General Notion of Inference

I. SOME DEFINITIONS

• INFERENCE = one of the ways to arrive at a truth.
o COHERENCE THEORY OF TRUTH

• INFERENCE (broad sense) = any process by which the mind proceeds from one or more propositions to other propositions seen to be implied in the former.

• INFERENCE (strict sense) = the operation by which the mind gets new knowledge by drawing out the implications of what is already known.

• INFERENCE = also applied to any series of propositions so arranged that one, called the CONSEQUENT, flows with logical necessity from one or more others, called the ANTECEDENT.

• ANTECEDENT (Latin, antecedo) = “that which goes before” o Defined as “that from which something is inferred”

• CONSEQUENT ( Latin, consequor) = “that which follows after” o Defined as “that which is inferred from the antecedent”

• N.B.
1. The ANTECEDENT AND CONSEQUENT of a VALID INFERENCE are so related that the TRUTH of the ANTECEDENT involves the TRUTH of the CONSEQUENT (but not vice versa).

2. The FALSITY of the CONSEQUENT involves the FALSITY of the ANTECEDENT (but not vice versa).

3. The connection by virtue of which the consequent flows with LOGICAL NECESSITY from the antecedent is known as CONSEQUENCE or simply SEQUENCE.

4. The SEQUENCE (which is signified by the so called CONCLUSION INDICATORS, e.g., therefore, consequently, accordingly, hence, thus, and so, for this reason, etc) is the VERY HEART of INFERENCE; and when we make an inference, our assent bears on it directly.

• A GENUINE SEQUENCE is called VALID; a PSEUDO SEQUENCE is called INVALID.

SYNOPTIC SCHEMA

ANTECEDENT (premises)

(connection between
INFERENCE the antecedent and
the consequent)

CONSEQUENT (conclusion)

FORMAL AND MATERIAL VALIDITY

• FORMAL VALIDITY = the sequence springs from the form of inference o Example: Every S is a P; therefore some P is an S. o N.B. We can substitute anything we want to for S and P, and the consequent will always be true if the antecedent is true. o Example:

▪ S = dog, P = animal: Every dog is an animal; therefore some animal is a dog. ▪ S = voter, P = citizen: Every voter is a citizen; therefore some citizen is a voter.

• MATERIAL VALIDITY = the sequence springs from the special character of the thought content. o Example: Every triangle is a plane figure bounded by three straight lines; therefore every plane figure bounded by three straight lines is a triangle. o Analysis:

▪ The inference is formally invalid for the consequent does not flow from the antecedent because of the form; but materially valid because it does flow from the antecedent due to the special character of the thought content. ▪ “Plane figure bounded by three straight lines” is a definition of “triangle” and is therefore interchangeable.

TRUTH AND FORMAL VALIDITY

• LOGICAL TRUTH = consists in the conformity of our minds with reality. o A proposition, as explained, is true if things are as the proposition says they are. • Logic studies reason as an instrument for acquiring truth, and the attainment of truth must ever remain the ultimate aim of the logician.

• N.B. We shall not be directly concerned with acquiring true data but rather with conserving the truth of our data as we draw inferences from them. o In other words, we shall aim at making such a transition from data to conclusion that if the data (antecedent, premises) are true, the conclusion (consequent) will necessarily be true. o Formal validity, correctness, rectitude, or consistency will be our immediate aim. o We shall not ask ourselves, ARE THE PREMISES TRUE?, but, DOES THE CONCLUSION FLOW FROM THE PREMISES so that IF the premises are true, the conclusion is necessarily true? o The following syllogism is correct in this technical sense although the premises and the conclusion are false: ▪ No plant is a living being; but every man is a plant; therefore no man is a living being. ▪ This syllogism is CORRECT FORMALLY.

• Why: because the conclusion really flows from the premises by virtue of the form or structure of the argument. IF the premises were true, the conclusion would also be true. o The following syllogism is not correct formally although the premises and the conclusion are true: ▪ Every dog is an animal; but no dog is a plant; therefore no plant is an animal. ▪ The syllogism is not correct because the conclusion does not really flow from the premises. ▪ For instance, we substitute “plant” with “cow”: • Every dog is an animal; but no dog is a cow; therefore no cow is an animal.

IMMEDIATE AND MEDIATE INFERENCE

▪ IMMEDIATE INFERENCE = consists in passing directly (that is, without the intermediacy of a middle term or a second proposition) from one proposition to a new proposition that is a partial or complete reformulation of the very same truth expressed in the original proposition.

▪ MEDIATE INFERENCE = draws a conclusion from two propositions (instead of one) and does involve an advance in knowledge. o It is mediate in either of two ways:
▪ Categorical syllogism = it unites, or separates, the subject and predicate of the conclusion through the intermediacy of a middle term; ▪ Hypothetical syllogism = the major premise “causes” the conclusion through the intermediacy of a second proposition. o Goal: not only a new proposition but also a new truth ▪ There is an advance in knowledge.

|SYNOPSIS | |IMMEDIATE INFERENCE |MEDIATE INFERENCE | |A. passes from one proposition |A. passes from two propositions | |B. without a medium |B. through a medium | |C. to a new proposition but not to a new truth |C. not only to a new proposition but also to a new truth |

DEDUCTION AND INDUCTION

• DEDUCTION = the process by which our minds proceed from a more universal truth to a less universal truth. o Example:
▪ All men are mortal; but Peter is a man; therefore Peter is mortal.

• INDUCTION = the process by which our minds proceed from sufficiently enumerated instances to a universal truth. o Example:
▪ This ruminant (hoofed-mammal) (a cow) is cloven-hoofed; this one ( a deer) is cloven-hoofed; and this one (a goat) and this (an antelope); therefore all ruminants are cloven-hoofed.

VENN DIAGRAM

• Aristotelian Standpoint = universal propositions about existing things imply the existence of the things talked about.
o Example:
o All Stephen King’s novels are thrillers.
• Implies the existence of at least one novel by Stephen King. ▪ All unicorns are one-horned animals.
• Does not imply the existence of unicorns. • Boolean Standpoint = universal propositions never imply the existence of the things talked about. ▪ All Stephen King’s novels are thrillers. • Does not imply the existence of any novels by Stephen King. ▪ All unicorns are one-horned animals.

• Does not imply the existence of unicorns. • Aristotelian and Boolean interpretations are the same for particular propositions. o Bot I and O propositions actually claim that the subject class contains at least one existing thing. ▪ “Some” = at least one exists.

SQUARE OF OPPOSITION
[pic]
• CONTRADICTION = cannot be true and false at the same time • CONTRARY = At least 1of the propositions is false.
• SUBCONTRARY = At least 1of the propositions is true. • SUBALTERNATION = truth flows down; falsity flows up.

SOURCE:

Bachhuber, Andrew H., S.J. Introduction to Logic. New York: Appleton-Century-Crofts, Inc., 1957.
SEQUENCE