LOGIC abd Critical Thinking

Proposition - an expression in language or signs of something that can be believed, doubted, or denied or is either true or false
~a proposition is identified ontologically as an idea, concept, or abstraction whose token instances are patterns of symbols, marks, sounds, or strings of words.

Inference - the act of passing from one proposition, statement, or judgment considered as true to another whose truth is believed to follow from that of the former.
~is the process of drawing a conclusion by applying clues (of logic, statistics etc.) to observations or hypotheses; or by interpolating the next logical step in an intuited pattern. The conclusion drawn is also called an inference.

Premise - In logic, an argument is a set of one or more declarative sentences (or "propositions") known as the premises along with another declarative sentence (or "proposition") known as the conclusion.

Antecedent is the first half of a hypothetical proposition.
Ex.

• If P, then Q.

This is a nonlogical formulation of a hypothetical proposition. In this case, the antecedent is P, and the consequent is Q.

• If X is a man, then X is mortal.
"X is a man" is the antecedent for this proposition.

• If men have walked on the moon, then I am the king of France.
Here, "men have walked on the moon" is the antecedent.

Consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then".

Ex.
• If P, then Q.
Q is the consequent of this hypothetical proposition.
• If X is a mammal, then X is an animal.

Here, "X is an animal" is the consequent.

• If computers can think, then they are alive.
"They are alive" is the consequent.

The consequent in a hypothetical proposition is not necessarily a consequence of the antecedent.

• If monkeys are purple, then fish speak Klingon.

"Fish speak Klingon" is the consequent here, but clearly is not a consequence of (nor has anything to do with) the claim made in the antecedent that "monkeys are purple".


Inductive reasoning, also known as induction or inductive logic, is a type of reasoning that involves moving from a set of specific facts to a general conclusion. It uses premises from objects that have been examined to establish a conclusion about an object that has not been examined. It can also be seen as a form of theory-building, in which specific facts are used to create a theory that explains relationships between the facts and allows prediction of future knowledge. The premises of an inductive logical argument indicate some degree of support (inductive probability) for the conclusion but do not entail it; i.e. they do not ensure its truth. Induction is used to ascribe properties or relations to types based on an observation instance (i.e., on a number of observations or experiences); or to formulate laws based on limited observations of recurring phenomenal patterns. Induction is employed, for example, in using specific propositions such as:

This ice is cold. (Or: All ice I have ever touched has been cold.)
This billiard ball moves when struck with a cue. (Or: Of one hundred billiard balls struck with a cue, all of them moved.)
...to infer general propositions such as:
All ice is cold.
All billiard balls move when struck with a cue.
Another example would be:

3+5=8 and eight is an even number. Therefore, an odd number added to another odd number will result in an even number.

Deductive Inferences
When an argument claims that the truth of its premises guarantees the truth of its conclusion, it is said to involve a deductive inference. Deductive reasoning holds to a very high standard of correctness. A deductive inference succeeds only if its premises provide such absolute and complete support for its conclusion that it would be utterly inconsistent to suppose that the premises are true but the conclusion false.

Notice that each argument either meets this standard or else it does not; there is no middle ground. Some deductive arguments are perfect, and if their premises are in fact true, then it follows that their conclusions must also be true, no matter what else may happen to be the case. All other deductive arguments are no good at all—their conclusions may be false even if their premises are true, and no amount of additional information can help them in the least.
This is an example of a valid argument. The first premise is false, yet the conclusion is still true.
1. Everyone who eats steak is a quarterback.
2. John eats steak.
3. [Therefore,] John is a quarterback.

Inductive Inferences
When an argument claims merely that the truth of its premises make it likely or probable that its conclusion is also true, it is said to involve an inductive inference. The standard of correctness for inductive reasoning is much more flexible than that for deduction. An inductive argument succeeds whenever its premises provide some legitimate evidence or support for the truth of its conclusion. Although it is therefore reasonable to accept the truth of that conclusion on these grounds, it would not be completely inconsistent to withhold judgment or even to deny it outright.
Inductive arguments, then, may meet their standard to a greater or to a lesser degree, depending upon the amount of support they supply. No inductive argument is either absolutely perfect or entirely useless, although one may be said to be relatively better or worse than another in the sense that it recommends its conclusion with a higher or lower degree of probability. In such cases, relevant additional information often affects the reliability of an inductive argument by providing other evidence that changes our estimation of the likelihood of the conclusion.
It should be possible to differentiate arguments of these two sorts with some accuracy already. Remember that deductive arguments claim to guarantee their conclusions, while inductive arguments merely recommend theirs. Or ask yourself whether the introduction of any additional information—short of changing or denying any of the premises—could make the conclusion seem more or less likely; if so, the pattern of reasoning is inductive.
An inductive inference apparatus comprises an input section for inputting a proposition, conditions for the proposition, and the tendency of each condition, a storage section for storing the proposition, and necessary and sufficient conditions of the proposition, a condition detecting section for forming the necessary and sufficient conditions for the truth or falsity of the proposition in accordance with the input proposition, the input conditions, and the input tendency of each condition, a judging section for, with respect to an example in which the truth or falsity of the proposition is unknown, judging the truth or falsity of the proposition using already stored necesary and sufficient conditions, and a control section for, when the truth or falsity of the proposition in a new example input to the input section is known, supplying the conditions of the proposition and the tendency of each condition to the condition detecting section to store the necessary and sufficient conditions formed by the condition detecting section in the storage section, and for, when the truth or falsity of the proposition in the example is unknown, supplying the conditions of the proposition to the judging section.

Truth and Validity
Since deductive reasoning requires such a strong relationship between premises and conclusion, we will spend the majority of this survey studying various patterns of deductive inference. It is therefore worthwhile to consider the standard of correctness for deductive arguments in some detail.
A deductive argument is said to be valid when the inference from premises to conclusion is perfect. Here are two equivalent ways of stating that standard:

• If the premises of a valid argument are true, then its conclusion must also be true.
• It is impossible for the conclusion of a valid argument to be false while its premises are true.

(Considering the premises as a set of propositions, we will say that the premises are true only on those occasions when each and every one of those propositions is true.) Any deductive argument that is not valid is invalid: it is possible for its conclusion to be false while its premises are true, so even if the premises are true, the conclusion may turn out to be either true or false.
Notice that the validity of the inference of a deductive argument is independent of the truth of its premises; both conditions must be met in order to be sure of the truth of the conclusion. Of the eight distinct possible combinations of truth and validity, only one is ruled out completely: