Introduction to Symbolic Logic



Introduction to Symbolic Logic

June 6, 2005

❖ Introductions

➢ Name, Year, Background, Goal

❖ Syllabus (read through)

❖ Basic Notions of Logic

➢ Truth Preservation

➢ Aristotle

▪ If the premises are true, then the conclusion must follow – Good(valid) Argument

▪ Examples

• All mammals are vertebrates. Some sea creatures are vertebrates. ( Some sea creatures are vertebrates

• All As are Bs. Some Cs are As. ( Some Cs are Bs.

▪ Drawbacks

• Limited to syllogisms, which must have exactly 2 premises and 1 conclusion

• Every sentence of a syllogism must be in one of the following forms: All As are Bs; No As are Bs; Some As are Bs; Some As are not Bs

• Best suited to reasoning about relations among groups

• Doesn’t accommodate reasoning that relies on relations such as transitivity

➢ Sentential Logic – natural language sentences can be generated from other sentences using connecting works like “or”, “and”, “if…then…”

▪ What problems are immediately apparent with a system of this kind?

▪ So why bother with SL?

➢ Truth Values

▪ ‘true’ and ‘false’ are properties of sentences

▪ True sentences have the truth-value T; false sentences have the truth-value F

▪ SL will deal only with those sentences that we know are true or false (though we might not know which)

• So, no questions, requests, commands, exclamations

□ One is the smallest prime number.

□ Beware of Greeks bearing gifts.

□ Who created these screwy examples?

□ This sentence is false.

• Moreover, we want to talk about arguments and we want them in standard form

□ Example not in standard form ( standard form

□ “Michael will not get the job, for whoever gets the job will have strong references, and Michael’s references are not strong.”

➢ Analyzing Arguments

▪ In order to use the skills you will pick up by studying symbolic logic to analyze natural language arguments, you will have to find the argument from within the discourse and put it in standard form

• Look for premise/conclusion indicator words

□ Premise words: since, because

□ Conclusion words: therefore, consequently

▪ Not all discourse will contain an argument

▪ Def. An argument is a set of two or more sentences, one of which is designated as the conclusion and the others as the premises

• Homer is a yellow dog. ( Today is Monday.

▪ Practice Exercises

• When Mike, Sharon, Sandy, and Vicky are all out of the office, no important decisions get made. Mike is off skiing, Sharon is in Spokane, Vicky is in Olympia, and Sandy is in Seattle. So no decision will be made today.

• Shelby and Noreen are wonderful in dealing with irate students and faculty. Stephanie is wonderful at managing the chancellor’s very demanding schedule, and Tina keeps everything moving and cheers everyone up.

➢ Properties of Arguments

▪ Deductive Validity

• Def. An argument is deductively valid if and only if it is not possible for the premises to be true and the conclusion false. An argument is deductively invalid if and only if it is not deductively valid.

▪ Deductive Soundness

• Def. An argument is deductively sound if and only if it is deductively valid and its premises are true. An argument is deductively unsound if and only if it is not deductively sound.

▪ {In 2004, Bush and Kerry were the only major party candidates in the presidential election. A major party candidate won. Kerry did not win.} ( Bush won the presidential election in 2004.

➢ Properties of Sentences

▪ Logical Consistency

• Def. A set of sentences is logically consistent if and only if it is possible for all the members of that set to be true. A set of sentences is logically inconsistent if and only if it is not logically consistent.

□ {Jess knows a hamster named Heidi. Boston is the capital of MA. Today is June 6.}

▪ Logical Truth and Falsity

• Generally, logic cannot be used to determine whether a particular sentence is true or false, except in special cases where the form or structure of the sentence determines its truth value

• Def. A sentence is logically true if and only if it is not possible for the sentence to be false.

□ Either Jess will finish her dissertation or she won’t.

□ If there’s traffic on the Pike, then there’s traffic on the pike.

• Def. A sentence is logically false if and only if it is not possible for the sentence to be true.

□ Heidi is a brown hamster and Heidi is not a brown hamster.

□ All dogs are friendly, but there are some dogs that are not friendly.

• Def. A sentence is logically indeterminate if and only if it is neither logically true or logically false.

□ Sentences that give us information about the world.

▪ Logical Equivalence

• Def. The member of a pair of sentences are logically equivalent if and only if it is not possible for one of the sentences to be true while the other sentence is false.

□ Both Sarah and Mandy will pass. Both Mandy and Sarah will pass.

□ Henry loves Sarah. Sarah loves Henry. (not equivalent)

➢ Homework 1

▪ Part 1 – 1.3E(2b,d,j,l,p), 1.4E(1, 2), 1.6E(2b,d,f,l, 4, 5)

▪ Part 2 -- Upcoming

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