Semantics of SL and Review

1
Semantics of SL and Review

• Part 1 What you need to know for test 2
• Part 2 The structure of definitions of truth
  functional notions
• Part 3 Rules when using truth tables to
  demonstrate that a truth functional notion does
  or does not apply in a specific case.
• Part 4 Meta theoretical questions
• Part 5 Your questions

2
Test 2 What you need to know

• Definitions of truth functional notions that
  apply to
• individual sentences (t-f truth, t-f falsity, t-f
  indeterminacy)
• pairs of sentences (t-f equivalence)
• arguments (t-f validity)
• sets of sentences (t-f consistency)
• some set of sentences and an individual sentence
  (t-f entailment)

3
Test 2 What you need to know

• How to construct truth tables
• How to use truth tables to show that a truth
  functional notion does or does not apply to some
  sentence or group of sentences
• When you need a full table and why
• When a shortened table will do and why
• When and which row, rows, or column, you need to
  circle.
• And how to construct an appropriately shortened
  truth table when directed to.

4
Test 2 What you need to know

• How to answer meta theoretical questions about SL
• Just as there are (denumerably) infinite ways to
  symbolize a sentence of natural language into SL,
  there are many possible ways to answer a specific
  meta theoretical question.
• As with symbolization, we recommend constructing
  the most straightforward answer to meta
  theoretical questions, as you are less likely to
  make a mistake.

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Part 2

• The structure of
• truth functional definitions

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The structure of truth functional definitions

• Truth functional ___________.
• For all but entailment
• A (or an) ___________ is truth functionally
  _________ IFF ________________ (specification
  that there is/are or is not/are not one or more
  truth value assignments on which ).
• A set ? truth functionally entails a sentence P
  iff there is no TVA on which all the members of ?
  are true and P is false.

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Part 3

• More on truth table conventions, full vs.
  appropriately shortened tables, constructing and
  using CMCs

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Using truth tables

• A full truth table is required to show
• That a sentence is truth functionally true or
  that a sentence is truth functionally false.
• That 2 sentences are truth functionally
  equivalent.
• That an argument is truth functionally valid.
• That a set is not truth functionally consistent.
• That a set ? truth functionally entails a
  sentence.

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Using truth tables

• A shortened truth table will suffice to show
• That a sentence is not truth functionally true or
  that a sentence is not truth functionally false.
• That a sentence is truth functionally
  indeterminate.
• That 2 sentences are not truth functionally
  equivalent.
• That an argument is not truth functionally valid.
• That a set is truth functionally consistent.
• That a set ? does not truth functionally entail a
  sentence.
• How many rows does each require?

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Using truth tables

• Additional tips
• When constructing a truth table to determine if
  an argument is truth functionally valid you need
  only to
• Determine the truth value of the conclusion on
  each TVA.
• Having done so, pay attention to only those rows
  in which it is false.
• Whenever you find 1 premise false as well, you
  need not do more with that row.
• When constructing a truth table to determine if
  some set ? P, you need only to
• Determine the truth value of P on each TVA.
• Then pay attention to only those rows on which P
  is false.
• Whenever you find 1 member of the set is false,
  you are done with that row.

11
Using truth tables

• Using an appropriately shortened truth table
  (when directed to) to demonstrate that some t-f
  notion does or does not apply in a particular
  case involves a different skill from constructing
  a full table. If so directed, do so.
• We circle one column the truth values under the
  main connective when a truth table establishes
  that a sentence is truth functionally true or
  truth functionally false.
• We circle 2 columns when a truth table
  establishes that 2 sentences are truth
  functionally equivalent.

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Using truth tables

• Additional tips
• If using a table to determine the truth
  functional status of a sentence, as soon as (if
  it occurs) you identify 1 TVA on which it is true
  and 1 TVA on which it is false, youve shown its
  t-f indeterminate.
• If using a table to determine if 2 sentences are
  t-f equivalent, if you find 1 TVA on which they
  have different truth values, youve shown they
  are not.
• To determine if a set is t-f consistent, if you
  find 1 TVA on which all its members are true, you
  have shown that it is.

