Semantics of SL and Review

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

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)

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.

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.

Part 2 The structure of truth functional definitions

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.

Part 3 More on truth table conventions, full vs. appropriately shortened tables, constructing and using CMC’s

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.

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?

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.

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.

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, you’ve shown it’s 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, you’ve 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.

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.

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 Pin just those cases that allow for reasoning from true statements to a false one.

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

Corresponding material conditionals • We can construct its corresponding material conditional, the antecedent of which is an iterated conjunction of the argument’s premises and the consequent of which is the argument’s 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.

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.

Part 4 The tasks and techniques required to answer meta theoretical questions

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 thata 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?

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?

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?

Meta theoretical questions • Show that ifQ 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?

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?

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?

Part 5 Review! Your turn…

Semantics of SL and Review