Game Theory and Logic

A Nash equilibrium is a set of strategies, one for each player, such that no player can improve their outcome by unilaterally changing their strategy while the other players keep their strategies unchanged.

The dominant strategy for each player in the Prisoner's dilemma is to defect, regardless of what the other player does. This is because each player can always improve their outcome by defecting, no matter what the other player does.

In a zero-sum game, the sum of the payoffs to all players is always zero. This is because for every winner, there must be a loser, and the amount that the winner gains is exactly equal to the amount that the loser loses.

The Pareto efficiency criterion is a social welfare criterion that states that a social state is Pareto efficient if it is impossible to make any one person better off without making someone else worse off. In other words, a Pareto efficient social state is one where it is impossible to improve the welfare of any one individual without harming the welfare of another individual.

Deductive logic is based on the principle of syllogism, which states that if two premises are true, then the conclusion must also be true. Inductive logic, on the other hand, is based on the principle of generalization, which states that if a certain pattern is observed in a number of cases, then it is likely to hold true in future cases as well.

A valid argument is one where the conclusion follows logically from the premises, regardless of whether the premises are true or not. A sound argument, on the other hand, is one where both the premises are true and the conclusion follows logically from the premises.

The fallacy of affirming the consequent is the logical fallacy of assuming that if the consequent of a conditional statement is true, then the antecedent must also be true. This is a fallacy because the truth of the consequent does not guarantee the truth of the antecedent.

The fallacy of denying the antecedent is the logical fallacy of assuming that if the antecedent of a conditional statement is false, then the consequent must also be false. This is a fallacy because the falsity of the antecedent does not guarantee the falsity of the consequent.

A categorical syllogism is one where the premises and conclusion are all categorical statements, which means that they are statements that make a general claim about a class of things. A hypothetical syllogism, on the other hand, is one where at least one of the premises or the conclusion is a hypothetical statement, which means that it is a statement that makes a claim about a conditional relationship between two things.

A deductive argument is one where the conclusion is guaranteed to be true if the premises are true. This is because the conclusion is a logical consequence of the premises. An inductive argument, on the other hand, is one where the conclusion is only probably true even if the premises are true. This is because the conclusion is not a logical consequence of the premises, but is instead based on the observation of a pattern in a number of cases.

A necessary condition is one that must be met in order for something to happen. In other words, if the necessary condition is not met, then the thing cannot happen. A sufficient condition, on the other hand, is one that is enough to ensure that something happens. In other words, if the sufficient condition is met, then the thing will definitely happen.

A tautology is a statement that is always true, regardless of the values of its variables. This is because the statement is true by virtue of its logical form. A contradiction, on the other hand, is a statement that is always false, regardless of the values of its variables. This is also because the statement is false by virtue of its logical form.

A universal quantifier asserts that something is true for all members of a set. This means that the statement is true regardless of which member of the set is chosen. An existential quantifier, on the other hand, asserts that something is true for at least one member of a set. This means that the statement is true if there is at least one member of the set for which the statement is true.

A deductive argument is one where the conclusion is guaranteed to be true if the premises are true. This is because the conclusion is a logical consequence of the premises. An inductive argument, on the other hand, is one where the conclusion is only probably true even if the premises are true. This is because the conclusion is not a logical consequence of the premises, but is instead based on the observation of a pattern in a number of cases.