13
Using truth tables

• We circle 1 row when a table demonstrates that an
  argument is not truth functionally valid, a set
  is truth functionally consistent, a set does not
  truth functionally entail a sentence, a sentence
  is not t-f true or not t-f false.
• We circle 2 rows when a table demonstrates that a
  sentence is truth functionally indeterminate.

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Corresponding material conditionals

• As we have seen (and noted) the if/then
  relationship is at the core of logic.
• It is reflected in the truth conditions for
  sentences of the form P ? Q
• It is reflected in the definition of truth
  functional validity (and the more general version
  of deductive validity we studied first).
• And it is reflected in the definition of truth
  functional entailment.
• Material conditions are not true, arguments are
  not truth functionally valid, and a set does not
  truth functionally entail a sentence P in just
  those cases that allow for reasoning from true
  statements to a false one.

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Corresponding material conditionals

• Accordingly, we use the notion of a
  corresponding material conditional to further
  illuminate the importance logic places on truth
  preservation.
• For any argument of SL
• P1
• P2
• .
• .
• PN
• ----
• Q

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Corresponding material conditionals

• We can construct its corresponding material
  conditional, the antecedent of which is an
  iterated conjunction of the arguments premises
  and the consequent of which is the arguments
  conclusion.
• (P1 P2) PN ? Q
• And it turns out that an argument is truth
  functionally valid IFF its corresponding material
  conditional is truth functionally true.

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Corresponding material conditionals

• Similarly, for any entailment relationships
  between some set ? and some sentence
• P1, P2, PN Q
• we can construct a corresponding material
  conditional
• (P1 P2) PN ? Q
• And if the set truth functionally entails the
  sentence, Q, the corresponding material
  conditional is truth functionally true.

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Part 4

• The tasks and techniques required
• to answer meta theoretical questions

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Meta theoretical questions

• All you need to know, but you do need to know,
  are the definitions of the truth functional
  notions. But there are strategies or techniques
  that serve to streamline the reasoning.
• Pay attention to the question asked Are you
  being asked to show that a claim holds and, if
  so, what kind of claim (is it of the form
  it/then or of the form if and only if? Or are
  you being asked to assume that something is the
  case and then asked to consider whether something
  else follows?

20
Meta theoretical questions

• Some simpler meta theoretical questions
• Why does it take a full truth table to establish
  that an argument is truth functionally valid? Why
  does it only take a shortened truth table to
  establish that an argument is truth functionally
  invalid (including how many rows and why that
  many)?
• Why can the truth functional status of a
  corresponding material conditional of an argument
  demonstrate that the argument is or is not truth
  functionally valid?

21
Meta theoretical questions

• Why if P is a truth functionally true sentence is
  P a truth functionally false sentence?
• Why if P is a truth functionally true sentence is
  P a truth functionally inconsistent set?
• Why if P is a truth functionally false sentence
  is P a truth functionally true sentence?
• Why if P is a truth functionally false sentence
  is P a truth functionally consistent set?

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Meta theoretical questions

• Show that if Q is truth functionally true, then
• P ? Q is truth functionally true.
• Step 1 If Q is truth functionally true, then
  what do we know about Q on any given truth value
  assignments? Why?
• Step 2 Given the answer to the above, could
  there be a truth value assignment on which P is
  true and Q is false? Why?
• Step 3 What does the answer to the above show
  about the truth functional status of P ? Q? Why?

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The steps to take

• In some cases you are asked to assume that
  something is the case, and asked given this
  assumption, whether something else is possible.
• For example
• Assume that the argument
• P
• --
• Q
• is truth functionally valid. Is it possible that
  the argument
• P
• --
• Q is also truth functionally valid?

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The steps to take

• Step 1 Given that the first argument is truth
  functionally valid, what kind of truth value
  assignment can there not be? Why?
• Step 2 If the second argument is truth
  functionally valid, what kind of truth value
  assignment can there not be? Why?
• Step 3 Given the answers to the above, what must
  the truth functional status of P be for both
  arguments to be truth functionally valid? Why?

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Part 5

• Review!
• Your turn