Syllogisms, Enthymemes, and Inductive Arguments

PHL 122 Logic Class 2025 Winter

1st Order Enthymemes

unstated major premise, assumption that the audience all agree one one of the premises, so obvious it doesn’t need to be stated

2nd Order Enthymemes

unstated minor premise

3rd Order Enthymemes

unstated conclusion

Conjunctive Syllogism

phrased by “both…and”, BOTH alternates must be true for the whole syllogism to be true,

example: I have a wife (P, true), I have children (Q, true), I have both a wife and children (P and Q true)

Conjunctive Syllogism Fallacy

affirming on the alternate, example: Gandalf is wise (P, true) therefore Gandalf is wise and old (P & Q, false)

Disjunctive Syllogism

phrased by “either P or Q or both”; you need to deny one, about validity not truth, example: either hotdogs or sushi [or both] (P v Q), not hotdogs (~ P), therefore sushi (:. Q)

Disjunctive Syllogism Fallacies

affirming one alternate (must deny one), sometimes they exclude the other (exception)

Hypothetical Syllogism

phrased by “if..then”; if the first part (the antecedent) is true, then the second part (the consequent) will also be true; four variations

Modus Ponens (Hypothetical Syllogism)

affirming the antecedent (first part); example: if you drive well, you steer well (P → Q), you drive well (P), therefore you steer well (:. Q)

Modus Tollens (Hypothetical Syllogism)

denying the consequent (second part); example: if stabbed in heart, then you die (P → Q), not dead (~ Q), therefore not stabbed in heart (:. ~ P)

Hypothetical Syllogism Proper

consists of more than one conditional proposition; example: if you are saved, then you believe in God (P → Q), if you believe in God, then you are a theist (Q → P), therefore, if saved, then theist (P → R)

Reductio ad absurdum (Hypothetical Syllogism)

variation of modus ponens, it means “a reduction to absurdity”; attempt to establish a conclusion by showing that its opposite is absurd or impossible; example: suppose Jesus is still dead (P), if Jesus is still dead, then 11 or the 12 disciples are very cowardly (P → Q), but they didn’t renounce Christ when their lives were on the line (~ Q), therefore Jesus is [likely] not still dead (:. ~ P)

Fallacies of the Hypothetical Syllogism

denying the antecedent (just because A implies B doesn’t mean that B cannot be without A); affirming the consequent (assuming the consequent is true without confirming the antecedent caused it)

“Only If” Language Note

that which follows immediately after the “only if” is not the sufficient condition (or antecedent clause) but the necessary condition (or consequent clause)

IFF Exception

“if and only if”; “bi-conditional” because it is a pair of conditionals lumped together; example: If P, then Q; if Q, then P; :. P ←→ Q

Inductive Arguments

long string of propositions giving evidence; particular to general or particular to particular; from effect to cause, a posteriori reasoning (after the evidence); the aim is probable conclusions

Induction by Simple Enumeration

reason arrives at a conclusion through a process of reasoning which begins with a number of particular truths as premises and ends with a universal truth as a probable conclusion; when giving examples in support of a generalization: give more than one example, use representative examples, background information is crucial, consider counter-examples

Induction through Statistical Generalizations

has to do with statistical evidence (data collected by polling and research studies), considering: characteristic of interest, target population, sample; be representative, random, and large enough

1st Canon of Causal Induction (JSM): Method of Agreement

a cause can be found by noticing that something “X” is the only factor always present when something “Y” occurs; it’s likely then that “X” caused “Y”; cause/effect relationship; “What did they all have in common as a cause that might result in this effect?”

2nd Canon of Causal Induction (JSM): Method of Difference

found by noticing that the only difference between the effect “Y” occurring or not is when one element “X” is different; absence is terms of the effect

3rd Canon of Causal Induction (JSM): Joint Method of Agreement and Difference

likely cause isolated when you identify relevant factors common to the occurrences of the phenomenon (method of agreement) and you discard any of these that are present even when there are no occurrences (method of difference)

4th Canon of Causal Induction (JSM): Method of Residues

if there are a range of causes thought linked to a range of effects, and we suspect act all the factors, except one factor, are the causes for all the effects, except one, then we should infer that C is the cause of remaining effect; example: you step on a scale (cause) and get your weight as 175 lbs (effect), but then you add your cat (another cause) and get 185 lbs (another effect), we can then infer that the residual weight of 10 lbs is due to the cat-cause.

5th Canon of Causal Induction (JSM): Method of Concomitant Variations

a bunch of possible causes that lead to some effect but we’re not quite sure (only suspect that there is) and are more interested in the degree of cause and effect relation over just the cause and effect

Arguments Using Authority

generalizations and probable conclusions can also be drawn from the testimony of experts who are individuals who have education, significant experience, or both in a given area; 5 things to keep in mind when dealing with the question of authority: sources should be cited, seek informed sources, seek impartial sources, cross-check sources, personal attacks do not disqualify an expert (Ad Hominem).