Deductive Reasoning

Deductive Reasoning• Inductive: premise offers support and

evidence• Deductive: premises offers proof that the

conclusion is true

• Deductive Absolutely follows

Necessarily follows

Defiantly the case

Deductive logic = If premises are true it proves the

conclusion is true

Inductive •Likely follows

•Probably follows

•Best possible result

Two types of deductive logic

• Classical Logic(Socratic) – Allows for simple arguments– Tries to place subjects in/out of

categories (predicate)– Written in a very specific fashion

(standard form)– Inference = Form

• Modern Logic (Symbolic)– Allows for more flexibility– Shows connections between

premises and conclusion – Inference = Flow

P v ~P

Therefore, Q

Classical Deductive LogicSocratic Symbolism

MathStandard Form

• Equation – Two things are equal– X – 7 = 0

• Linear Equation – Equation for a line on graph– 3x – y = -2

• Algebraic Polynomial– Expression of mathematical terms– 4x + 3x – 7 (depends on variables)

Logic tries to put language into expressions and equations as well.

• Any statement made must be in a proper form (format) = Standard Form

• All statements must have a relationship:– Subject: The what/whom the statement is about– Predicate: Tells us something about the subject

All apple are redAll you is the best student everAll Wisconsin is in the United States

What category does the subject fit into? Has to be ALL, None or Some

This means you may have to rewrite statement to fit this format

English to Standard Form

Putting into standard form• "

• Ships are beautiful" translates to"All ships are beautiful things”

• "The whale is a mammal" translates to"All whales are mammals."

• "Whoever is a child is silly" translates to"All children are silly creatures."

• "Snakes coil" translates to"All snakes are coiling things."

Standard Form- Structure

• Can only have 3 lines or (statements)

– Two lines are premises & one is a conclusion• Can only have 3 terms (subject/predicate relationship)

– Cannot switch meaning of words– 1 term must be in both premises– 2 terms in conclusion must be in one of the premise

• If either premise is negative than the conclusion must be negative

Why do this? By doing this we guarantee an inference in our

argument (Glue)

• Categorical proposition (statement)– Statement that is asserting an inclusion or exclusion into

a catagory

• Class– Objects that have same characteristics in common

• Quantity (All/None/Some)– Subject is all, none or some of a certain class

• Quality (Affirmative or negative)– “Copula”– is or is not– The verb in the sentence

Standard Form – Statement Rules

Standard Form- Categorical Statements

Only 4 types of statements you can have given standard format rules.

– A: All S are P– E: No S are P– I: Some S are P– O: Some S are not P

Quantifier + Subject Term + Copula (quality)+ Predicate Term

Standard Form-Examples

• Universal Affirmative- A– All S are P (All Oaks are Trees)

• Universal Negative- E– No S are P (No Oaks are fish)

• Particular affirmative- I– Some S are P (Some Oaks are big)

• Particular negative- O– Some S are not P (Some oaks are not big)

Standard Form- Argument

All arguments will have a – Major term:

• predicate of the conclusion– Minor term:

• subject of the conclusion

Each one of these premises will share a part of the overall conclusionConclusion: All mortals are Greeks

– Middle term:• term in both premises

Middle Term

All human are Greeks (A)All mortal are humans (A)Conclusion: All mortal are Greeks (A)

Middle Term= The inference in the premisesNotice that the connecting term does not appear in the conclusion. It is the connection between the two premises. It is the subject in one and the predicate in the other

Standard Form-Inferences

All humans are Greeks (A)All mortals are human (A)

∴ All mortals are Greeks (A)–Major term–Minor term–Middle term

Valid argument-Clear line of inference

GreeksHumansMortals

All human are Greeks (A)All mortals are humans (A)Conclusion: All mortals are Greeks (A)

Invalid Argument

Has no inference to connect argument terms and statements together

All dogs are mortalAll cats are strange

∴ All dogs are cats

Invalid = No middle term

Deductive ArgumentValid

•Conclusion must be true if both premises are true •Conclusion can only be false when one of the premises are false•The form must be valid (Standard Form)•Fill in any variable term and the structure is valid

All X are YAll Z are X

Therefore, all Z are YThen talk about soundness (truth of premises)

Standard Form-Valid

• Only 15 valid standard form arguments– That are in valid form– If both premises are true than it forces you to

accept the conclusion as true– Because of the structure…the set up…the

form…the inferences

–True premises?

Standard Form-Sound

• If the set up of argument is in standard form (validity)

• All premises are true (truth)• Forces a true conclusion (truth)

If an argument is valid and has all true statement then it is a sound argument

Modern Deductive logic

Rules of modern logic• Allows for many lines of logic

– Only one line is the conclusion• Inferences

– Is shown through 9 rules of deductive logic– Rules are valid inferences between premises or

between premises and conclusion– Rules will be valid 100% of the time

• Shows the flow of the argument -or- reasoning– How do the reasons presented “flow” to the

conclusion presented• These offer proofs – like math (geometry)

– Not based on form of argument

DisjunctiveEither you are red or you are green. You are not

Red. Therefore, you are green

P v Q~P/Q

• Also deny the second term (~q)• Be careful for Fallacy of False Dilemma

Modus PonensIf you repent, then you will go to heaven. You have repented. So you will go to heaven.

P QP

/Q

If the IF part is true then the THEN part must be true as well

Modus TollenIf there is smoke, then there is fire. There is not fire, so there is no smoke.

P Q ~Q

~P

If the THEN part is false then the IF part must be FALSE as well

Hypothetical Syllogism If something is a tree (P), then it is green

(Q). If it is green (Q), then it is a plant (R). Therefore, if something is a tree (P), then it is a plant (R).

P Q Q R. /P R.

Hypothetical Syllogism- ChainIf something is a tree (P), then it is green

(Q). If it is green (Q), then it is a plant (R). If it is a plant (R), then it needs the sun (Z). Therefore, if something is a tree (P), then it needs the sun (Z).

P Q. Q R. R Z/P Z.

1. I (C v K)2. I3. ~C4. K (~S E)5. M S /Therefore ~M6. C v K (1)(2) MP7. K (6)(3) DS8. ~S E (4)(7) MP9. ~S (8) Simp.10. ~M (5)(9) MT

If your reasoning has this “flow” than it is valid. You can put in any term and it will flow- validly- to the conclusion. Then argue if sound or not (truth)