Connectives and Syllogisms

• Proposition


  A declarative sentence that 'declares' a fact or facts
• Propositions can be assigned a truth value of either true or false, but not both
• Types of logical connectives

  • Conjunction ("and")
  • Disjunction ("or")
  • Negation ("not")
  • Conditional ("if . . . then")
  • Biconditional ("if and only if")
• Negation


  The symbol ∼ is used to indicate the negation. If there is a proposition p, then the negation of p will also be a proposition, which contains the following properties: When p is true, then the negation of p will be false. When p is false, then the negation of p will be true.
• Conjunction


  The conjunction is indicated by the symbol ∧. If there are two propositions, p and q, then the conjunction of p and q will also be a proposition, which contains the following properties: When p and q are true, then the conjunction of them will be true. When p and q are false, then the conjunction of them will be false.
• Disjunction


  Disjunction is indicated by the symbol ∨. If there are two propositions, p and q, then the disjunction of p and q will also be a proposition, which contains the following properties: When p and q are false, then the disjunction of them will be false. When either p or q or both are true, then the disjunction of them will be true.
• Conditional


  The conditional propositional is also known as the implication proposition. It is indicated by the symbol →. If there are two propositions, p and q, then the conditional of p and q will also be a proposition, which contains the following properties: If there is a proposition that has the form "if p then q", then that type of proposition will be known as the implication or conditional proposition. When p is false, or p and q are true, then the implication of them will be true. When p is true, and q is false, then the implication of them will be false.
• Biconditional


  The bi-conditional propositional is also known as the bi-implication proposition. It is indicated by the symbol ↔. If there are two propositions, p and q, then the bi-conditional of p and q will also be a proposition, which contains the following properties: If there is a proposition that has the form "p if and only if q", then that type of proposition will be known as a bi-implication or bi-conditional proposition. When both p and q are true, or p and q both are false, then the bi-implication of them will be true. In all the other cases, then the bi-conditional of them will be false.
• Syllogism


  A syllogism is a three-part logical argument, based on deductive reasoning, in which two premises are combined to arrive at a conclusion. So, long as the premises of the syllogism are true and the syllogism is correctly structured, the conclusion will be true.
• Major premise


  The general premise in a syllogism
• Minor premise


  The premise in a syllogism that specifically scales down the major premise
• Conclusion


  The part of a syllogism that joins the two premises
• Syllogism examples


  • All frogs are amphibians
  • All toads are frogs
  • Therefore, all toads are amphibians
  • All flowers need light
  • Tulips are flowers
  • Therefore, tulips need light
• There are 256 standard form categorical syllogistic forms, but only 15 of these forms are valid
• Moods of syllogism


  • Universal Affirmative (A)
  • Universal Negative (E)
  • Particular Affirmative (I)
  • Particular Negative (O)
• Figure #1


  The middle term (M) can occur as the subject term of the major premise and the predicate term of the minor premise
• Figure #1 example


  • All Micronesians are Pacific Islanders
  • All Kosraens are Micronesians
  • All Kosraens are Pacific Islanders
• Figure #2


  The middle term (M) can occur as the predicate terms of both the major and minor premises
• Figure #2 example


  • All lizards are reptiles
  • All frill lizards are reptiles
  • All frill lizards are lizards
• Figure #3


  The middle term (M) can occur as the subject terms of both the major premise and the minor premise
• Figure #3 example


  • All rabbits are long eared
  • No rabbits are cats
  • No cats are long eared
• Figure #4


  The middle term can occur as the predicate term of the major premise and the subject term of the minor premise
• Figure #4 example


  • All beef dishes are savory food
  • Some savory food are not spicy food
  • Some spicy food are not beef dishes
• Rules for testing categorical syllogism

  • Must have only three terms, each of which designates the same class throughout (Fallacy of Illicit Terms)
  • Cannot have two negative premises (Fallacy of Two Negatives)
  • Must have a negative conclusion if either premise is negative (Fallacy of Illicit Quality)
  • Cannot have a conclusion with a existential quantity if both premises are universal in quantity (Existential Fallacy)
  • Must distribute the major term in the major premise if the major term is distributed in the conclusion (Fallacy of Illicit Major)
  • Must have a distributed minor term in the minor premise if the minor term is distributed in the conclusion (Fallacy of Illicit Minor)
  • Must have a distributed middle term in at least one premise (Fallacy of Illicit Middle)
• Conclusion is the statement that follows from the premises.
• Premise is any proposition or set of propositions used to support another proposition, especially one in an argument.
• A syllogism is a form of deductive reasoning consisting of two statements (premises) followed by a conclusion.
• Premise is a proposition or assertion made to support another proposition or argument.
• Major Premise is the first premise in a syllogism; it contains the major term.
• Distributive property states that when multiplying an expression by a sum, we can distribute it across all addends.
• Minor Premise is the second premise in a syllogism; it contains the minor term.
• Syllogistic reasoning is a type of logical reasoning where conclusions follow necessarily from given premises.
• Induction is a method of reasoning that involves making generalizations about a population based on observations made within it.
• Middle Term is the term common to the major and minor premises.
• Deduction is a process of drawing valid conclusions based on established principles or facts.
• In a syllogistic argument, there must be exactly three terms involved: the major term, the minor term, and the middle term.
• Associative Property states that when adding or subtracting expressions, we can change their groupings without changing the result.
• The conclusion is the final statement reached through logical reasoning based on given information.
• The first step in constructing a valid syllogism is to identify the subject and predicate of each premise and the conclusion.
• Abduction is a form of non-deductive reasoning that seeks explanations for observed phenomena.