Foundations of Logical Reasoning

Universal Affirmative

("A") a type of categorical claim expressing an inclusive relation between an entire category of things and another category of things. For example: All X are Y, or All men are mortal.

Universal Negative

("E") this is the negative form of universal affirmative, which is a syllogism of the form: No X is Y, or as example: No humans are perfect. This syllogism type is exactly the opposite of proposition "A" explained above.

Particular Affirmative

("I") a type of categorical claim expressing an inclusive relation between a portion of a category of things and another category of things. This type only influences some people and not the whole population. This syllogism is of the form: Some X are Y.

Particular Negative

("O") the opposite of proposition "I" is proposition "O" which is of the form: Some X are not Y. an example of this would be: some cars are not green.

Sufficient Condition

(P) Something that, if true, Guarentees that something else is true. The Antecedent is a "sufficient" condition for the Consequent.

Necessary Condition

(Q) Something that must be the case, if something else is true. The Consequent is a "necessary" condition of the Antecedent.

False Dilemma

A False Dilemma is a fallacy in which a person uses the following pattern of "reasoning": Either claim X is true or claim Y is true. When it is possible that both X and Y could both be false.Claim Y is false. Therefore claim X is true.

Syllogism

A deductive scheme of a formal argument consisting of a major and a minor premise and a conclusion (as in "every virtue is laudable; kindness is a virtue; therefore kindness is laudable")

Universal Quantifier

A logical quantifier of a proposition that asserts that the proposition is true for all members of a class of things

Existential Quantifier

A logical quantifier of a proposition that asserts the existence of at least one thing for which the proposition is true

Conditional Proposition

A proposition which takes the form "if p, then q". P is referred to as the Antecedent, while Q is referred to as the Consequent.

Logical Falsity

A sentence is logically false if and only if, it is not possible for the sentence to be true.

Logical Indeterminacy

A sentence is logically indeterminate if and only if, it is neither logically true nor logically false.

Logical Truth

A sentence is logically true if and only if, it is not possible for the sentence to be false.

Logical Consistency

A set of sentences is logically consistent if and only if, it is possible for all the members of that set to be true. A set of sentences is logically inconsistent if and only if it is not logically consistent.

Affirming the Consequent

Affirming the consequent, sometimes called converse error, is a formal fallacy, committed by reasoning in the form: if P then Q, Q; Therefore: P.

Appeal to False Authority

An argument from the fact that a person judged to be an authority affirms a proposition to the claim that the proposition is true. The fallacy occurs only when the authority cited either (a) is not an authority, or (b) is not an authority on the subject on which he is being cited.

Inductive Strength

An argument has inductive strength to the extent that the conclusion is probable given the premises.

Argument

An argument is a set of two or more sentences, one of which is designated as the conclusion and the others as the premises.

Deductively Invalid

An argument is deductively invalid if and only if, it is possible for the premises to be true and the conclusion to be false.

Deductive Soundness

An argument is deductively sound if and only if, it is deductively valid and all its premises are true.

Deductively Unsound

An argument is deductively unsound if and only if, it is deductively valid and has one or more premises that are false, or is invalid.

Deductively Valid (validity)

An argument is deductively valid if and only if, it is not possible for the premises to be true and the conclusion to be false.

Denying the Antecedent

An invalid form of a deductive argument. For example: if p, then q; not p; therefore, not q. This is invalid because you are denying the antecedent or P.

Fallacies of Ambiguity

Appear to support their conclusions only due to their imprecise use of language. Once terms are clarified, fallacies of ambiguity are exposed. It is to avoid fallacies of this type that philosophers often carefully define their terms before launching into an argument.

Logically Equivalent:

Are related conditional statements that have the same truth value. The members of a pair of sentences are logically equivalent if and only if, it is not possible for one of the sentences to be true while the other sentence is false.

Appeal to Pity

Attempts to persuade using emotion, specifically: sympathy rather than evidence or reason. Playing on the pity that someone feels for an individual or group, can certainly affect what that person thinks about the group. Hence, why it is considered a fallacy of relevance.

Conclusion Indicators

Conclusion indicators are: Therefore, Thus, So, Consequently, As a result, and It follows that.

Common Connectives For Not P

English connectives: Not P , Paraphrases: It Is Not The Case That P, Symbolizations: ( ~ P )

Common Connectives For P or Q

English connectives: P or Q, Paraphrases: either P or Q, Symbolizations: [ P v Q ]

Common Connectives For P & Q

English connectives: [ P And Q, P But Q, P However Q, P Although Q, P Nevertheless Q, P Nonetheless Q, P Moreover Q. ], Paraphrases: both P and Q, Symbolizations: P & Q.

Common Connectives For P if and only if Q

English connectives: [ P if and only if Q, P if but only if Q, P just in case Q.] Paraphrases: P if and only if Q, Symbolizations: [ P ≡ Q. ]

Common Connectives For If P > Q

English connectives: [ if P then Q, P only if Q, Q if P, Q provided that P, Q given P. ] Paraphrases: if P then Q, Symbolizations: [ P > Q. ]

Fallacies of Presumption

Fallacies of Presumption begin with a false, or at least unwarranted assumption; and so fail to establish their conclusion.

Fallacies of Relevance

Fallacies of relevance are attempts to prove a conclusion by offering considerations that simply don't bear on its truth. Arguments that commit fallacies of relevance offer considerations in support of their conclusion that are irrelevant to determining whether that conclusion is true.

Red Herring Fallacy

Is a fallacy in which an irrelevant topic is presented in order to divert attention from the original issue. The basic idea is to "win" an argument by leading attention away from the argument and to another topic.

Begging the Question

Is a fallacy in which the premises include the claim that the conclusion is true or; directly or indirectly assume that the conclusion is true.

