Post on 23-Feb-2016
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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)