serves to establish universal syllogism, the second premise leads to predicate instantiation, as it equates the two sides of as triangle to the “same”, before reaffirming the antecedent, which is done by restating the initial premise, and relating it to the fact that the two sides are both equal to the same, essentially making them equal. Euclid’s statement is therefore deductively valid as it meets the necessary conditions, with acceptance of the two premises as true the only requirement for the conclusion to be accepted. It would not be logical to accept the first two statements as true and reject the conclusion.
The tortoise is unwilling to accept the conclusion because it does not fully believe in the two premises provided by Euclid’s assertion. In fact, it is perhaps plausible to claim that the tortoise does not believe in the logical truth of Euclid’s premises, which essentially means that he does not operate within the logical constants that the two premises provide. This further means that the tortoise is therefore, in a position to create other possible conclusions that are not within the logical constants provided by the two premise statements. In essence, the tortoise is not operating within the boundaries of logic, as he admits that he is willing to accept the assertions and aspersions made by the two premises but not the conclusion. As as such, it is possible to argue that the tortoise is not willing to accept not just the conclusions provided by Euclid, but also the use of logical argument. By not being bound by a need to abide by logical thought, the tortoise is not likely to agree with the logical constants applied in the argument, as well as the logical consequences normally necessary for a conclusion to be made in cases of logical arguments.
Lewis Carroll attempts to highlight the futility of attempting to engage in a logical argument with an entity that does not believe in operating with the logical constants usually applied in logical argument. The manner with which the tortoise continually convinces Achilles to keep on writing new premises with each attempt to convince him is proof of the fact that the tortoise was not really willing to engage in logical argument, specifically to recognize the concept of logical consequences. Furthermore, at the initial stage when the tortoise tells Achilles at the beginning of their argument, that he shall need all the leaves of the notebook, which at the beginning is almost empty. This is clear proof that the mind of the tortoise is essentially made up, and there is nothing really that Achilles can do, to convince the tortoise of the fact that by accepting the two premises, he is already reaffirming belief in the logical argument put forth by Euclid. The fact that even after reaching 1001 attempts, the tortoise still sees them proceeding with no change in opinion, is perhaps the clearest display of a lack of faith and belief in the mantra of logical argument.
The story of Achilles and the Tortoise casts doubt on the rational acceptability of deductive logic, because the tortoise while accepting that the first two premises are true, proceeds to reject the conclusion arising from it is a clear indication that it does not believe in the concept of logical consequence. Secondly, the tortoise’s insistence on Achille’s writing whatever premise they arrive at is most likely to demonstrated to Achille’s the futility of his attempts at convincing him using logical argument or reasons. The argument essentially served to remind all of the most important part of any argument. First, there is a huge need to establish the ground rules of argument. Without this, putting up a convincing argument, particularly using logical reasoning, can be very difficult. In essence, one comes to the realization that providing a clear connection between the premises offered and the conclusion arrived at is very important. as without proper explanation, proving proof for logical arguments and reaffirmation of the claims can only be done by appealing to the individuals belief in the process that is logical argument. This in particular, is the greatest problem faced by Achilles, as he has no tangible evidence he could use to substantiate claims within the conclusion. At the same time, being able to convince an individual through a logical argument, requires that one is conversant with the deductive logical approach, which can same plenty of time.
Carroll, L. (1895) What the Tortoise said toAchilles.