Appeal to Force

Is a fallacy of relevance that attempts to persuade using threats. Its Latin name, "argumentum ad baculum", literally means "argument with a cudgel". Disbelief, such arguments go, will be met with sanctions, perhaps physical abuse; therefore, you'd better believe.

Appeal to Consequence

Is a fallacy of relevance that attempts to: motivate belief with an appeal either to the good consequences of believing, or the bad consequences of disbelieving.

Appeal to Tradition

Is a fallacy that occurs when it is assumed that something is better or correct simply because it is older, traditional, or "always has been done."

Inductive Argument

Is an argument that is intended by the arguer merely to establish the probability of its conclusion. The argument's strength is variant based upon the truth values of the premises.

Deductive Argument

Is an argument that is intended by the arguer to be (deductively) valid, that is, to provide a guarantee of the truth of the conclusion provided that the argument's premises (assumptions) are true.

Fallacy of Composition

Is committed when a conclusion is drawn about a whole based on the features of its constituents when, in fact, there is no justification provided for the inference. There are actually two types of this fallacy. The first type arises when a person reasons from the characteristics of individual members of a class or group to a conclusion regarding the characteristics of the entire class or group. The second type is committed when it is concluded that what is true of the parts of a whole must be true of the whole without there being adequate justification for the claim.

Straw Man Fallacy

Is committed when a person simply ignores a person's actual position and substitutes a distorted, exaggerated or misrepresented version of that position.

Perception

Learning about a fact through direct experience

Testimony

Learning about the world through the assertion, claim, or teaching of a person of reliable authority.

Inference

Mental process which moves from premises to a conclusion. The conclusion is inferred from the premises.

Modus Ponens Argument

Modus Ponens is a Logically Valid argument. It is also known as the argument of Affirming the Antecedent. For example: If the Sky is Blue, then the Sun is Shining. The Sky is Blue. Therefore: The Sun is Shining.

Modus Tollens Argument

Modus Tollens is a Logically Valid argument, that is also known as the argument of Denying the Consequent. For example: If the cake is made with sugar, then it is sweet. The cake is not sweet. Therefore: The cake is not made with sugar.

Truth Table For P & Q

P | Q | P & Q T | T | T T | F | F F | T | F F | F | F

Truth Table For P > Q

P | Q | P > Q T | T | T T | F | F F | T | T F | F | T

Truth Table For A Tautology

P | Q | P > Q | Q > P | ( P > Q ) v ( Q > P ) T | T | T | T | T T | F | F | T | T F | T | T | F | T F | F | T | T | T

Truth Table For P v Q

P | Q | P v Q T | T | T T | F | T F | T | T F | F | F

Truth Table For P if and only if Q

P | Q | P ≡ Q T | T | T T | F | F F | T | F F | F | F

Truth Table For 3 Variables

P | Q | R | T | T | T | T | T | F | T | F | T | T | F | F | F | T | T | F | T | F | F | F | T | F | F | F |

Truth Table For ~ P & ( P > Q )

P | Q | ~ P | P > Q | ~ P & ( P > Q ) T | T | F | T | F T | T | F | F | F F | T | T | T | T F | T | T | F | F

Truth Table For Negation

P | ~ P T | F F | T

Premise Indicators

Premise indicators are: Since, Because, As, For, Given that, and Assuming that.

Logic

Studies the relationship between indicator facts, and target facts.

Categorical Syllogism

The third and most commonly used type of syllogisms are the categorical syllogisms. The basic for this syllogism type is: if A is a part of C, then B is a part of C (A and B are members of C). An example of this syllogism type will clarify the above: Major premise: All men are mortal. Minor premise: Socrates is a man. Conclusion: Socrates is mortal. Both premises are known to be valid, by observation or historical facts. Because the two premises are valid, the conclusion must be valid as well. Categorical Syllogisms are divided into four different types, A: Universal Affirmative, E: Universal Negative, I: Particular Affirmative, O: Particular Negative.

Disjunctive Syllogism

These syllogism types do not actually state that a certain premise (major or minor) is correct, but is does states that one of the premises is correct. The basic type for this syllogism is: Either A or B is true, but they can't be true at the same time. Example: Major premise: Either the meeting is at school or at home. Minor premise: The meeting is not at home. Conclusion: Therefore the meeting is at school. The conclusion of the syllogism type may be given, however most of the times the conclusion can be drawn based up on own conclusions.

Equivocation

This Fallacy occurs when a term is used in two or more different senses within a single argument. In an argument words must have the same meaning each time they appear in its premises or conclusion. Arguments that switch between different meanings of words equivocate. Therefore: they are invalid.

Hasty Generalization

This fallacy is committed when a person draws a conclusion about a population based on a sample that is not large enough.

Post Hoc Ergo Propter Hoc

This has been traditionally interpreted as "After this, therefore because of this." This fallacy is committed when it is concluded that one event causes another simply because the proposed cause occurred before the proposed effect. More formally, the fallacy involves concluding that A causes or caused B because A occurs before B and there is not sufficient evidence to actually warrant such a claim.

Ad Hominem

a Fallacy of relevance that attempts to discredit a point of view by discrediting the person that holds it.However, the character of the person that holds a view, entails nothing about the truth of that view.

Hypothetical Syllogism

a deductive argument that contains two premises, at least one of which is a conditional statement. If P > Q, if Q > R, Therefore: if P > R. [ ( >)=Then. ]

Slippery Slope

is a fallacy in which a person asserts that some event must inevitably follow from another without any argument for the inevitability of the event in question. In most cases, there are a series of steps or gradations between one event and the one in question and no reason is given as to why the intervening steps or gradations will simply be bypassed.